Title: READ: HL Theorem Reading
Course: Geometry (A)

Proof Using HL

Learning Objectives

- Identify the distinct characteristics and properties of right triangles.
- Understand and apply the HL Congruence Theorem.
- Understand that SSA does not necessarily prove triangles are congruent.

Introduction

You have already seen three different ways to prove that two triangles are congruent (without measuring six angles and six sides). Since triangle congruence plays such an important role in geometry, it is important to know all of the different theorems and postulates that can prove congruence, and it is important to know which combinations of sides and angles do not prove congruence.

Right Triangles

So far, the congruence postulates we have examined work on any triangle you can imagine. As you know, there are a number of types of triangles. Acute triangles have all angles measuring less than Obtuse triangles have one angle measuring between and Equilateral triangles have congruent sides, and all angles measure Right triangles have one angle measuring exactly

In right triangles, the sides have special names. The two sides adjacent to the right angle are called legs and the side opposite the right angle is called the hypotenuse.

Example 1

Which side of right triangle is the hypotenuse?

Looking at , you can identify as a right angle (remember the little square tells us the angle is a right angle). By definition, the hypotenuse of a right triangle is opposite the right angle. So, side is the hypotenuse.

HL Congruence

There is one special case when SSA does prove that two triangles are congruent-When the triangles you are comparing are right triangles. In any two right triangles you know that they have at least one pair of congruent angles, the right angles.

Though you will learn more about it later, there is a special property of right triangles referred to as the Pythagorean theorem. It isn’t important for you to be able to fully understand and apply this theorem in this context, but it is helpful to know what it is. The Pythagorean Theorem states that for any right triangle with legs that measure and and hypotenuse measuring units, the following equation is true.

In other words, if you know the lengths of two sides of a right triangle, then the length of the third side can be determined using the equation. This is similar in theory to how the Triangle Sum Theorem relates angles. You know that if you have two angles, you can find the third.

Because of the Pythagorean Theorem, if you know the length of the hypotenuse and a leg of a right triangle, you can calculate the length of the missing leg. Therefore, if the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, you could prove the triangles congruent by the SSS congruence postulate. So, the last in our list of theorems and postulates proving congruence is called the HL Congruence Theorem. The “H” and “L” stand for hypotenuse and leg.

HL Congruence Theorem: If the hypotenuse and leg in one right triangle are congruent to the hypotenuse and leg in another right triangle, then the two triangles are congruent.

The proof of this theorem is omitted because we have not yet proven the Pythagorean Theorem.

Example 2

What information would you need to prove that these two triangles were congruent using the HL theorem?

A. the measures of sides and

B. the measures of sides and

C. the measures of angles and

D. the measures of angles and

Since these are right triangles, you only need one leg and the hypotenuse to prove congruence. Legs and are congruent, so you need to find the lengths of the hypotenuses. The hypotenuse of is . The hypotenuse of is . So, you need to find the measures of sides and . The correct answer is A.

Points to Consider

The HL congruence theorem shows that sometimes SSA is sufficient to prove that two triangles are congruent. You have also seen that sometimes it is not. In trigonometry you will study this in more depth. For now, you might try playing with objects or you may try using geometric software to explore under which conditions SSA does provide enough information to infer that two triangles are congruent.

Lesson Summary

In this lesson, we explored triangle sums. Specifically, we have learned:

- How to identify the distinct characteristics and properties of right triangles.
- How to understand and apply the HL Congruence Theorem.
- That SSA does not necessarily prove triangles are congruent.

These skills will help you understand issues of congruence involving triangles. Always look for triangles in diagrams, maps, and other mathematical representations.

The following questions are for your own benefit. The answers are listed below in order for you to check your work and understanding.

Review Questions

Use the following diagram for exercises 1-3.

1. Complete the following congruence statement, if possible _________.
2. What postulate allows you to make the congruence statement in 1, or, if it is not possible to make a congruence statement explain why.
3. Given the marked congruent parts in the triangles above, what other congruence statements do you now know based on your answers to 1 and 2?

Use the following diagram below for exercises 4-6 .

1. Complete the following congruence statement, if possible _________.
2. What postulate allows you to make the congruence statement in 4, or, if it is not possible to make a congruence statement explain why.
3. Given the marked congruent parts in the triangles above, what other congruence statements do you now know based on your answers to 4 and 5?

Use the following diagram below for exercises 7-9.

1. Complete the following congruence statement, if possible ________.
2. What postulate allows you to make the congruence statement in 7, or, if it is not possible to make a congruence statement explain why.
3. Given the marked congruent parts in the triangles above, what other congruence statements do you now know based on your answers to 7 and 8?
4. Write one or two sentences and use a diagram to show why AAA is not a triangle congruence postulate.
5. Do the following proof using a two-column format.

Given: and intersect at , and

Prove:

Statement | Reason
1. | 1. Given
2. (Finish the proof using more steps!) | 2.

Review Answers

2. HL triangle congruence postulate
3. and
5. SAS triangle congruence postulate
7. No triangle congruence statement is possible
8. SSA is not a valid triangle congruence postulate
9. No other congruence statements are possible
10. One counterexample is to consider two equiangular triangles. If AAA were a valid triangle congruence postulate, than all equiangular (and equilateral) triangles would be congruent. But this is not the case. Below are two equiangular triangles that are not congruent:
These triangles are not congruent.

11. Statement | Reason
1. | 1. Given
2. | 2. Given
3. and intersect at | 3. Given
4. and are vertical angles | 4. Definition of vertical angles
5. | 5. Vertical angles theorem
6. | 6. SAS triangle congruence postulate
7. | 7. Definition of congruent triangles (CPCTC)