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Question

Triangle upper A upper B upper C is drawn next to a smaller triangle, triangle upper M upper N upper P. The interior angles by vertices upper A and upper P are marked by single arcs and are given as 56 and 62 degrees, respectively.

Given two isosceles triangles, △ABC and △MNP, is △ABC similar to △MNP if ∠A=56° and ∠P=62°? If so, which criterion helps you prove this conclusion?

(1 point)
Responses

This does not matter because ∠A and ∠P are not corresponding angles.
This does not matter because angle upper A  and angle upper P  are not corresponding angles.

Yes, △ABC is similar to △MNP by the AA Criterion.
Yes, triangle upper A upper B upper C  is similar to  triangle upper M upper N upper P by the AA Criterion.

Yes, △ABC is similar to △MNP by the SSS Criterion.
Yes, triangle upper A upper B upper C  is similar to triangle upper M upper N upper P  by the SSS Criterion.

No, △ABC is not similar to △MNP because ∠A=56° is not equal to ∠P=62°.
No,  triangle upper A upper B upper C is not similar to triangle upper M upper N upper P  because angle upper A equals 56 degrees  is not equal to angle upper P equals 62 degrees .
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GPT-4o mini · Answer

Yes, triangle ABC is similar to triangle MNP by the AA Criterion.

Since the triangles are isosceles and you know one angle in each triangle (∠A = 56° and ∠P = 62°), you can find the third angle of each triangle using the fact that the sum of all angles in a triangle is 180°.

For triangle ABC:

If ∠A = 56° and the other two angles are equal (let's call them x), then: 56° + x + x = 180° 2x = 180° - 56° 2x = 124° x = 62°

So the angles in triangle ABC are 56°, 62°, and 62°.

For triangle MNP:

If ∠P = 62° and the other two angles (let's call them y) are equal, then: 62° + y + y = 180° 2y = 180° - 62° 2y = 118° y = 59°

So the angles in triangle MNP are 62°, 59°, and 59°.

Now we can see that:

Triangle ABC has angles 56°, 62°, and 62°.
Triangle MNP has angles 62°, 59°, and 59°.

Since they have two pairs of corresponding angles that are equal (the 62° angles), by the AA (Angle-Angle) criterion, triangle ABC is similar to triangle MNP.