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ASA and AAS Triangle Congruence

Two sets of corresponding angles and any corresponding set of sides prove congruent triangles.

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ASA and AAS Triangle Congruence

What if your parents changed their minds at the last second about their kitchen layout? Now, they have decided they to have the distance between the sink and the fridge be 3 ft, the angle at the sink and the angle at the fridge is . You used your protractor to measure the angle at the stove and sink at your neighbor’s house. Are the kitchen triangles congruent now? 

ASA and AAS Triangle Congruence 

Consider the question: If I have two angles that are and and the side between them is 5 in, can I construct only one triangle? We will investigate it here.

Investigation: Constructing a Triangle Given Two Angles and Included Side

Tools Needed: protractor, pencil, ruler, and paper

  1. Draw the side (5 in) horizontally, 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 angle. Mark this measurement and draw a ray from the left endpoint through the mark.
  3. At the right endpoint of your line segment, use the protractor to measure the angle. Mark this measurement and draw a ray from the left endpoint through the 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? The answer is NO. Only one triangle can be created from any given two angle measures and the INCLUDED side.

Angle-Side-Angle (ASA) Triangle 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.

The markings in the picture are enough to say .

A variation on ASA is AAS, which is Angle-Angle-Side. Recall that for ASA you need two angles and the side between them. But, if you know two pairs of angles are congruent, then the third pair will also be congruent by the Third Angle Theorem. Therefore, you can prove a triangle is congruent whenever you have any two angles and a side.

Be careful to note the placement of the side for ASA and AAS. As shown in the pictures above, the side is between the two angles for ASA and it is not for AAS.

 

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

Proof of AAS Theorem:

Given:

Prove:

Statement Reason
1. Given
2. Third Angle Theorem
3. ASA

 

Using the ASA Postulate 

What information would you need to prove that these two triangles are congruent using the ASA Postulate?

a)

b)

c)

d)

For ASA, we need the side between the two given angles, which is and . The answer is b.

Writing a Two-Column Proof 

Write a two-column proof.

Given:

Prove:

Statement Reason
1. Given
2. Reflexive PoC
3. ASA

Information Necessary to Prove Congruency 

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

 

a) ASA?

For ASA, we need the angles on the other side of and . Therefore, we would need .

b) AAS?

For AAS, we would need the angle on the other side of and . .

 

Kitchen Problem 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 . The missing angle in your neighbor’s kitchen is . From this, we can conclude that the two kitchens are now congruent, either by ASA or AAS.

Examples

Example 1

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

Even though , they are not corresponding. Look at the angles around and . has one arc and is unmarked. The angles around are and . has two arcs and is unmarked. In order to use AAS, needs to be congruent to . They are not congruent because the arcs marks are different. Therefore, we cannot conclude that these two triangles are congruent.

Example 2

Write a 2-column proof.

Given: is an angle bisector of

Prove:

Here is the proof:

Statement Reason
1. is an angle bisector of Given
2. Definition of an Angle Bisector
3. Reflexive PoC
3. AAS

Example 3

Write a two-column proof.

Given:

Prove:

First, prove that the triangles are congruent. Once you have proved they are congruent, you need one more step to show that the corresponding pair of sides must be congruent. Remember that CPCTC stands for corresponding parts of congruent triangles are congruent.

Statement Reason
1. Given
2. Alternate Interior Angles Theorem
3. ASA
4. CPCTC

Review

For questions 1-3, determine if the triangles are congruent. If they are, write the congruence statement and which congruence postulate or theorem you used.

For questions 4-8, use the picture and the given information below.

Given: is the angle bisector of

  1. From , which angles are congruent and why?
  2. Because is the angle bisector of , 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 two-column proof to prove , using #4-6.
  5. What would be your reason for ?

For questions 9-13, use the picture and the given information.

Given:

  1. From , which angles are congruent and why?
  2. From looking at the picture, what additional piece of information can you conclude?
  3. Write a two-column proof to prove .
  4. What would be your reason for ?
  5. Fill in the blanks for the proof below. Use the given from above. Prove: is the midpoint of .
Statement Reason
1. 1. Given
2. 2. Alternate Interior Angles
3. 3. ASA
4. 4.
5. is the midpoint of . 5.

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

Review (Answers)

To view the Review answers, open this PDF file and look for section 4.8. 

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Vocabulary

AAS (Angle-Angle-Side)

If two angles and a non-included side in one triangle are congruent to two angles and the corresponding non-included side in another triangle, then the triangles are congruent.

Angle Side Angle Triangle

The term 'angle-side-angle triangle' refers to a triangle with known measures of two angles and the length of the side between them.

ASA

ASA, angle-side-angle, refers to two known angles in a triangle with one known side between the known angles.

Congruent

Congruent figures are identical in size, shape and measure.

Triangle Congruence

Triangle congruence occurs if 3 sides in one triangle are congruent to 3 sides in another triangle.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

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