Question

Triangles A C B and M Q R are shown. Sides A B and M R are congruent. Angles C A B and M R Q are 42 degrees. Angle C B A is 53 degrees. Angle M Q R is 85 degrees.
Are the triangles congruent? Why or why not?

Yes, all the angles of each of the triangles are acute.
Yes, they are congruent by either ASA or AAS.
No, AngleB is not congruent to AngleQ.
No, the congruent sides do not correspond.

Answer 1

To determine if triangles ACB and MQR are congruent, we can use the information given about their sides and angles.

1. **Congruent Sides**: We know that sides AB and MR are congruent.

2. **Angles**:
- Angle CAB = 42 degrees (for triangle ACB)
- Angle MRQ = 42 degrees (for triangle MQR)
- Angle CBA = 53 degrees (for triangle ACB)
- Angle MQR = 85 degrees (for triangle MQR)

Now we can find the third angle for triangle ACB and triangle MQR:

- For triangle ACB:
- The sum of angles in a triangle is 180 degrees.
- Angle ACB = 180 - (Angle CAB + Angle CBA)
- Angle ACB = 180 - (42 + 53) = 180 - 95 = 85 degrees

- For triangle MQR:
- Angle QRM can be found likewise.
- Angle QRM = 180 - (Angle MRQ + Angle MQR)
- Angle QRM = 180 - (42 + 85) = 180 - 127 = 53 degrees

Now we have:
- Triangle ACB:
- Angle CAB = 42 degrees
- Angle CBA = 53 degrees
- Angle ACB = 85 degrees

- Triangle MQR:
- Angle MRQ = 42 degrees
- Angle QRM = 53 degrees
- Angle MQR = 85 degrees

We observe that:
- Angle CAB ≅ Angle MRQ (42 degrees)
- Angle CBA ≅ Angle QRM (53 degrees)
- Angle ACB ≅ Angle MQR (85 degrees)

This gives two angles in each triangle and the included side (AB ≅ MR), which makes the triangles congruent by the ASA (Angle-Side-Angle) criteria.

So the correct conclusion is:
**Yes, they are congruent by either ASA or AAS.**