Question

is △ pqr ~ △ xyz? if so, name which similarity postulate or theorem applies.
a. similar - aa
b. similar - sss
c. similar - sas
d. cannot be determined

Explanation:

Step1: Analyze Angle and Side Ratios

First, check the given angles: ∠Q = 95° and ∠Y = 95°, so one pair of angles is equal. Now check the sides: PQ and XY (or related sides) – the side lengths are 22 and 11, so the ratio is $\frac{22}{11} = 2$. But for SAS similarity, we need two sides in proportion and the included angle equal. Wait, but we only know one angle (95°) and one pair of sides in ratio. Wait, no, maybe I misread. Wait, the triangles: let's assume the sides adjacent to the 95° angle. If ∠Q and ∠Y are equal, and the sides around them are in proportion (22/11 = 2), but do we have another angle? Wait, no, maybe the problem is that we have one angle equal and one pair of sides in ratio, but for SAS, the angle must be included. Wait, maybe the triangles have two sides in proportion and the included angle equal. Wait, the side PQ is 22, XY is 11, and the angle between them (∠Q and ∠Y) is 95°, so if the other sides (like QR and YZ) are also in ratio 2, but we don't know. Wait, no, maybe the problem is that we have one angle equal and the sides adjacent to it in ratio. Wait, actually, the SAS similarity postulate states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. Here, ∠Q = ∠Y = 95°, and the sides PQ/XY = 22/11 = 2, and if the other sides (let's say QR/YZ) are also in ratio 2, but we only see one side. Wait, maybe the diagram shows that the sides adjacent to the 95° angle are in ratio 2, so SAS applies. Wait, no, maybe I made a mistake. Wait, the options: A is AA, B is SSS, C is SAS, D is cannot be determined. Wait, we have one angle equal (95°), and one pair of sides in ratio 2. But for SAS, we need two sides and included angle. Wait, maybe the triangles have two sides in proportion (22/11 = 2) and the included angle (95°) equal, so SAS similarity. So the answer should be C. Similar - SAS.

Step2: Confirm the Similarity Criterion

Check the SAS similarity postulate: two sides in proportion, included angle equal. Here, ∠Q = ∠Y = 95° (included angle), and the sides PQ/XY = 22/11 = 2. Assuming the other pair of sides (e.g., QR/YZ) is also in ratio 2 (from the diagram, maybe), so SAS applies. Thus, the triangles are similar by SAS.

Answer:

C. Similar - SAS