If two angles of one triangle are congruent to two angles of another triangle, what can be concluded about the third angles?

If two angles of one triangle are congruent to two angles of another triangle, what can be concluded about the third angles?

• A. The triangles are congruent
• B. The sum of the interior angles is 180 degrees
• C. The third angles are congruent
• D. The two triangles have equal perimeters

Answer: (C)

Two triangles sharing two equal angles force the third angles to be equal as well. The sum of the interior angles in any triangle is 180 degrees. If two angles in one triangle match two angles in the other, then the remaining angles must satisfy the same 180-degree total, so the third angles are equal: C = 180 − (A + B) = 180 − (A' + B') = C'. This not only shows the third angles are congruent but also establishes similarity by the AA criterion—the triangles have two equal angles, so their shapes match up proportionally even if the sides aren’t all equal. It doesn’t automatically mean the triangles are congruent, since equal angles with potentially different side lengths still yield similar, not necessarily congruent, triangles. And equal perimeters aren’t guaranteed either. So the conclusion about the third angles is that they are congruent.

Two triangles sharing two equal angles force the third angles to be equal as well. The sum of the interior angles in any triangle is 180 degrees. If two angles in one triangle match two angles in the other, then the remaining angles must satisfy the same 180-degree total, so the third angles are equal: C = 180 − (A + B) = 180 − (A' + B') = C'. This not only shows the third angles are congruent but also establishes similarity by the AA criterion—the triangles have two equal angles, so their shapes match up proportionally even if the sides aren’t all equal. It doesn’t automatically mean the triangles are congruent, since equal angles with potentially different side lengths still yield similar, not necessarily congruent, triangles. And equal perimeters aren’t guaranteed either. So the conclusion about the third angles is that they are congruent.