Question

1. determine if each pair of triangles is similar by aa~ or by showing all 3 corresponding sides are proportional.

a.

b.

c.

d.

Explanation:

Step1: Analyze Triangle Pair A

In triangle pair A, we check the angles. The first triangle has angles \(36^\circ\), \(47^\circ\), so the third angle is \(180 - 36 - 47 = 97^\circ\). The second triangle has angles \(36^\circ\), \(47^\circ\), so the third angle is also \(180 - 36 - 47 = 97^\circ\). By AA (Angle - Angle) similarity criterion, if two angles of one triangle are equal to two angles of another triangle, the triangles are similar. So triangle pair A is similar by AA.

Step2: Analyze Triangle Pair B

For triangle pair B, we check the ratios of corresponding sides. Let's identify the sides: first triangle sides \(9\), \(24\), \(32\); second triangle sides \(4.5\), \(12\), \(16\). Calculate the ratios: \(\frac{9}{4.5}=2\), \(\frac{24}{12} = 2\), \(\frac{32}{16}=2\). All three corresponding sides are proportional (ratio \(2\) for each pair), so triangle pair B is similar by SSS (Side - Side - Side) similarity (since all corresponding sides are proportional).

Step3: Analyze Triangle Pair C

For triangle pair C, the sides of the first triangle are \(10\), \(12\), \(15\); the second triangle are \(7\), \(9\), \(12\). Calculate the ratios: \(\frac{10}{7}\approx1.428\), \(\frac{12}{9}=\frac{4}{3}\approx1.333\), \(\frac{15}{12}=\frac{5}{4} = 1.25\). The ratios are not equal, so triangle pair C is not similar.

Step4: Analyze Triangle Pair D

In triangle pair D, there is a line parallel to the base of the triangle (by the arrow marks). By the Basic Proportionality Theorem (Thales' theorem), if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. Also, the corresponding angles are equal (alternate interior angles). So by AA similarity (since the corresponding angles are equal due to the parallel line), the two triangles (the smaller one and the larger one) are similar.

Answer:

• A: Similar by AA.
• B: Similar by SSS (all sides proportional, ratio \(2\)).
• C: Not similar (side ratios not equal).
• D: Similar by AA (due to parallel line, equal corresponding angles).