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# 4.4: Triangle Congruence using ASA, AAS, and HL

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Use and understand the ASA, AAS, and HL Congruence Postulate.
• Complete two-column proofs using SSS, SAS, ASA and AAS.

## Review Queue

1. What sides are marked congruent?
2. Is third side congruent? Why?
3. Write the congruence statement for the two triangles. Why are they congruent?
1. From the parallel lines, what angles are congruent?
2. How do you know the third angle is congruent?
3. Are any sides congruent? How do you know?
4. Are the two triangles congruent? Why or why not?
1. If \begin{align*}\triangle DEF \cong \triangle PQR\end{align*}, can it be assumed that:
1. \begin{align*}\angle F \cong \angle R\end{align*}? Why or why not?
2. \begin{align*}\overline{EF} \cong \overline{PR}\end{align*}? Why or why not?

Know What? Your parents changed their minds at the last second about their kitchen layout. Now, the measurements are in the triangle on the left, below. Your neighbor’s kitchen is in blue on the right. Are the kitchen triangles congruent now?

## ASA Congruence

ASA refers to Angle-Side-Angle. The placement of the word Side is important because it indicates that the side that you are given is between the two angles.

Consider the question: If I have two angles that are \begin{align*}45^\circ\end{align*} and \begin{align*}60^\circ\end{align*} and the side between them is 5 in, can I construct only one triangle?

Investigation 4-4: Constructing a Triangle Given Two Angles and Included Side

Tools Needed: protractor, pencil, ruler, and paper

1. Draw the side (5 in) horizontally, about halfway down the page.

The drawings in this investigation are to scale.

2. At the left endpoint of your line segment, use the protractor to measure the \begin{align*}45^\circ\end{align*} angle. Mark this measurement and draw a ray from the left endpoint through the \begin{align*}45^\circ\end{align*} mark.

3. At the right endpoint of your line segment, use the protractor to measure the \begin{align*}60^\circ\end{align*} angle. Mark this measurement and draw a ray from the left endpoint through the \begin{align*}60^\circ\end{align*} mark. Extend this ray so that it crosses through the ray from Step 2.

4. Erase the extra parts of the rays from Steps 2 and 3 to leave only the triangle.

Can you draw another triangle, with these measurements that looks different? NO. Only one triangle can be created from any given two angle measures and the INCLUDED side.

Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent.

\begin{align*}\angle A \cong \angle X, \ \angle B \cong \angle Y\end{align*}, and \begin{align*}\overline{AB} \cong \overline{XY}\end{align*}, then \begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}.

Example 1: What information do you need to prove that these two triangles are congruent using the ASA Postulate?

a) \begin{align*}\overline{AB} \cong \overline{UT}\end{align*}

b) \begin{align*}\overline{AC} \cong \overline{UV}\end{align*}

c) \begin{align*}\overline{BC} \cong \overline{TV}\end{align*}

d) \begin{align*}\angle B \cong \angle T\end{align*}

Solution: For ASA, we need the side between the two given angles, which is \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{UV}\end{align*}. The answer is b.

Example 2: Write a 2-column proof.

Given: \begin{align*}\angle C \cong \angle E, \ \overline{AC} \cong \overline{AE}\end{align*}

Prove: \begin{align*}\triangle ACF \cong \triangle AEB\end{align*}

Solution:

Statement Reason
1. \begin{align*}\angle C \cong \angle E, \ \overline{AC} \cong \overline{AE}\end{align*} Given
2. \begin{align*}\angle A \cong \angle A\end{align*} Reflexive PoC
3. \begin{align*}\triangle ACF \cong \triangle AEB\end{align*} ASA

## AAS Congruence

A variation on ASA is AAS, which is Angle-Angle-Side. For ASA you need two angles and the side between them. But, if you know two pairs of angles are congruent, the third pair will also be congruent by the \begin{align*}3^{rd}\end{align*} Angle Theorem. This means you can prove two triangles are congruent when you have any two pairs of corresponding angles and a pair of sides.

ASA

AAS

Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side in one triangle are congruent to two angles and a non-included side in another triangle, then the triangles are congruent.

## Proof of AAS Theorem

Given: \begin{align*}\angle A \cong \angle Y, \ \angle B \cong \angle Z, \ \overline{AC} \cong \overline{XY}\end{align*}

Prove: \begin{align*}\triangle ABC \cong \triangle YZX\end{align*}

Statement Reason
1. \begin{align*}\angle A \cong \angle Y, \ \angle B \cong \angle Z, \ \overline{AC} \cong \overline{XY}\end{align*} Given
2. \begin{align*}\angle C \cong \angle X\end{align*} \begin{align*}3^{rd}\end{align*} Angle Theorem
3. \begin{align*}\triangle ABC \cong \triangle YZX\end{align*} ASA

By proving \begin{align*}\triangle ABC \cong \triangle YZX\end{align*} with ASA, we have also proved that the AAS Theorem is true.

Example 3: What information do you need to prove that these two triangles are congruent using:

a) ASA?

b) AAS?

c) SAS?

Solution:

a) For ASA, we need the angles on the other side of \begin{align*}\overline{EF}\end{align*} and \begin{align*}\overline{QR}\end{align*}. \begin{align*}\angle F \cong \angle Q\end{align*}

b) For AAS, we would need the other angle. \begin{align*}\angle G \cong \angle P\end{align*}

c) For SAS, we need the side on the other side of \begin{align*}\angle E\end{align*} and \begin{align*}\angle R\end{align*}. \begin{align*}\overline{EG} \cong \overline{RP}\end{align*}

Example 4: Can you prove that the following triangles are congruent? Why or why not?

Solution: We cannot show the triangles are congruent because \begin{align*}\overline{KL}\end{align*} and \begin{align*}\overline{ST}\end{align*} are not corresponding, even though they are congruent. To determine if \begin{align*}\overline{KL}\end{align*} and \begin{align*}\overline{ST}\end{align*} are corresponding, look at the angles around them, \begin{align*}\angle K\end{align*} and \begin{align*}\angle L\end{align*} and \begin{align*}\angle S\end{align*} and \begin{align*}\angle T\end{align*}. \begin{align*}\angle K\end{align*} has one arc and \begin{align*}\angle L\end{align*} is unmarked. \begin{align*}\angle S\end{align*} has two arcs and \begin{align*}\angle T\end{align*} is unmarked. In order to use AAS, \begin{align*}\angle S\end{align*} needs to be congruent to \begin{align*}\angle K\end{align*}.

Example 5: Write a 2-column proof.

Given: \begin{align*}\overline{BD}\end{align*} is an angle bisector of \begin{align*}\angle CDA, \ \angle C \cong \angle A\end{align*}

Prove: \begin{align*}\triangle CBD \cong \angle ABD\end{align*}

Solution:

Statement Reason
1. \begin{align*}\overline{BD}\end{align*} is an angle bisector of \begin{align*}\angle CDA, \ \angle C \cong \angle A\end{align*} Given
2. \begin{align*}\angle CDB \cong \angle ADB\end{align*} Definition of an Angle Bisector
3. \begin{align*}\overline{DB} \cong \overline{DB}\end{align*} Reflexive PoC
4. \begin{align*}\triangle CBD \cong \triangle ABD\end{align*} AAS

## Hypotenuse-Leg

So far, the congruence postulates we have used will work for any triangle. The last congruence theorem can only be used on right triangles. A right triangle has exactly one right angle. The two sides adjacent to the right angle are called legs and the side opposite the right angle is called the hypotenuse.

You may or may not know the Pythagorean Theorem, which says, for any right triangle, this equation is true:

\begin{align*}(leg)^2 + (leg)^2 = (hypotenuse)^2\end{align*}

What this means is that if you are given two sides of a right triangle, you can always find the third. Therefore, if you have two sides of a right triangle are congruent to two sides of another right triangle; you can conclude that third sides are also congruent.

The Hypotenuse-Leg (HL) Congruence Theorem is a shortcut of this process.

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.

\begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle XYZ\end{align*} are both right triangles and \begin{align*}\overline{AB} \cong \overline{XY}\end{align*} and \begin{align*}\overline{BC} \cong \overline{YZ}\end{align*} then \begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}.

Example 6: What information would you need to prove that these two triangles were congruent using the:

a) HL Theorem?

b) SAS Theorem?

Solution:

a) For HL, you need the hypotenuses to be congruent. \begin{align*}\overline{AC} \cong \overline{MN}\end{align*}.

b) To use SAS, we would need the other legs to be congruent. \begin{align*}\overline{AB} \cong \overline{ML}\end{align*}.

AAA and SSA Relationships There are two other side-angle relationships that we have not discussed: AAA and SSA.

AAA implies that all the angles are congruent.

As you can see, \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle PRQ\end{align*} are not congruent, even though all the angles are.

SSA relationships do not prove congruence either. See \begin{align*}\triangle ABC\end{align*} and \begin{align*}\triangle DEF\end{align*} below.

Because \begin{align*}\angle B\end{align*} and \begin{align*}\angle D\end{align*} are not the included angles between the congruent sides, we cannot prove that these two triangles are congruent.

## Recap

Side-Angle Relationship Picture Determine Congruence?
SSS

Yes

\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}

SAS

Yes

\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}

ASA

Yes

\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}

AAS (or SAA)

Yes

\begin{align*}\triangle ABC \cong \triangle YZX\end{align*}

HL

Yes, Right Triangles Only

\begin{align*}\triangle ABC \cong \triangle XYZ\end{align*}

SSA NO
AAA NO

Example 7: Write a 2-column proof.

Given: \begin{align*}\overline{AB} || \overline{ED}, \ \angle C \cong \angle F, \ \overline{AB} \cong \overline{ED}\end{align*}

Prove: \begin{align*}\overline{AF} \cong \overline{CD}\end{align*}

Solution:

Statement Reason
1. \begin{align*}\overline{AB} || \overline{ED}, \ \angle C \cong \angle F, \ \overline{AB} \cong \overline{ED}\end{align*} Given
2. \begin{align*}\angle ABE \cong \angle DDB\end{align*} Alternate Interior Angles Theorem
3. \begin{align*}\triangle ABF \cong \triangle DEC\end{align*} ASA
4. \begin{align*}\overline{AF} \cong \overline{CD}\end{align*} CPCTC

Prove Move: At the beginning of this chapter we introduced CPCTC. Now, it can be used in a proof once two triangles are proved congruent. It is used to prove the parts of congruent triangles are congruent.

Know What? Revisited Even though we do not know all of the angle measures in the two triangles, we can find the missing angles by using the Third Angle Theorem. In your parents’ kitchen, the missing angle is \begin{align*}39^\circ\end{align*}. The missing angle in your neighbor’s kitchen is \begin{align*}50^\circ\end{align*}. From this, we can conclude that the two kitchens are now congruent, either by ASA or AAS.

## Review Questions

• Questions 1-10 are similar to Examples 1, 3, 4, and 6.
• Questions 11-20 are review and use the definitions and theorems explained in this section.
• Question 21-26 are similar to Examples 1, 3, 4 and 6.
• Questions 27 and 28 are similar to Examples 2 and 5.
• Questions 29-31 are similar to Example 4 and Investigation 4-4.

For questions 1-10, determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used.

For questions 11-15, use the picture to the right and the given information below.

Given: \begin{align*}\overline{DB} \perp \overline{AC}, \ \overline{DB}\end{align*} is the angle bisector of \begin{align*}\angle CDA\end{align*}

1. From \begin{align*}\overline{DB} \perp \overline{AC}\end{align*}, which angles are congruent and why?
2. Because \begin{align*}\overline{DB}\end{align*} is the angle bisector of \begin{align*}\angle CDA\end{align*}, what two angles are congruent?
3. From looking at the picture, what additional piece of information are you given? Is this enough to prove the two triangles are congruent?
4. Write a 2-column proof to prove \begin{align*}\triangle CDB \cong \triangle ADB\end{align*}, using #11-13.
5. What would be your reason for \begin{align*}\angle C \cong \angle A\end{align*}?

For questions 16-20, use the picture to the right and the given information.

Given: \begin{align*}\overline{LP}||\overline{NO}, \ \overline{LP} \cong \overline{NO}\end{align*}

1. From \begin{align*}\overline{LP}||\overline{NO}\end{align*}, which angles are congruent and why?
2. From looking at the picture, what additional piece of information can you conclude?
3. Write a 2-column proof to prove \begin{align*}\triangle LMP \cong \triangle OMN\end{align*}.
4. What would be your reason for \begin{align*}\overline{LM} \cong \overline{MO}\end{align*}?
5. Fill in the blanks for the proof below. Use the given from above. Prove: \begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{PN}\end{align*}.
Statement Reason
1. \begin{align*}\overline{LP} || \overline{NO}, \ \overline{LP} \cong \overline{NO}\end{align*} Given
2. Alternate Interior Angles
3. ASA
4. \begin{align*}\overline{LM} \cong \overline{MO}\end{align*}
5. \begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{PN}\end{align*}.

Determine the additional piece of information needed to show the two triangles are congruent by the given postulate.

1. AAS
2. ASA
3. ASA
4. AAS
5. HL
6. SAS

Fill in the blanks in the proofs below.

1. Given: \begin{align*}\overline{SV} \perp \overline{WU}\end{align*} \begin{align*}T\end{align*} is the midpoint of \begin{align*}\overline{SV}\end{align*} and \begin{align*}\overline{WU}\end{align*} Prove: \begin{align*}\overline{WS} \cong \overline{UV}\end{align*}
Statement Reason
1.
2. \begin{align*}\angle STW\end{align*} and \begin{align*}\angle UTV\end{align*} are right angles
3.
4. \begin{align*}\overline{ST} \cong \overline{TV}, \ \overline{WT} \cong \overline{TU}\end{align*}
5. \begin{align*}\triangle STW \cong \triangle UTV\end{align*}
6. \begin{align*}\overline{WS} \cong \overline{UV}\end{align*}
1. Given: \begin{align*}\angle K \cong \angle T\end{align*}, \begin{align*}\overline{EI}\end{align*} is the angle bisector of \begin{align*}\angle KET\end{align*} Prove: \begin{align*}\overline{EI}\end{align*} is the angle bisector of \begin{align*}\angle KIT\end{align*}
Statement Reason
1.
2. Definition of an angle bisector
3. \begin{align*}\overline{EI} \cong \overline{EI}\end{align*}
4. \begin{align*}\triangle{KEI} \cong \triangle{TEI}\end{align*}
5. \begin{align*}\angle KIE \cong \angle TIE\end{align*}
6. \begin{align*}\overline{EI}\end{align*} is the angle bisector of \begin{align*}\angle KIT\end{align*}

Construction Let’s see if we can construct two different triangles like \begin{align*}\triangle KLM\end{align*} and \begin{align*}\triangle STU\end{align*} from Example 4.

1. Look at \begin{align*}\triangle KLM\end{align*}.
1. If \begin{align*}m\angle K = 70^\circ\end{align*} and \begin{align*}m\angle M = 60^\circ\end{align*}, what is \begin{align*}m\angle L\end{align*}?
2. If \begin{align*}KL = 2 \ in\end{align*}, construct \begin{align*}\triangle KLM\end{align*} using \begin{align*}\angle L, \ \angle K, \ \overline{KL}\end{align*} and Investigation 4-4 (ASA Triangle construction).
2. Look at \begin{align*}\triangle STU\end{align*}.
1. If \begin{align*}m\angle S = 60^\circ\end{align*} and \begin{align*}m\angle U = 70^\circ\end{align*}, what is \begin{align*}m\angle T\end{align*}?
2. If \begin{align*}ST = 2 \ in\end{align*}, construct \begin{align*}\triangle STU\end{align*} using \begin{align*}\angle S, \ \angle T, \ \overline{ST}\end{align*} and Investigation 4-4 (ASA Triangle construction).
3. Are the two triangles congruent?

1. \begin{align*}\overline{AD} \cong \overline{DC}, \overline{AB} \cong \overline{BC}\end{align*}
2. Yes, by the Reflexive Property
3. \begin{align*}\triangle DAB \cong \triangle DCB\end{align*} by SSS
1. \begin{align*}\angle L \cong \angle N\end{align*} and \begin{align*}\angle M \cong \angle P\end{align*} by the Alternate Interior Angles Theorem
2. \begin{align*}\angle PON \cong \angle LOM\end{align*} by Vertical Angles or the \begin{align*}3^{rd}\end{align*} Angle Theorem
3. No, no markings or midpoints
4. No, no congruent sides.
1. Yes, CPCTC
2. No, these sides do not line up in the congruence statement.

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