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

The answer would be the 2nd one because of the triangle sum property:

Given:
AB ≅ MR
∠CAB = 42°
∠MRQ = 42°
∠CBA = 53°
∠MQR = 85°

In a Triangle sum of the measures of all the angles of a triangle is 180°:

∠MRQ+∠MQR+∠RMQ=180°
42+85+∠RMQ=180
127+∠RMQ=180
∠RMQ=53°

This means that due to the Transitive Property..


∠CBA ≅ ∠RMQ = 53°

Now, in ΔACB and Δ RQM:

∠CAB ≅∠MRQ = 42° (given)

AB ≅ MR (given)

∠CBA ≅ ∠RMQ = 53° (from last problem)

ΔACB ≅ ΔRQM (due to ASA/AAS congruence)



Therefore, the Triangles ACB and RQM are Congruent by either AAS congruence property or ASA congruence property.

Answer 2

Therefore, triangles ACB and RQM*. Not "the Triangles" my bad ;-;