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How to identify reflectional and rotational symmetry in figures

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The Isosceles Triangle Theorem | MathHelp.com

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Need a custom math course? Visit http://www.MathHelp.com. This lesson covers the isosceles triangle theorem. Students learn that an isosceles triangle is composed of a base, two congruent legs, two congruent base angles, and a vertex angle.

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If two sides of a triangle are congruent, then the angles opposite to these sides are congruent.

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The sum of the three angles in any triangle sum to 180 degrees. The importance of this fact in Geometry cannot be emphasized enough. The triangle angle sum theorem is used in almost every missing angle problem, in the exterior angle theorem, and in the polygon angle sum formula.

One of the most important properties of triangles that we use all over Geometry is this triangle, angle, sum. What this is talking about is the 3 angles in a triangle and what they're always going to add up to. We're going to look at this two different ways, the first one using a physical model of a triangle secondly we're going to walk over here and we're going to look at this triangle and prove it on the board using some Algebra. So let's start off with this triangle, what's a way that I can determine what these 3 angles of the scaling triangle have to add up to?
Well, what I could do, is that I could tear off each of these 3 angles. So here I've got 1 piece and I'm going tear off another piece and what I've done is I've created 3 different angles. So what I'm going to do, is I'm going too take these 3 angles which I've already precut out and taped and I'm going to tape them on the board to see if we can figure anything out. So here's angle 1 and I'm going to tape on the board and I'm going to grab angle 2 and so you can make a difference between the different angles I'm going to leave a little bit of space here, so here's angle 2, so that way you can see that there's a little bit of a difference and last angle 3, and again leaving a little bit of space so you can see that we have 3 different angles and they form a straight line.
We know that a straight line is always 180 degrees, so what we can say is that angle 1 plus angle 2 plus angle 3 sum to 180 degrees. But does this always work? It will work for any triangle. So let's move over to the Algebraic explanation of this, right here we have a triangle with 3 angles labeled 1, 2 and 3, and for this proof I'm going to draw in 3 auxiliary lines. The first auxiliary line or helping line is going to be parallel to this base containing 1 and 3. Now to extend our 2 other auxiliary lines I'm going to think of these as transversals that is a line that intersects 2 parallel lines. So we've got 1 transversal right there and we have our second transversal right here.
I'm also going to add in 2 more angles, angle 4 and angle 5. So what can we say about 4, 2 and 5, well it's pretty clear that they are all forming one line. So I'm going to say that angle 4 plus angle 2 plus angle 5 sum to 180 degrees. If I compare my 2 equations, one equation has our 3 angles 1, 2, 3 so this is where we want to end up, the other one has 4 and 5 instead of 1 and 3. So how can we rectify this problem, well I see that angel 4 and angle 1 are alternate interior angles. So what I'm going to do, I'm going to say angle 4 is congruent to angle 1 by alternate interior angles. By the same token we can say that 5 and 3 are alternate interior angles as well. Which means angle 5 is congruent to angle 3 by alternate interior angles. So what I'm going to is I'm going to substitute in for 4 with angle 1 and in for angle 5 angle 3. So I'm going to write angle 1 plus angle 2 plus angle 3 sum to 180 degrees. So we've proved it two ways, one by tearing up the 3 pieces of a triangle and seeing that they form a line and secondly using this Algebraic proof where we drew this parallel line that helped us prove that the 3 angles in a triangle will sum to 180 degrees.

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Triangle Angle Sum - Concept - Geometry Video by Brightstorm

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The sum of the three angles in any triangle sum to 180 degrees. The importance of this fact in Geometry cannot be emphasized enough. The triangle angle sum theorem is used in almost every missing angle problem, in the exterior angle theorem, and in the polygon angle sum formula.

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Angle sum theorem: The angle measures in any triangles add up to 180 degrees.

Key concept: Alternate interior angles are equal. We will accept this fact without a proof.

The figure above shows two pairs of alternate interior angles.

For the pair in red, angle 1 = angle 2. For the pair in blue, angle 3 = angle 4

Now, take a close look at the figure below. I claim that angle x is equal to 85 degrees so the sum is 180 degrees.

To see why this is so, draw a line parallel to AC at vertex B

Angle a = 65 degrees because it alternates with the angle inside the triangle that measures 65 degrees

Angle b = 30 degrees because it alternates with the angle inside the triangle that measures 30 degrees

Looking at the figure again, it is easy to see why angle x is 85.

Since the 65 degrees angle, the angle x, and the 30 degrees angle make a straight line together, the sum must be 180 degrees

Since, 65 + angle x + 30 = 180, angle x must be 85

This is not a proof yet. This just shows that it works for one specific example

Proof of the angle sum theorem:

Start with the following triangle with arbitrary values for the angles:

Since angle a, angle b, and angle c make a straight line,

angle a + angle b + angle c = 180 degrees

Since alternate interior angles are equal, angle a = angle x and angle b = angle y

Therefore, angle x + angle y + angle c = 180 degrees

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Angle sum theorem: The angle measures in any triangles add up to 180 degrees. Key concept: Alternate interior angles are equal. We will accept this fact without a proof. The figure above shows two pairs of alternate interior angles. For the pair in red, angle 1 = angle 2.

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Essential Question: How can the coordinate plane help me understand properties of reflections, translations, and rotations?

Lesson

Figures in a plane can be reflected, rotated, translated or dilated to produce new figures. The new figure is called the image, and the original figure is called the preimage. The operation that maps, or moves, the preimage onto the image is called a transformation.

Some transformations involve labels. When naming an image, take the corresponding point of the preimage and add a prime symbol. For instance, if the preimage is A, the image is A′, read as "A prime." If A′ has a second transformation applied, its image becomes A′′, read as "A double-prime."

Transformations can be described by writing "∆ABC is mapped onto ∆DEF." Another notation often used is ∆ABC → ∆DEF. The order in which the vertices are listed specifies the correspondence. Either of the descriptions implies that A → D, B → E and C → F.

An isometry is a transformation that preserves lengths. Isometries also preserve angle measures, parallel lines, and distances between points.
Rigid Transformations
Transformations that are isometries are called rigid transformations. Three types of isometries are shown below. The preimages are shown in blue and the images in red.

Reflection in a line: A transformation that uses a line that acts like a mirror, called the line of reflection, with an image reflected in the line. Since all points move across the line of reflection, the image of each point will be the same distance away from the line as the preimage.

Rotation about a point: A transformation in which a figure is turned about a fixed point, called the center of rotation.

Translation: A translation is a transformation that slides each point of a figure the same distance in the same direction. The lines connecting each point of the preimage to its image are congruent.
Non-Rigid Transformation
One type of transformation that is not an isometry is called a dilation.

Dilation: A dilation is an enlargement or reduction of an image by a factor of k. Even if the image is smaller than the preimage it is still called a dilation, the dilation factor would be less than 1.

When two or more transformations are combined to produce a single transformation, the result is called a composition of the transformations. When a translation (glide) and a reflection are both performed one after the other, the result is called a glide reflection. As long as the line of reflection is parallel to the direction of the translation, it does not matter which is done first, the glide or the reflection. The composition of two or more isometries is an isometry. In the figure below, the red segment is the image of the blue segment, and the green segment is the image of the red segment (the transformation maps blue → red → green). The blue segment glides along a line parallel to k, to map onto the red segment, and then reflects in line k to map onto the green segment.

When determining the type of transformation, pay attention to the labels. In a reflection, since all points move across the line of reflection, the image of each point will be the same distance away from the line as the preimage. In a translation, the distance between each point on the preimage and its image is the same. In a rotation, there is a clockwise or counter clockwise motion of each point.

To determine the type of isometry, find at least three pairs of corresponding points. Check the orientation of the preimage and image. If one is clockwise and one is counter-clockwise, then the orientation is reversed and the isometry must be a reflection. If the orientation is not reversed, then the isometry is a translation or a rotation. Draw lines connecting the preimages to the images of the matching points. If they are the same length, the isometry is a translation. Otherwise, it is a rotation.

Examples

Example 1. Identify the transformation that maps the lighter checkered flag (preimage) onto the darker checkered flag (image).

Solution: Every point of the preimage slides the same distance in the same direction. The transformation is a translation. Shown below is a diagram of some of the lines of translation. Notice that they are all congruent and parallel.

Example 2. Identify the transformation that maps the blue figure (preimage) onto the red figure (image).

Solution: This transformation uses a line that acts like a mirror, with the image reflected in the line, so it is a reflection. If the object was printed on paper, folded along the line of reflection, and held up to the light, only one shape would be seen. The preimage and image would lie behind one another.

Example 3. Identify the transformation that maps the blue pickup truck (preimage) onto the red pickup truck (image).

Solution: The figure is turned about a fixed point, so it is a rotation. The figure below shows the center of rotation and the preimage and image of a random point to show the 180º rotation. With a 180º rotation, the image appears "upside down."

Example 4. Which of the following transformations appear to be isometries?

Solution: Figures a and c appear to preserve lengths, angle measures, parallel lines, and distances between points. The image for Figure b appears to be smaller than its preimage, so it is not an isometry.

Example 5. Identify the transformation that maps the blue figure (preimage) onto the red figure (image).

Solution: Pay close attention to the labeling. Since is A →A′, B →B′ and C →C′, determine the transformation in relation to each vertex. This transformation uses a line that acts like a mirror, with its image reflected in the line. It is a reflection.

Example 6. In the diagram, is the image of and is the image of . Describe the combination of transformations → → .

Solution: The transformation → is a translation, all points slide vertically. The transformation → is a reflection in the y-axis. The transformation is a translation followed by a reflection (can be referred to as a glide reflection).

Self-Check

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Essential Question: How can the coordinate plane help me understand properties of reflections, translations, and rotations? Figures in a plane can be reflected, rotated, translated or dilated to produce new figures. The new figure is called the image, and the original figure is called the preimage.

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Transformations and Isometries - Concept - Geometry Video by Brightstorm

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Time-saving video that describes all the different types of transformations in Geometry: dilations, rotations, translations, and reflections. Isometry is also described. Concept explanation.

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Properties of parallelograms (Geometry, Quadrilaterals) - Mathplanet

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If we have a quadrilateral where one pair and only one pair of sides are parallel then we have what is called a trapezoid. The parallel sides are called bases while the nonparallel sides are called legs. If the legs are congruent we have what is called an isosceles trapezoid.

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Parallel Lines

Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember:

Always the same distance apart and never touching.

The red line is parallel to the blue line in each of these examples:

Parallel lines also point in the same direction.

Parallel lines have so much in common. It's a shame they will never meet!

Try it yourself:

Pairs of Angles

When parallel lines get crossed by another line (which is called a Transversal), you can see that many angles are the same, as in this example:

These angles can be made into pairs of angles which have special names.

Click on each name to see it highlighted:

Now play with it here. Try dragging the points, and choosing different angle types. You can also turn "Parallel" off or on:

Testing for Parallel Lines

Some of those special pairs of angles can be used to test if lines really are parallel:

Examples

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Lines are parallel if they are always the same distance apart (called "equidistant"), and will never meet. Just remember: The red line is parallel to the blue line in each of these examples: Parallel lines also point in the same direction. Parallel lines have so much in common.

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Angles Formed by Parallel Lines and Transversals

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How to identify the angles formed by parallel lines and transversals, including consecutive and alternate interior and exterior angles and corresponding angles

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Reflecting points in the coordinate plane (practice) | Khan Academy

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Practice reflecting points across axes on the xy coordinate plane.

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The point negative 8 comma, 5 is reflected across the y-axis. Plot negative 8 comma 5 and its reflection across the y-axis. So first let's plot negative 8 comma 5. So its x-coordinate is negative 8, so I'll just use this one right over here. So the x-coordinate is negative 8, and the y-coordinate is 5, so I'll go up 5. So the y-coordinate is 5 right over here. You see negative 8 and 5. We've gone 8 to the left because it's negative, and then we've gone 5 up, because it's a positive 5. So we've plotted negative 8 comma 5. Now we have to plot its reflection across the y-axis. And so you can imagine if this was some type of lake or something and you were to see its reflection, and this is, say, like the moon, you would see its reflection roughly around here. You would see an equal distance away from the y-axis. So you would see it at 8 to the right of the y-axis, which would be at positive 8, and still 5 above the x-axis. So that's its reflection right over here. It's reflection is the point 8 comma 5. Let's do a couple more of these. The point negative 6 comma negative 7 is reflec-- this should say "reflected" across the x-axis. Plot negative 6 comma negative 7 and its reflection across the x-axis. So negative 6 comma negative 7, so we're going to go 6 to the left of the origin, and we're going to go down 7. So there we go. Negative 6 comma negative 7 is right there. And we are reflecting across the x-axis. So, once again, if you imagine that this is some type of a lake, or maybe some type of an upside-down lake, or a mirror, where would we think we see its reflection? Well, its reflection would be the same distance. We're reflecting across the x-axis, so it would be the same distance, but now above the x-axis. So this was 7 below. Now we're going to go 7 above the x-axis, and it's going to be at the same x-coordinate. So there you have it right over here. We reflected this point to right up here, because we reflected across the x-axis. Let's check our answer. Let's do one more. The point B is a reflection of point A across which axis? So let's think about this right over here. This is at the point negative 6 comma 5. This is at the point negative 5 comma 6. Let's see. It doesn't look like it's only one axis. If I were to reflect this point across the y-axis, it would go all the way to positive 6, 5. So it would go all the way right over here. So if I reflect A just across the y-axis, it would go there. And then if I reflected that point across the x-axis, then I would end up at 5 below the x-axis at an x-coordinate of 6. So to go from A to B, you could reflect across the y and then the x, or you could reflect across the x, and it would get you right over here. It would get you to negative 6 comma 5, and then reflect across the y. So it's really reflecting across both axes. So we would reflect across the x-axis and then the y-axis. It would have also been legitimate if we said the y-axis and then the x-axis. Let's check our answer. We got it right.

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Reflecting points in the coordinate plane (video) | Khan Academy

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The point negative 8 comma, 5 is reflected across the y-axis. Plot negative 8 comma 5 and its reflection across the y-axis. So first let's plot negative 8 comma 5. So its x-coordinate is negative 8, so I'll just use this one right over here.

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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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Geometry - Constructions 2 - Copying An Angle

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Copying an angle using a compass and straight-edge.

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Perpendicular to a line at a point on the line construction

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Perpendicular to a line at a point on the line construction

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Perpendicular to a line at a point on the line construction

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Perpendicular to a line at a point on the line construction

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Constructing a Perpendicular at a Point on a Line

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Perpendicular to a line at a point on the line construction

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Perpendicular to a line at a point on the line construction

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How to construct (draw) a perpendicular at a point on a line - Math Open Reference

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This page shows how to draw a perpendicular at a point on a line with compass and straightedge or ruler. It works by effectively creating two congruent triangles and then drawing a line between their vertices. A Euclidean construction.

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Prove lines parallel given supplementary angles

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corresponding parts or congruent triangles

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CPCTC - Concept - Geometry Video by Brightstorm

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Time-saving video that defines and describes what CPCTC means: corresponding parts of congruent triangles are congruent. Example problems show corresponding parts. Concept explanation.

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Cpctc - math word definition - Math Open Reference

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Definition and meaning of the math word cpctc. Includes a list of other pages that refer to this word

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Geometry Proofs | Congruent Triangles & CPCTC

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Geometry - 4 - Proving Triangles Congruent

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Math explained in easy language, plus puzzles, games, quizzes, worksheets and a forum. For K-12 kids, teachers and parents.