Triangle Congruence Postulates - Congruence and Theorems - High School Geometry Unlocked (2016)

High School Geometry Unlocked (2016)

Chapter 2. Congruence and Theorems

GOALS

By the end of this chapter, you will be able to:


•Identify congruent triangles using the SSS, AAS, ASA, SAS postulates

•Use the Pythagorean theorem or the Third Side Rule to solve for an unknown side in a right triangle

•Understand how to write a formal or informal proof

•Construct figures with a compass and straightedge (parallel and perpendicular lines, angle bisector, angle copy, equilateral triangle, square, and regular hexagon)

Lesson 2.1. Triangle Congruence Postulates

REVIEW

POSTULATES: REVIEW THE FOLLOWING POSTULATES, WHICH YOU MAY FIND HELPFUL FOR THIS CHAPTER.

Vertical angles are congruent (intersection of any two lines)

Alternate interior angles are congruent (parallel lines cut by a transversal)

Alternate exterior angles are congruent (parallel lines cut by a transversal)

Corresponding angles are congruent (parallel lines cut by a transversal)

In geometry, we often have to use limited information about a problem in order to figure out several unknowns. Mathematicians call it a proof when they use known information to derive other facts and unknowns in a problem. In fact, many geometry classes spend a great deal of time writing exhaustively detailed proofs for problems and figures. We won’t make you write full proofs in this book! However, we will show you proofs and sometimes ask you questions about them.

One of the more common facts you’ll need to prove is whether or not two figures are congruent. This is one of the most important topics in mathematics, and in fact, a great number of geometric rules and theorems are based on congruence. We deal with this in real life as well—if you’re building a structure, you need to make sure that various parts of the structure are congruent (e.g. opposite walls, the ceiling and the floor, etc.) if you don’t want the structure to be lopsided. Another example is the tires on your car—if the tires are not exactly congruent (even if one has just a little more or less air in it compared to the others), you’d feel a lot of bumps and shakes as you drive, and it may put excessive wear and tear on your car.

To recap, if two figures are congruent, it means that they have the same shape and size. If two polygons are congruent, it means that all of their side lengths and angles are congruent too. Additionally, recall that if a figure is translated, rotated, or reflected, the image and pre-image will be congruent.

Example of Congruent Figures

Throughout history, mathematicians have developed several rules that we now use to solve problems. The words theorem and postulate both describe rules. These terms are often used interchangeably, and you probably shouldn’t worry about the differences. But just so you know: a postulate (also known as an axiom) is something that everyone agrees is true, while a theorem is something that needs to be proven true using logical steps. Many of the “theorems” that we discuss today are effectively postulates, because they’ve been proven true in the past and we no longer need to doubt their accuracy.

In the case of triangles, we can often derive a lot of information when we know even just a couple of facts (e.g. angle measures or side lengths). In this lesson, you’ll learn and understand several basic theorems regarding congruent triangles.

SSS (SIDE-SIDE-SIDE) POSTULATE

If the triangles above have three side lengths in common, are the triangles necessarily congruent?

If a triangle has three specified side lengths, then both its size and shape are fixed. Those three sides can meet only at specified angles. Any way of “rearranging” the given sides would result in a translation, reflection, or rotation of the same triangle.

For instance, what if we try making angle A smaller?

If we make angle A smaller, the figure will not be able to connect at the same vertices. We can try moving it around in any way we can think of, but the only way to make a triangle with those specific side lengths is if it has the same angles as the original triangle.

Recall from Lesson 1.1 that
if a figure is translated,
reflected, or rotated,
then the image and pre-
image are congruent.

That’s the basic idea behind each of these postulates—once you know certain facts about a triangle, you know that there’s only one form that the triangle can take.

SSS (Side-Side-Side) Postulate

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

Try it yourself! You can
demonstrate this postulate
using physical objects,
such as straws, to form
the sides of the triangle.

SSS Postulate Exercise

For each of the pairs of triangles below, write “yes” if the triangles can be proved to be congruent to the information provided; otherwise, write “no.”

Complete the exercise on your own. Answers are at the end of the chapter.

1. ______

2. ______

3. ______

4. ______

5. ______

6. ______

AAS (ANGLE-ANGLE-SIDE) AND ASA (ANGLE-SIDE-ANGLE) POSTULATES

If the triangles above have two angles and one side length in common, are the triangles necessarily congruent?

First, let’s check our unknowns in the example above. If two angles are specified, then really, the third angle is specified also. That’s because all triangles have angles that add up to 180° (so we can just subtract the two given angles from 180° to find the third one).

If all three angles of a triangle are known, then the shape of the triangle is fixed, but not its size. You can make the triangle bigger or smaller, while keeping the angles the same.

AAA (Angle-Angle-Angle) Similarity Postulate

The AAA (Angle-Angle-Angle) postulate states that if two triangles have all three angle measures in common, then the two triangles are similar. However, they are not necessarily congruent. Hence, this is often referred to as the “AAA Similarity” postulate.

If all three angles are known, then we know the triangle’s shape. If we also specify a side length, then we know the triangle’s size.

Therefore, we have two congruence postulates that we can use when triangles have two (actually three!) angle measures and one side length in common.

ASA (Angle-Side-Angle) Postulate

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

Congruent angles are
often identified with
marks like these. Look
for matching marks
(such as a single
hash) to identify
corresponding angles.

AAS (Angle-Angle-Side) Postulate

If two angles and a non-included side of one triangle—that is, a side not are congruent to the corresponding parts in another triangle, then the triangles are congruent.

There’s an important catch that you must remember—the known side must be in the same place, relative to the angles, in both triangles. That is, if the known side is opposite to the smallest angle in the first figure, then that must also be true for the second figure.

Another way to think of this is that order matters—as you go around the triangle in either direction, you have an angle, side, and angle (or an angle, angle, and side), one after the other, in that order. If the second triangle has those same parts with the same measures in the same order, then the two triangles are congruent.

That’s the reason that there are two different postulates specified as AAS and ASA—so we don’t mix up the rules and we make sure that the given parts are in the right place.

AAS Postulate Exercise

For each of the pairs of triangles below, write “yes” if the triangles can be proved to be congruent to the information provided; otherwise, write “no.”

Complete the exercise on your own. Answers are at the end of the chapter.

1. ______

2. ______

3. ______

4. ______

5. ______

6. ______

SAS (SIDE-ANGLE-SIDE) POSTULATE

If the triangles above have two sides and one angle in common, are the triangles necessarily congruent?

If a triangle had only two side lengths specified, and no other information, you wouldn’t be able to determine the triangle’s shape or size. The two given side lengths could come together in any number of ways.

However, if you can specify the angle that joins the two given side lengths, then the triangle is fixed. That angle measure tells you how far apart the other two vertices are, and it specifies the length of the third side of the triangle. (You’ll learn how to calculate that length in Chapter 4!)

Therefore, the SAS postulate can be used when you have two triangles with two side lengths in common, as well as the angle that joins those two sides.

SAS (Side-Angle-Side) Postulate

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

Congruent Triangles

Again, order matters with this postulate, as it does with the ASA and AAS postulates. As you go around the triangle in either direction, you have a side, angle, and side, one after the other, in that order. If the second triangle has those same parts with the same measures in the same order, then the two triangles are congruent.

Additionally, you may wonder if there is an SSA (Side-Side-Angle) postulate. This particular combination does not work for triangle congruence, so there is no SSA postulate. The reason is that if you have two side lengths and a non-included angle specified, then there are exactly two triangles that can exist with that combination.

In the example above, there are two different ways the side with two hash marks can fit into the space created by the marked angle.

OTHER IMPORTANT THEOREMS AND POSTULATES FOR TRIANGLES

The following theorems are used very frequently in geometry. You may know them already!

Pythagorean Theorem

The Pythagorean theorem is definitely one of the most important triangle theorems that you should know. It is used to solve for an unknown side length in a right triangle.

Pythagorean Theorem

In any right triangle with legs a and b and hypotenuse c:

a2 + b2 = c2

In the triangle above, what is the value of c ?

To solve, use the Pythagorean theorem. Plug in the two given side lengths, and solve for the unknown side.

In this case, the given lengths are for the legs of the triangle, which are a and b in the equation. The unknown side is the hypotenuse, or c in the equation.

Plug in:

32 + 42 = c2

9 + 16 = c2

25 = c2

5 = c

Take the square root of both sides of the equation.

The value of c is 5.

You might have recognized this example as a Pythagorean triple—one of the combinations of integers that satisfy the Pythagorean equation. Try memorizing the following Pythagorean triples to make right triangle problems easier. There are many more, but these are some of the most common ones (and the lowest values) that you’ll encounter:

Third Side Rule

Use the Third Side Rule if you know two side lengths, but no other information. This will give you a limited range of values for the third, unknown side. It’s not a single, exact answer, but sometimes a range is all we have!

The Third Side Rule is also known as the triangle inequality rule.

Third Side Rule for Triangles

In any triangle, the length of one side must be greater than the difference and less than the sum of the other two sides.

In other words, find the difference of the two known sides, and also find the sum of the two known sides. The length of the third side has to be between these two values.

To see how the Third
Side Rule is tested on
the ACT, access your
Student Tools online.

In the triangle above, what is the range of values for x ?

Use the Third Side Rule to solve for x.

First, find the difference of the two known sides:

8 − 3 = 5

Then, find the sum of the two known sides:

8 + 3 = 11

Therefore, the value of x must be between 5 and 11.

5 < x < 11

Here is how you may see the Pythagorean theorem on the ACT.

Greg is making a triangular sail for a boat, shaped like a right triangle and shown below.

To determine how much trim to buy for the sail, Greg calculated the sail’s perimeter. What is the sail’s perimeter?

A.275

B.300

C.290

D.220

E.170