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ASA and AAS Triangle Congruence

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ASA and AAS Triangle Congruence
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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 71^\circ and the angle at the fridge is 50^\circ . You used your protractor to measure the angle at the stove and sink at your neighbor’s house. Are the kitchen triangles congruent now? After completing this Concept, you'll be able to use a congruence shortcut to help you answer this question.

Watch This

CK-12 Foundation: Chapter4ASAandAASTriangleCongruenceA

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

James Sousa: Example 2: Prove Two Triangles are Congruent

Guidance

Consider the question: If I have two angles that are 45^\circ and 60^\circ 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 45^\circ angle. Mark this measurement and draw a ray from the left endpoint through the 45^\circ mark.
  3. At the right endpoint of your line segment, use the protractor to measure the 60^\circ angle. Mark this measurement and draw a ray from the left endpoint through the 60^\circ 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 \triangle ABC \cong \triangle XYZ .

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 : \angle A \cong \angle Y, \angle B \cong \angle Z, \overline{AC} \cong \overline{XY}

Prove : \triangle ABC \cong \triangle YZX

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

Example A

What information would 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 two-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} Given
2. \angle A \cong \angle A Reflexive PoC
3. \triangle ACF \cong \triangle AEB ASA

Example C

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

a) ASA?

b) AAS?

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

b) For AAS, we would need the angle on the other side of \angle E and \angle R . \angle G \cong \angle P .

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter4ASAandAASTriangleCongruenceB

Concept 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 39^\circ . The missing angle in your neighbor’s kitchen is 50^\circ . From this, we can conclude that the two kitchens are now congruent, either by ASA or AAS.

Vocabulary

Two figures are congruent if they have exactly the same size and shape. By definition, two triangles are congruent if the three corresponding angles and sides are congruent. The symbol \cong means congruent. There are shortcuts for proving that triangles are congruent. The ASA Triangle Congruence Postulate states that 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 AAS Triangle Congruence Theorem states that 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. CPCTC refers to Corresponding Parts of Congruent Triangles are Congruent . It is used to show two sides or two angles in triangles are congruent after having proved that the triangles are congruent.

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 two-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. Even though \overline{KL} \cong \overline{ST} , they are not corresponding. Look at the angles around \overline{KL}, \angle K and \angle L . \angle K has one arc and \angle L is unmarked. The angles around \overline{ST} are \angle S and \angle T . \angle S has two arcs and \angle T is unmarked. In order to use AAS, \angle S needs to be congruent to \angle K . They are not congruent because the arcs marks are different. Therefore, we cannot conclude that these two triangles are congruent.

2. Here is the proof:

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

3. 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. \overline{AB} \ || \ \overline{ED}, \angle C \cong \angle F, \overline{AB} \cong \overline{ED} Given
2. \angle ABE \cong \angle DEB Alternate Interior Angles Theorem
3. \triangle ABF \cong \triangle DEC ASA
4. \overline{AF} \cong \overline{CD} CPCTC

Explore More

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 two-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 two-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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