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ASA and AAS Triangle Congruence

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ASA and AAS Triangle Congruence

What if you were given two triangles and provided with only the measure of two of their angles and one of their side lengths? How could you determine if the two triangles were congruent? After completing this Concept, you'll be able to use the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) shortcuts to prove triangle congruency.

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CK-12 ASA and AAS Triangle Congruence

Watch the portions of the following two videos that deal with ASA and AAS triangle congruence.

James Sousa: Introduction to Congruent Triangles

James Sousa: Determining If Two Triangles are Congruent

Finally, watch this video.

James Sousa: Example 2: Prove Two Triangles are Congruent

Guidance

If two angles and one side in one triangle are congruent to the corresponding two angles and one side in another triangle, then the two triangles are congruent. This idea encompasses two triangle congruence shortcuts: Angle-Side-Angle and Angle-Angle-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.

Angle-Angle-Side (AAS) Congruence Theorem: 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.

The placement of the word Side is important because it indicates where the side that you are given is in relation to the angles. The pictures below help to show the difference between the two shortcuts.

ASA

AAS

Example A

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

a) \overline{AB} \cong \overline{UT}

b) \overline{AC} \cong \overline{UV}

c) \overline{BC} \cong \overline{TV}

d) \angle B \cong \angle T

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

Example B

Write a 2-column proof.

Given : \angle C \cong \angle E, \ \overline{AC} \cong \overline{AE}

Prove : \triangle ACF \cong \triangle AEB

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

Example C

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

a) ASA?

b) AAS?

Solution:

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

b) For AAS, we would need the other angle. \angle G \cong \angle P

CK-12 ASA and AAS Triangle Congruence

Guided Practice

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

2. Write a 2-column proof.

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

Prove : \triangle CBD \cong \angle ABD

3. Write a 2-column proof.

Given : \overline{AB} || \overline{ED}, \ \angle C \cong \angle F, \ \overline{AB} \cong \overline{ED}

Prove : \overline{AF} \cong \overline{CD}

Answers:

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

2.

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

3.

Statement Reason
1. \overline{AB} || \overline{ED}, \ \angle C \cong \angle F, \ \overline{AB} \cong \overline{ED} 1. Given
2. \angle ABE \cong \angle DEB 2. Alternate Interior Angles Theorem
3. \triangle ABF \cong \triangle DEC 3. ASA
4. \overline{AF} \cong \overline{CD} 4. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Practice

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 : \overline{DB} \perp \overline{AC}, \ \overline{DB} is the angle bisector of \angle CDA

  1. From \overline{DB} \perp \overline{AC} , which angles are congruent and why?
  2. Because \overline{DB} is the angle bisector of \angle CDA , 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 \triangle CDB \cong \triangle ADB , using #4-6.
  5. What would be your reason for \angle C \cong \angle A ?

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

Given : \overline{LP}||\overline{NO}, \ \overline{LP} \cong \overline{NO}

  1. From \overline{LP}||\overline{NO} , 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 \triangle LMP \cong \triangle OMN .
  4. What would be your reason for \overline{LM} \cong \overline{MO} ?
  5. Fill in the blanks for the proof below. Use the given from above. Prove : M is the midpoint of \overline{PN} .
Statement Reason
1. \overline{LP} || \overline{NO}, \ \overline{LP} \cong \overline{NO} 1. Given
2. 2. Alternate Interior Angles
3. 3. ASA
4. \overline{LM} \cong \overline{MO} 4.
5. M is the midpoint of \overline{PN} . 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

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