Question

which descriptions from the list below accurately describe the relationship between △ qrs and △ tuv? check all that apply. which descriptions from the list below accurately describe the relationship between △ qrs and △ tuv? check all that apply. two triangles with angles and side lengths: △ qrs has angles 50°, 65°, and side lengths 22, 23, 30; △ tuv has angles 50°, 65°, and side lengths 22, 23, 30. options: same size, congruent, similar, same shape

Explanation:

Step1: Analyze triangle angles

For $\triangle QRS$ and $\triangle TUV$, check angle measures. In $\triangle QRS$: angles are $63^\circ$, $50^\circ$, and $180 - 63 - 50 = 67^\circ$? Wait, no, looking at the triangles, both have angles $63^\circ$, $50^\circ$, and $47^\circ$ (since $180 - 63 - 50 = 67$? Wait, no, the given angles: one triangle has $63^\circ$, $50^\circ$, and the third angle is $180 - 63 - 50 = 67$? Wait, no, the other angle is $47^\circ$? Wait, maybe I misread. Wait, the triangles have sides and angles: let's check angle - angle - angle (AAA) similarity. Both triangles have the same angle measures (since the angles are $63^\circ$, $50^\circ$, and $47^\circ$? Wait, $63 + 50 + 47 = 160$? No, that's wrong. Wait, $63 + 50 = 113$, $180 - 113 = 67$. Wait, maybe the angles are $63^\circ$, $50^\circ$, and $67^\circ$? But the other triangle also has the same angles? Wait, no, the problem shows two triangles with angles $63^\circ$, $50^\circ$, and the third angle should be equal. Also, check side lengths: if the sides are proportional? Wait, the sides of $\triangle QRS$: $QR = 22$, $RS = 23$, $QS = 30$? Wait, no, the other triangle $\triangle TUV$: $TU = 22$, $UV = 23$, $TV = 30$? So the sides are equal. So first, check congruence: if all corresponding sides are equal and all corresponding angles are equal, then they are congruent. Congruent triangles are also similar (since similarity requires AA/SSS/SAS, and congruent triangles satisfy SSS for similarity). Also, same size (since congruent means same size) and same shape (since similar means same shape, and congruent implies similar).

Step2: Evaluate each option

• Same size: Congruent triangles have the same size, so this applies.
• Congruent: Since all corresponding sides and angles are equal (sides: 22, 23, 30; angles: $63^\circ$, $50^\circ$, and the third angle equal), so they are congruent.
• Similar: Congruent triangles are a special case of similar triangles (SSS similarity, since all sides are proportional with ratio 1), so this applies.
• Same shape: Similar triangles have the same shape, and congruent triangles are similar, so this applies.

Answer:

Same size, Congruent, Similar, Same shape (all four options apply as the triangles are congruent, hence satisfy all these relationships)