# data[].attributes.body
I can use the SAS Triangle Congruence Theorem to determine if triangles are congruent.

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When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent.

When triangles are congruent, six facts are always true.

Corresponding sides are congruent.

Corresponding angles are congruent.

The good news is that when proving triangles congruent, it is not necessary to prove all six facts to show congruency. There are certain ordered combinations of these facts that are sufficient to prove triangles congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape (be unique), thus insuring congruency.

Methods that Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts
will be sufficient to prove triangles congruent.

If three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
This is an extension of ASA. In ASA, since you know two sets of angles are congruent, you automatically know the third sets are also congruent since there are 180º in each triangle.

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

The "included angle" in SAS is the angle formed by the two sides of the triangle being used.
The "included side" in ASA is the side between the angles being used. It is the side where the rays of the angles overlap.
The "non-included" side in AAS can be either of the two sides that are not directly between the two angles being used.

Once triangles are proven congruent, the corresponding leftover "parts" that were not used in SSS, SAS, ASA, AAS and HL, are also congruent.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent.

Methods that DO NOT Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts will NOT be sufficient to prove triangles congruent. Let's see why these combinations DO NOT work!

This method will NOT always prove triangles congruent!

The AAA combination will show that the triangles are the same SHAPE (similar), but will NOT show that the triangles are the same size.

Example:Consider these two equilateral triangles that satisfy the AAA combination. They are the same shape, but are not the same size. Thus, they are not congruent. (They are similar.)

Yes, it is possible that the sides "could" be the same length and the triangles would be congruent, but this would be the exception, not the rule.

SSA or ASS
Side-Side-Angle

This method will NOT always prove triangles congruent!

The SSA (or ASS) combination deals with two sides and the non-included angle. This combination is humorously referred to as the "Donkey Theorem". SSA (or ASS) is NOT a universal method to prove triangles congruent since it cannot guarantee that the shapes of the triangles formed will always be the same.

Let's look at the problems associated with SSA:
When using SSA (or ASS), the lengths of the sides being used, and their position in relation to the angle, will determine if congruent triangles will always be created. Let's see which special conditions will create congruent triangles, and which ones will not.

Note: This specific case of SSA is the basis for the acceptable method HL (Hypotenuse Leg) which applies only in right triangles.

If the side which lies on one ray of the angle is longer than the other side, and the other side is greater than the minimum distance needed to create a triangle, the two triangles will not necessarily be congruent.

If EF is greater than EG, the diagram below shows how it is possible for to "swing" to either side of point G, creating two non-congruent triangles using SSA.

If the side which lies on one ray of the angle is longer than the other side, and the other side is less than the minimum distance needed to create a triangle, no triangle can be drawn. Since no triangles are possible, no congruent triangles are possible.

No Congruent Triangles

Since the SSA (or ASS) method allows for the possibility of creating triangles of various shapes (or even no triangles at all), this method is not an universal method for proving triangles congruent.

and is not considered "fair use" for educators. Please read the " NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
Terms of Use".

# data[].attributes.attachment_link.title
Methods of Proving Triangle Congruent - MathBitsNotebook(Geo - CCSS Math)

# data[].attributes.attachment_link.description
When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent. When triangles are congruent, six facts are always true. Corresponding sides are congruent. Corresponding angles are congruent.

# data[].attributes.attachment_link.readable_text
Two triangles are congruent if they have:

exactly the same three sides and

exactly the same three angles.

But we don't have to know all three sides and all three angles ...usually three out of the six is enough.

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

(See Solving SSS Triangles to find out more)
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:
(See Solving SAS Triangles to find out more)
If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:
(See Solving ASA Triangles to find out more)
If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:
(See Solving AAS Triangles to find out more)
If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL (hypotenuse, leg)

This one applies only to right angled-triangles!

or

HL stands for " Hypotenuse, L eg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

the same length of hypotenuse and

the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:
(See Pythagoras' Theorem to find out more)
If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.
This is not enough information to decide if two triangles are congruent!
Because the triangles can have the same angles but be different sizes:

Without knowing at least one side, we can't be sure if two triangles are congruent.

# data[].attributes.attachment_link.description
Two triangles are congruent if they have: exactly the same three sides and exactly the same three angles. But we don't have to know all three sides and all three angles ...usually three out of the six is enough. There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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SAS Congruent Triangles - Math Open Reference

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Congruent triangles - Two sides and included angle. SAS

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SSS and SAS Congruence - Worksheet and Key

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I can use the SSS Triangle Congruence Theorem to determine if triangles are congruent.

# data[].attributes.attachment_link.readable_text
When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent.

When triangles are congruent, six facts are always true.

Corresponding sides are congruent.

Corresponding angles are congruent.

The good news is that when proving triangles congruent, it is not necessary to prove all six facts to show congruency. There are certain ordered combinations of these facts that are sufficient to prove triangles congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape (be unique), thus insuring congruency.

Methods that Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts
will be sufficient to prove triangles congruent.

If three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
This is an extension of ASA. In ASA, since you know two sets of angles are congruent, you automatically know the third sets are also congruent since there are 180º in each triangle.

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

The "included angle" in SAS is the angle formed by the two sides of the triangle being used.
The "included side" in ASA is the side between the angles being used. It is the side where the rays of the angles overlap.
The "non-included" side in AAS can be either of the two sides that are not directly between the two angles being used.

Once triangles are proven congruent, the corresponding leftover "parts" that were not used in SSS, SAS, ASA, AAS and HL, are also congruent.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent.

Methods that DO NOT Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts will NOT be sufficient to prove triangles congruent. Let's see why these combinations DO NOT work!

This method will NOT always prove triangles congruent!

The AAA combination will show that the triangles are the same SHAPE (similar), but will NOT show that the triangles are the same size.

Example:Consider these two equilateral triangles that satisfy the AAA combination. They are the same shape, but are not the same size. Thus, they are not congruent. (They are similar.)

Yes, it is possible that the sides "could" be the same length and the triangles would be congruent, but this would be the exception, not the rule.

SSA or ASS
Side-Side-Angle

This method will NOT always prove triangles congruent!

The SSA (or ASS) combination deals with two sides and the non-included angle. This combination is humorously referred to as the "Donkey Theorem". SSA (or ASS) is NOT a universal method to prove triangles congruent since it cannot guarantee that the shapes of the triangles formed will always be the same.

Let's look at the problems associated with SSA:
When using SSA (or ASS), the lengths of the sides being used, and their position in relation to the angle, will determine if congruent triangles will always be created. Let's see which special conditions will create congruent triangles, and which ones will not.

Note: This specific case of SSA is the basis for the acceptable method HL (Hypotenuse Leg) which applies only in right triangles.

If the side which lies on one ray of the angle is longer than the other side, and the other side is greater than the minimum distance needed to create a triangle, the two triangles will not necessarily be congruent.

If EF is greater than EG, the diagram below shows how it is possible for to "swing" to either side of point G, creating two non-congruent triangles using SSA.

If the side which lies on one ray of the angle is longer than the other side, and the other side is less than the minimum distance needed to create a triangle, no triangle can be drawn. Since no triangles are possible, no congruent triangles are possible.

No Congruent Triangles

Since the SSA (or ASS) method allows for the possibility of creating triangles of various shapes (or even no triangles at all), this method is not an universal method for proving triangles congruent.

and is not considered "fair use" for educators. Please read the " NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
Terms of Use".

# data[].attributes.attachment_link.title
Methods of Proving Triangle Congruent - MathBitsNotebook(Geo - CCSS Math)

# data[].attributes.attachment_link.description
When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent. When triangles are congruent, six facts are always true. Corresponding sides are congruent. Corresponding angles are congruent.

# data[].attributes.attachment_link.readable_text
Two triangles are congruent if they have:

exactly the same three sides and

exactly the same three angles.

But we don't have to know all three sides and all three angles ...usually three out of the six is enough.

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

(See Solving SSS Triangles to find out more)
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:
(See Solving SAS Triangles to find out more)
If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:
(See Solving ASA Triangles to find out more)
If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:
(See Solving AAS Triangles to find out more)
If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL (hypotenuse, leg)

This one applies only to right angled-triangles!

or

HL stands for " Hypotenuse, L eg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

the same length of hypotenuse and

the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:
(See Pythagoras' Theorem to find out more)
If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.
This is not enough information to decide if two triangles are congruent!
Because the triangles can have the same angles but be different sizes:

Without knowing at least one side, we can't be sure if two triangles are congruent.

# data[].attributes.attachment_link.description
Two triangles are congruent if they have: exactly the same three sides and exactly the same three angles. But we don't have to know all three sides and all three angles ...usually three out of the six is enough. There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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SSS and SAS Congruence - Worksheet and Key

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6.3: AAA and ASA Triangle Congruence Theorems

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6.3: AAA and ASA Triangle Congruence Theorems

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I can explain the difference between the AAS and ASA congruence theorems.

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When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent.

When triangles are congruent, six facts are always true.

Corresponding sides are congruent.

Corresponding angles are congruent.

The good news is that when proving triangles congruent, it is not necessary to prove all six facts to show congruency. There are certain ordered combinations of these facts that are sufficient to prove triangles congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape (be unique), thus insuring congruency.

Methods that Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts
will be sufficient to prove triangles congruent.

If three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
This is an extension of ASA. In ASA, since you know two sets of angles are congruent, you automatically know the third sets are also congruent since there are 180º in each triangle.

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

The "included angle" in SAS is the angle formed by the two sides of the triangle being used.
The "included side" in ASA is the side between the angles being used. It is the side where the rays of the angles overlap.
The "non-included" side in AAS can be either of the two sides that are not directly between the two angles being used.

Once triangles are proven congruent, the corresponding leftover "parts" that were not used in SSS, SAS, ASA, AAS and HL, are also congruent.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent.

Methods that DO NOT Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts will NOT be sufficient to prove triangles congruent. Let's see why these combinations DO NOT work!

This method will NOT always prove triangles congruent!

The AAA combination will show that the triangles are the same SHAPE (similar), but will NOT show that the triangles are the same size.

Example:Consider these two equilateral triangles that satisfy the AAA combination. They are the same shape, but are not the same size. Thus, they are not congruent. (They are similar.)

Yes, it is possible that the sides "could" be the same length and the triangles would be congruent, but this would be the exception, not the rule.

SSA or ASS
Side-Side-Angle

This method will NOT always prove triangles congruent!

The SSA (or ASS) combination deals with two sides and the non-included angle. This combination is humorously referred to as the "Donkey Theorem". SSA (or ASS) is NOT a universal method to prove triangles congruent since it cannot guarantee that the shapes of the triangles formed will always be the same.

Let's look at the problems associated with SSA:
When using SSA (or ASS), the lengths of the sides being used, and their position in relation to the angle, will determine if congruent triangles will always be created. Let's see which special conditions will create congruent triangles, and which ones will not.

Note: This specific case of SSA is the basis for the acceptable method HL (Hypotenuse Leg) which applies only in right triangles.

If the side which lies on one ray of the angle is longer than the other side, and the other side is greater than the minimum distance needed to create a triangle, the two triangles will not necessarily be congruent.

If EF is greater than EG, the diagram below shows how it is possible for to "swing" to either side of point G, creating two non-congruent triangles using SSA.

If the side which lies on one ray of the angle is longer than the other side, and the other side is less than the minimum distance needed to create a triangle, no triangle can be drawn. Since no triangles are possible, no congruent triangles are possible.

No Congruent Triangles

Since the SSA (or ASS) method allows for the possibility of creating triangles of various shapes (or even no triangles at all), this method is not an universal method for proving triangles congruent.

and is not considered "fair use" for educators. Please read the " NOTE: The re-posting of materials (in part or whole) from this site to the Internet is copyright violation
Terms of Use".

# data[].attributes.attachment_link.title
Methods of Proving Triangle Congruent - MathBitsNotebook(Geo - CCSS Math)

# data[].attributes.attachment_link.description
When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent. When triangles are congruent, six facts are always true. Corresponding sides are congruent. Corresponding angles are congruent.

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Two triangles are congruent if they have:

exactly the same three sides and

exactly the same three angles.

But we don't have to know all three sides and all three angles ...usually three out of the six is enough.

There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS (side, side, side)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.

For example:

is congruent to:

(See Solving SSS Triangles to find out more)
If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS (side, angle, side)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.

For example:
(See Solving SAS Triangles to find out more)
If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent.

3. ASA (angle, side, angle)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.

For example:
(See Solving ASA Triangles to find out more)
If two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

4. AAS (angle, angle, side)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:
(See Solving AAS Triangles to find out more)
If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent.

5. HL (hypotenuse, leg)

This one applies only to right angled-triangles!

or

HL stands for " Hypotenuse, L eg" (the longest side of a right-angled triangle is called the "hypotenuse", the other two sides are called "legs")

It means we have two right-angled triangles with

the same length of hypotenuse and

the same length for one of the other two legs.

It doesn't matter which leg since the triangles could be rotated.

For example:
(See Pythagoras' Theorem to find out more)
If the hypotenuse and one leg of one right-angled triangle are equal to the corresponding hypotenuse and leg of another right-angled triangle, the two triangles are congruent.

Caution! Don't Use "AAA"

AAA means we are given all three angles of a triangle, but no sides.
This is not enough information to decide if two triangles are congruent!
Because the triangles can have the same angles but be different sizes:

Without knowing at least one side, we can't be sure if two triangles are congruent.

# data[].attributes.attachment_link.description
Two triangles are congruent if they have: exactly the same three sides and exactly the same three angles. But we don't have to know all three sides and all three angles ...usually three out of the six is enough. There are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

# data[].attributes.attachment_link.description
Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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ASA and AAS Congruence - Worksheet and Key

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SAS, SSS, AAS, and ASA - Worksheet and Key

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SAS, SSS, AAS, and ASA - Worksheet and Key

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SSS, SAS, ASA, and AAS Congruence - Worksheet and Key

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I can use geometric tools to construct a copy of an angle.

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How to copy an angle with compass and straightedge or ruler - Math Open Reference

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Given an angle formed by two lines with a common vertex, this page shows how to construct another angle from it that has the same angle measure using a compass and straightedge or ruler. It works by creating two congruent triangles. A proof is shown below. A Euclidean construction

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We're asked to construct a 10 degree angle. So we have this little angle tool here that we can use to construct an angle. So just like that. And then they give us a protractor to actually measure the angle. So let's set it up. Let's put the protractor here. Let's put the vertex of the angle at the center of the protractor, in other words, center the protractor at the center of the vertex. Looks like the protractor's a little bit easier to manipulate. So let's do that there. Now let's put one of the rays here at 0 degrees. And now I'm going to put the other ray at 10 degrees. And it looks like I am done. I have constructed a 10 degree angle. And you want to be careful here when you use this tool because the angle in question-- let me move the protractor show you what I'm talking about-- that this right over here is the angle that we're talking about. If I'd switched these two rays, the way the tool is set up, it might have interpreted the angle as this outer angle right over here. So be careful to look at which angle we're actually measuring. But this looks like I did the 10 degree angle. So let me check my answer. And just to be clear what I'm talking about. If you did the 10 degree angle like this, then it definitely would have marked it wrong even though the interior angle right over here or this angle right here might be 10 degrees. The way the tool is set up, you see from this circle, that it thinks that you're looking at this outer angle. So it's important to make sure, at least for the sake of this exercise, that the system, that the computer program knows which angle you're going to talk about. Let's do one more of these. 155 degree angle. And this one's interesting because this is an obtuse angle. So once again, let us put the protractor at the vertex of the angle. So just like that. And now that seems pretty good. Now let's take one ray and put it at 0 degrees, and then let's take the other one and put it at 155. So once again, 10, 20, 30, 40, 50, 60, 70, 80, 90-- that gets us to a right angle. Then we'll start getting into obtuse angles, 100, 110, 120, 130, 140, 150. And just to make sure that blue arc is measuring this angle right over here, not the outer one. And let me move the protractor out of the way so we can get a good look at it. And we got it wrong. So let's see what we-- oh, 155 degree angle, not 150 degree angle. Let me fix that. 155 degree angle. Now we got it right.

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Constructing angles (video) | Khan Academy

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We're asked to construct a 10 degree angle. So we have this little angle tool here that we can use to construct an angle. So just like that. And then they give us a protractor to actually measure the angle. So let's set it up. Let's put the protractor here.

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Use only your compass and straight edge when drawing a construction. No free-hand drawing!

The constructions associated with copying a segment and copying an angle require that you have a "place" to begin your copy. It is customary to draw a straight line upon which you then produce your copy. Such a line is called a "reference line".

STEPS:
1. Using a straightedge, draw a reference line, if one is not provided.
2. Draw a dot on the reference line to mark your starting point for the construction.
3. Place the point of the compass on point A on the given figure.
4. Stretch the compass so that the pencil is exactly on B. Make a small arc through B.
(This small arc will show that you measured the length of the segment with your compass.)
5. Without changing the span of the compass, place the compass point on the starting point (dot) on the reference line and swing the pencil to create an arc crossing the reference line.
6. Label your copy.

Proof of Construction: The compass was used as a measuring tool to obtain (and copy) the length of the given segment. Since the given segment and the copy are the same length, the segments are congruent.
Note: You may also think of the length from A to B as being the radius of a circle with center at A. By copying the circle at A', the radii segments will be congruent.

STEPS:
1. Using a straightedge, draw a reference line, if one is not provided.
2. Place a dot (starting point) on the reference line.
3. Place the point of the compass on the vertex of the given angle, ∠ ABC (vertex at point B).
4. Stretch the compass to any length that will stay "on" the angle.
5. Swing an arc so the pencil will cross BOTH sides (rays) of the angle.
6. Without changing the size of the compass, place the compass point on the starting point (dot) on the reference line and swing an arc that will intersect the reference line and go above the reference line.
7. Go back to the given angle ∠ ABC and measure the span (width) of the arc from where it crosses one side of the angle to where it crosses the other side of the angle. (Place a small arc to show you measured this distance.)
8. Using this width, place the compass point on the reference line where the previous arc crosses the reference line and mark off this new width on your new arc.
9. Connect this new intersection point to the starting point (dot) on your reference line.
10. Label your copy.

Proof of Construction: When your construction is finished, draw line segments connecting where the first arcs cross the sides of the angles. We know these lengths are the same in both drawings since they represent the measured spans of the same arcs. These addition of these segments will create two triangles that have 3 sets of congruent (equal) sides. Since the triangles are congruent by SSS and any leftover corresponding parts will also be congruent (by CPCTC), we know that ∠ ABC is congruent to ∠ A'B'C'.
( Note: These triangles are not necessarily Δ ABC and Δ A'B'C', since the first arc in the construction need not pass through A and/or C on the given angle.)

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Construction to Copy Segment and Angle - MathBitsNotebook (Geo - CCSS Math)

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Use only your compass and straight edge when drawing a construction. No free-hand drawing! The constructions associated with copying a segment and copying an angle require that you have a "place" to begin your copy. It is customary to draw a straight line upon which you then produce your copy.

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6.5: Proving Triangles are Congruent or Similar

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6.5: Proving Triangles are Congruent or Similar

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I can prove triangular relationships using congruence and similarity criteria for triangles.

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Proving Triangles are Congruent - Packet

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Proving Triangles are Congruent - Packet

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Proving Triangles are Congruent - Packet

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Definition: Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other.

In the simple case below, the two triangles PQR and LMN are congruent because every corresponding side has the same length, and every corresponding angle has the same measure. The angle at P has the same measure (in degrees) as the angle at L, the side PQ is the same length as the side LM etc.

Try this Drag any orange dot at P,Q,R. The other triangle LMN will change to remain congruent to it.

In the diagram above, the triangles are drawn next to each other and it is obvious they are identical. However, one triangle may be rotated, flipped over (reflected), or the two triangles may share a common side. These cases are discussed further on other pages:

Imagine the triangles are cardboard

One way to think about triangle congruence is to imagine they are made of cardboard. They are congruent if you can slide them around, rotate them, and flip them over in various ways so they make a pile where they exactly fit over each other.

How to tell if triangles are congruent
Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of them to show that two triangles are congruent. Various groups of three will do. Triangles are congruent if:

1. SSS (side side side)
All three corresponding sides are equal in length.
SeeTriangle Congruence (side side side).

2. SAS (side angle side)
A pair of corresponding sides and the included angle are equal.
SeeTriangle Congruence (side angle side).

3. ASA (angle side angle)
A pair of corresponding angles and the included side are equal.
SeeTriangle Congruence (angle side angle).

4. AAS (angle angle side)
A pair of corresponding angles and a non-included side are equal.
SeeTriangle Congruence (angle angle side).

5. HL (hypotenuse leg of a right triangle)
Two right triangles are congruent if the hypotenuse and one leg are equal.
SeeTriangle Congruence (hypotenuse leg).

AAA does not work.

If all the corresponding angles of a triangle are the same, the triangles will be the same shape, but not necessarily the same size. For more on this see Why AAA doesn't work.

They are called similar triangles (See Similar Triangles).

SSA does not work.

Given two sides and a non-included angle, it is possible to draw two different triangles that satisfy the the values. It is therefore not sufficient to prove congruence. See Why SSA doesn't work.

Constructions
Another way to think about the above is to ask if it is possible to construct a unique triangle given what you know. For example, If you were given the lengths of two sides and the included angle (SAS), there is only one possible triangle you could draw. If you drew two of them, they would be the same shape and size - the definition of congruent. For more on constructions, see Introduction to Constructions
Properties of Congruent Triangles

If two triangles are congruent, then each part of the triangle (side or angle) is congruent to the corresponding part in the other triangle. This is the true value of the concept; once you have proved two triangles are congruent, you can find the angles or sides of one of them from the other.

To remember this important idea, some find it helpful to use the acronym CPCTC, which stands for " Corresponding Parts of Congruent Triangles are C ongruent".

In addition to sides and angles, all other properties of the triangle are the same also, such as area, perimeter, location of centers, circles etc.

Other congruence topics

Congruent Triangles

Congruent Polygons

(C) 2011 Copyright Math Open Reference. All rights reserved

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Definition: Triangles are congruent when all corresponding sides and interior angles are congruent. The triangles will have the same shape and size, but one may be a mirror image of the other. In the simple case below, the two triangles PQR and LMN are congruent because every corresponding side has the same length, and every corresponding angle has the same measure.

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Proving Triangles are Congruent - Examples and Practice Problems

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Proving Triangles are Congruent - Examples and Practice Problems

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Day 2: Proving Triangles Congruent

Objective: Prove two triangles congruent by using the SSS, SAS, and the ASA Postulates.
SSS Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

By the SSS Postulate, triangle ABC is congruent to triangle FGH.
SAS Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

By the SAS Postulate, triangle ABC is congruent to triangle FGH.
ASA Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

By the ASA Postulate, triangle ABC is congruent to triangle FGH.
Proving Triangles Congruent Using the SAS Postulate:
Prove: Triangle MOK is congruent to triangle TOK

Statements

Reasons

1. Segment OK bisects angle MOT

1. Given

2. angle 1 is congruent to angle 2

2. Definition of angle bisector

3. Segment OK is congruent to segment OK

3. Reflexive

4. Segment OM is congruent to segment OT

4. Given

5. Triangle MOK is congruent to triangle TOK

5. SAS Postulate

Proving Triangles Congruent Using the ASA Postulate:
Given: Segment BA is perpendicular to segment YZ and segment BA bisects angle YBZ

Statements
Reasons

1. Segment BA is perpendicular to segmt YZ

1. Given

2. Angle 1 is congruent to angle 2

2. If 2 lines are perpendicular, they form congruent adjacent angles

3. Segment BA bisects angle YBZ

3. Given

4. Angle 3 is congruent to angle 4

4. Def. of angle bisector

5. Segment AB is congruent to segment AB

5. Reflexive

6. Triangle AYB is congruent to triang AZB

6. ASA Postulate

Exercises
Supply the missing reasons.

1. Given: Segment AB is parallel to segment DC; Segment AB is congruent to segment DC

Statements
Reasons

1. Segment AB is congruent to segment DC

1. ?

2. Segment AC is congruent to segment AC

2. ?

3. Segment AB is parallel to segment DC

3. ?

4. Angle BAC is congruent to angle DCA

4. ?

5. Triangle ABC is congruent to triangle CDA

5 ?

2. Given: Segment RS is perpendicular to segment ST; Segment TU is perpendicular to segment ST; V is the midpoint of segment ST.

Statements
Reasons

1. Segment RS is perpend to ST; Segment TU is perpend to ST

1. ?

2. Angle S =90; Angle ? =90

2. ?

3. Angle S is congruent to Angle T

3. ?

4. V is the midpoint of ST

4. ?

5. Segment SV is congruent to ?

5. ?

6. Angle RVS is congruent to angle ?

6. ?

7. Triangle ? is congruent to triangle ?

7. ?

Write proofs in two-column form.

3. Given: Segment TM is congruent to PR; Segment TM is parallel to RP

4. Given: E is the midpoint of segment TP; E is the midpoint of segment M

Return to Instructional Unit

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Day 2: Proving Triangles Congruent Objective: Prove two triangles congruent by using the SSS, SAS, and the ASA Postulates. SSS Postulate If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent. By the SSS Postulate, triangle ABC is congruent to triangle FGH.

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Notes - Methods of Proving Triangles are Similar

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Notes - Methods of Proving Triangles are Similar

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When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent.

When triangles are congruent, six facts are always true.

Corresponding sides are congruent.

Corresponding angles are congruent.

The good news is that when proving triangles congruent, it is not necessary to prove all six facts to show congruency. There are certain ordered combinations of these facts that are sufficient to prove triangles congruent. These combinations guarantee that, given these facts, it will be possible to draw triangles which will take on only one shape (be unique), thus insuring congruency.

Methods that Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts
will be sufficient to prove triangles congruent.

If three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.

If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
This is an extension of ASA. In ASA, since you know two sets of angles are congruent, you automatically know the third sets are also congruent since there are 180º in each triangle.

If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent.

The "included angle" in SAS is the angle formed by the two sides of the triangle being used.
The "included side" in ASA is the side between the angles being used. It is the side where the rays of the angles overlap.
The "non-included" side in AAS can be either of the two sides that are not directly between the two angles being used.

Once triangles are proven congruent, the corresponding leftover "parts" that were not used in SSS, SAS, ASA, AAS and HL, are also congruent.
CPCTC
Corresponding Parts of Congruent Triangles are Congruent.

Methods that DO NOT Prove Triangles Congruent

The following ordered combinations of the congruent triangle facts will NOT be sufficient to prove triangles congruent. Let's see why these combinations DO NOT work!

This method will NOT always prove triangles congruent!

The AAA combination will show that the triangles are the same SHAPE (similar), but will NOT show that the triangles are the same size.

Example:Consider these two equilateral triangles that satisfy the AAA combination. They are the same shape, but are not the same size. Thus, they are not congruent. (They are similar.)

Yes, it is possible that the sides "could" be the same length and the triangles would be congruent, but this would be the exception, not the rule.

SSA or ASS
Side-Side-Angle

This method will NOT always prove triangles congruent!

The SSA (or ASS) combination deals with two sides and the non-included angle. This combination is humorously referred to as the "Donkey Theorem". SSA (or ASS) is NOT a universal method to prove triangles congruent since it cannot guarantee that the shapes of the triangles formed will always be the same.

Let's look at the problems associated with SSA:
When using SSA (or ASS), the lengths of the sides being used, and their position in relation to the angle, will determine if congruent triangles will always be created. Let's see which special conditions will create congruent triangles, and which ones will not.

Note: This specific case of SSA is the basis for the acceptable method HL (Hypotenuse Leg) which applies only in right triangles.

If the side which lies on one ray of the angle is longer than the other side, and the other side is greater than the minimum distance needed to create a triangle, the two triangles will not necessarily be congruent.

If EF is greater than EG, the diagram below shows how it is possible for to "swing" to either side of point G, creating two non-congruent triangles using SSA.

If the side which lies on one ray of the angle is longer than the other side, and the other side is less than the minimum distance needed to create a triangle, no triangle can be drawn. Since no triangles are possible, no congruent triangles are possible.

No Congruent Triangles

Since the SSA (or ASS) method allows for the possibility of creating triangles of various shapes (or even no triangles at all), this method is not an universal method for proving triangles congruent.

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Methods of Proving Triangle Congruent - MathBitsNotebook(Geo - CCSS Math)

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When triangles are congruent, one triangle can be moved (through one, or more, rigid motions) to coincide with the other triangle. All corresponding sides and angles will be congruent. When triangles are congruent, six facts are always true. Corresponding sides are congruent. Corresponding angles are congruent.

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Similar Triangle Flowcharts - Examples and Practice Problems