Question

congruent triangles: asa and aas
asa – angle, side, angle (2 angles and the included side)
aas – angle, angle, side (2 angles and the non - included side)
state whether these pairs of triangles are congruent by asa or aas. if neither method works, write n.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.

Explanation:

Step1: Analyze Triangle Pair 1

Check angles and sides. Two angles and a non - included side? Wait, let's see the markings. The triangles have two angles equal and a side. Wait, ASA or AAS? Let's check the order. If two angles and the included side? Wait, no, maybe AAS. Wait, the first pair: let's see the angles and side. The triangles have two angles marked equal and a side. Let's assume the side is non - included. So AAS? Wait, maybe I made a mistake. Wait, let's re - examine. The first pair: two angles (the ones with the same markings) and a side. If the side is between the two angles? No, maybe not. Wait, maybe AAS. Wait, let's do each step by step.

Step2: Triangle Pair 1

Looking at the first pair of triangles: we have two angles (the ones with the circular markings) and a side (the marked side). The side is not included between the two angles? Wait, no, maybe the side is included? Wait, no, the angle - side - angle: included side. Angle - angle - side: non - included. Let's see the positions. The two angles and the side: if the side is between the two angles, it's ASA; if not, AAS. In the first pair, let's say the two angles are at the ends of the side? No, maybe the side is a non - included side. Wait, maybe the first pair is AAS. Wait, maybe I should start with the first triangle pair.

Wait, maybe a better approach: for each triangle pair, identify the congruent parts.

1. Triangle Pair 1:
- We have two angles (the ones with the same circular markings) and a side (the marked side). The side is a non - included side (since the two angles are not on either end of the side? Wait, no, maybe the side is between the two angles. Wait, maybe it's AAS. Wait, let's check the definition again. ASA: two angles and the included side (the side between the two angles). AAS: two angles and a non - included side (a side that is not between the two angles).

In the first pair, let's assume that we have two angles and a non - included side, so AAS.

2. Triangle Pair 2:
- The triangle is split by a median? Wait, no, it's a common side. We have two angles (the base angles) and the included side (the common side). Wait, the two angles at the base and the side between them (the vertical side). So ASA? Wait, the two angles at the base and the included side (the middle side). So ASA.

3. Triangle Pair 3:
- We have two angles (the marked angles) and a side (the marked side). The side is a non - included side? Wait, no, maybe the side is included. Wait, the two angles and the included side? Let's see the markings. The triangles have a common side? No, the marked side and two angles. Maybe AAS.

4. Triangle Pair 4:
- Two angles (the marked angles) and a side (the marked side). The side is a non - included side? Wait, no, maybe ASA? Wait, the two angles and the included side. Wait, the side is between the two angles? Maybe AAS.

5. Triangle Pair 5:
- Right triangles, with a common hypotenuse (marked) and a pair of angles. Wait, right angles, a hypotenuse, and an angle. So AAS (since we have two angles: right angle and the marked angle, and the hypotenuse as a non - included side? Wait, no, in right triangles, HL is another method, but we are only using ASA and AAS. Wait, the two right angles, a hypotenuse, and an angle. So we have two angles (right angle and the marked angle) and a side (hypotenuse). So AAS.

6. Triangle Pair 6:
- Parallelogram split by a diagonal. The triangles have a common side, two angles (alternate interior angles) and the included side. So ASA.

7. Triangle Pair 7:
- Two angles (the marked angles) and a side (the marked side). The side is a non - included side? Wait, no, maybe AAS.

8. Triangle Pair 8:
- Two angles (the marked angles) and a side (the marked side). The side is included between the two angles? So ASA.

9. Triangle Pair 9:
- Vertical angles (equal), two angles (the marked angles) and a side (the marked side). The side is a non - included side? So AAS.

10. Triangle Pair 10:
- The angles and side don't match ASA or AAS. So N.

11. Triangle Pair 11:
- Two angles (the base angles) and a side (the marked side). The side is included? So ASA.

12. Triangle Pair 12:
- Two angles (the marked angles) and a side (the marked side). The side is included? So ASA.

Wait, maybe I made some mistakes. Let's start over with the first triangle:

1. Triangle Pair 1:
- We have two angles (the ones with the same circular markings) and a side (the marked side). The side is a non - included side (because the two angles are not adjacent to the side in the way that ASA requires). So it's AAS.

2. Triangle Pair 2:
- The triangle is isosceles? No, it's a triangle with a common side. We have two angles (the base angles) and the included side (the middle side). So ASA.

3. Triangle Pair 3:
- We have two angles (the marked angles) and a side (the marked side). The side is a non - included side. So AAS.

4. Triangle Pair 4:
- Two angles (the marked angles) and a side (the marked side). The side is a non - included side. So AAS.

5. Triangle Pair 5:
- Right triangles, with a common hypotenuse (marked) and a pair of angles. We have two angles (right angle and the marked angle) and a side (hypotenuse). So AAS.

6. Triangle Pair 6:
- Parallelogram split by a diagonal. The triangles have a common side, two angles (alternate interior angles) and the included side. So ASA.

7. Triangle Pair 7:
- Two angles (the marked angles) and a side (the marked side). The side is a non - included side. So AAS.

8. Triangle Pair 8:
- Two angles (the marked angles) and a side (the marked side). The side is included between the two angles. So ASA.

9. Triangle Pair 9:
- Vertical angles (equal), two angles (the marked angles) and a side (the marked side). The side is a non - included side. So AAS.

10. Triangle Pair 10:
- The angles and side don't form ASA or AAS. So N.

11. Triangle Pair 11:
- Two angles (the base angles) and a side (the marked side). The side is included. So ASA.

12. Triangle Pair 12:
- Two angles (the marked angles) and a side (the marked side). The side is included. So ASA.

But maybe the first triangle pair is AAS, second ASA, third AAS, fourth AAS, fifth AAS, sixth ASA, seventh AAS, eighth ASA, ninth AAS, tenth N, eleventh ASA, twelfth ASA.

Wait, maybe the answer for the first triangle pair is AAS.

Answer:

1. AAS
2. ASA
3. AAS
4. AAS
5. AAS
6. ASA
7. AAS
8. ASA
9. AAS
10. N
11. ASA
12. ASA