# data[].attributes.body
http://whistleralley.com/construction/reference.htm

# data[].attributes.attachment_link.readable_text
Constructions: The drawing of various shapes using only a pair of compasses and straightedge or ruler. No measurement of lengths or angles is allowed.

The word construction in geometry has a very specific meaning: the drawing of geometric items such as lines and circles using only compasses and straightedge or ruler. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler.

Compasses

Compasses are a drawing instrument used for drawing circles and arcs. It has two legs, one with a point and the other with a pencil or lead. You can adjust the distance between the point and the pencil and that setting will remain until you change it.

This kind of compass has nothing to do with the kind used find the north direction when you are lost. A compass used to find the north direction is usually referred to in the singular - a compass. The kind we are talking about here is usually referred to in the plural - compasses. This plural reference is similar to the way we talk about scissors - with an 's' on the end.

Straightedge

A straightedge is simply a guide for the pencil when drawing straight lines. In most cases you will use a ruler for this, since it is the most likely to be available, but you must not use the markings on the ruler during constructions. If possible, turn the ruler over so you cannot see them.

Why we learn about constructions

The Greeks formulated much of what we think of as geometry over 2000 years ago. In particular, the mathematician Euclid documented it in his book titled "Elements", which is still regarded as an authoritative geometry reference. In that work, he uses these construction techniques extensively, and so they have become a part of the geometry field of study. They also provide insight into geometric concepts and give us tools to draw things when direct measurement is not appropriate.

Why did Euclid do it this way?

Why didn't Euclid just measure things with a ruler and calculate lengths? For example, one of the basic constructions is bisecting a line (dividing it into two equal parts). Why not just measure it with a ruler and divide by two?

One theory is the the Greeks could not easily do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. Also, their numbers did not use a positional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In short, it was quite difficult to do useful arithmetic.

So, faced with the problem of finding the midpoint of a line, it was very difficult to do the obvious - measure it and divide by two. This led to the constructions using compass and straightedge or ruler. It is also why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes instead of using arithmetic.

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions

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

# data[].attributes.attachment_link.title
Constructions Introduction. Drawing shapes with compasses and straightedge - Math Open Reference

# data[].attributes.attachment_link.description
A brief introduction to constructions - creating various geometric objects with only compasses and straightedge or ruler. History and origins

# data[].attributes.attachment_link.description
This is an interactive course on [b][color=#1551b5]geometric constructions[/color][/b], a fascinating topic that has been ignored by the mainstream m...

# data[].attributes.attachment_link.title
BBC Bitesize - Higher Maths - Rectilinear shapes - Revision 8

# data[].attributes.attachment_link.description
Find gradients, equations and intersections of medians, altitudes and perpendicular bisectors for the topic on straight line in Higher Maths.

# data[].attributes.body
Game code: 092657
Or look for other transformations quizzes!

# data[].attributes.attachment_link.description
Join an activity with your class and find or create your own quizzes and flashcards.

# 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.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.

# data[].attributes.attachment_link.description
Get access to files anywhere through secure cloud storage and file backup for your photos, videos, files and more with Google Drive.