ASA and AAS Congruence

Show your work for each problem.

1.
In the ASA congruence theorem, what does the 'S' stand for?
1. a)Sum
2. b)Side (between the angles)
3. c)Straight
4. d)Similar
Answer: Side (between the angles)
ASA stands for Angle-Side-Angle. The 'S' represents a Side that is included between (located between) the two angles.
2.
What is the key difference between ASA and AAS?
1. a)Position of the known side relative to the angles
2. b)Size of the angles
3. c)Type of triangle
4. d)Number of angles known
Answer: Position of the known side relative to the angles
In ASA, the known side is between the two known angles (included side). In AAS, the known side is not between the two angles (non-included side).
3.
Why can't AAA (three angles) prove triangle congruence?
1. a)Angles don't determine shape
2. b)Three angles are not enough information
3. c)It only works for right triangles
4. d)Triangles can have equal angles but different sizes
Answer: Triangles can have equal angles but different sizes
AAA only proves similarity, not congruence. Similar triangles have the same angles but can be different sizes (like a small and large version of the same shape). You need at least one side to prove congruence.
4.
In triangle , and . What is ?
Answer: 65
. This is why knowing two angles automatically gives us the third angle!
5.
Triangle has and . What is ?
Answer: 45
. The angles in any triangle always sum to 180 degrees.
6.
Triangle has , , cm. Triangle has , , cm. Which theorem proves congruence?
1. a)ASA
2. b)SAS
3. c)AAS
4. d)SSS
Answer: ASA
Side is between angles and (it's the included side). Similarly, is between angles and . This is ASA: two angles and the included side.
7.
Triangle has , , cm. Triangle has , , cm. Which theorem proves congruence?
1. a)ASA
2. b)SSS
3. c)AAS
4. d)SAS
Answer: AAS
Side is NOT between angles and - it's opposite to angle . We know angles and , and a non-included side . This is AAS.
8.
If by ASA with , , what is in degrees?
Answer: 65
First, . Since , corresponding angles are equal: .
9.
Determine if the triangles are congruent: with , , cm and with , , cm.
Answer: ASA
1. Are the two given angles in each triangle equal? yes
2. Side AC connects which two angles? A and C
3. Is the side between the two known angles (included)? yes
4. Which congruence theorem applies? ASA
10.
Determine the theorem: with , , cm and with , , cm.
Answer: AAS
1. Do the angles M and P match? And N and Q? yes
2. Side NO connects vertices N and O. Is angle M at vertex N or O? neither
3. So is side NO between the known angles M and N? no
4. Two angles and a non-included side means which theorem? AAS
11.
Triangle has , . Triangle has , . If , can you prove the triangles congruent?
1. a)Yes, by SAS
2. b)No, the angles don't match
3. c)Yes, by ASA
4. d)Yes, by AAS
Answer: Yes, by ASA
In : . In : . Both triangles have angles 55°, 65°, 60°. Side is between angles and . Side is between angles and . Since and the adjacent angles match (55° and 65° on both sides), the triangles are congruent by ASA.
12.
Two triangles are congruent by AAS. Triangle 1 has angles and with the side opposite the angle measuring cm. What is the length of the corresponding side in Triangle 2?
Answer: 9
If triangles are congruent, ALL corresponding parts are equal. The side opposite the angle in Triangle 1 corresponds to the side opposite the angle in Triangle 2. Since they're congruent, both sides are cm.
13.
Given: and where , , and . Prove the triangles are congruent and identify the theorem used.
Answer: AAS
1. How many pairs of congruent angles do we have? 2
2. Side BC connects which two vertices of triangle ABC? B and C
3. Is angle A at vertex B or vertex C? neither
4. So is BC between angles A and B, or is it a non-included side? non-included
5. What theorem proves this congruence? AAS
14.
In , , , cm. In , , , cm. Find all side lengths of if cm and cm.
Answer: XY = 6, YZ = 9, XZ = 7
1. Which congruence theorem applies here? ASA
2. Which vertex of XYZ corresponds to A? X
3. Which side of XYZ corresponds to BC? YZ
4. What is YZ? 9
5. Which side of XYZ corresponds to AC? XZ
6. What is XZ? 7
15.
Why is AAS actually just a special case of ASA?
1. a)Both only need two measurements
2. b)They use the same number of pieces of information
3. c)Knowing two angles determines the third, so you actually know the included angle
4. d)They are completely different theorems
Answer: Knowing two angles determines the third, so you actually know the included angle
Since the sum of angles in a triangle is always , if you know any two angles, you automatically know the third. In AAS, once you know the two given angles, you also know the third angle. This means you effectively know all three angles plus a side. The side that was "non-included" for the two given angles is actually "included" between one of those angles and the third angle you calculated. So AAS reduces to ASA!