Question

1. in triangle abc, ab = 4 in, bc = 7 in, and m∠b = 40°. in triangle xyz, xy = 10 in, yz = 17.5 in, and m∠y = 40°. are the two triangles similar? if so, state the theorem or postulate that proves similarity. (draw a picture)

Explanation:

Step1: Check the ratios of corresponding sides

First, we find the ratio of \( AB \) to \( XY \) and the ratio of \( BC \) to \( YZ \).
\( \frac{AB}{XY} = \frac{4}{10} = \frac{2}{5} \)
\( \frac{BC}{YZ} = \frac{7}{17.5} = \frac{7}{\frac{35}{2}} = 7\times\frac{2}{35} = \frac{2}{5} \)

Step2: Check the included angle

We know that \( m\angle B = 40^\circ \) and \( m\angle Y = 40^\circ \), so \( \angle B \cong \angle Y \).

Step3: Apply the SAS Similarity Theorem

Since the ratio of two pairs of corresponding sides are equal (\( \frac{AB}{XY} = \frac{BC}{YZ} = \frac{2}{5} \)) and the included angles (\( \angle B \) and \( \angle Y \)) are congruent, by the Side - Angle - Side (SAS) Similarity Theorem, \( \triangle ABC \sim \triangle XYZ \).

Answer:

Yes, the two triangles are similar by the SAS (Side - Angle - Side) Similarity Theorem. The ratio of \( AB \) to \( XY \) is \( \frac{4}{10}=\frac{2}{5} \), the ratio of \( BC \) to \( YZ \) is \( \frac{7}{17.5}=\frac{2}{5} \), and \( \angle B\cong\angle Y \) (both \( 40^\circ \)).