Question

1. each pair of figures are similar. use this fact to solve for the missing side. show all work.
2. determine if the following triangles are similar.

if they are similar: complete the similarity statement and the condition that makes them similar.
show all work (- half a point for missing work).
if they are not similar: check not similar.

Explanation:

Step1: Analyze Triangle Similarity for Part 2a

For two triangles to be similar, the AA (Angle - Angle) similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
In \(\triangle ABC\) and \(\triangle KLM\), \(\angle A=\angle M = 47^{\circ}\) and \(\angle C=\angle K = 64^{\circ}\).
By the AA similarity criterion, \(\triangle ABC\sim\triangle MLK\) (the order of the letters matters based on the corresponding angles).

Step2: Analyze Triangle Similarity for Part 2b

For the SSS (Side - Side - Side) similarity criterion, if the ratios of the corresponding sides of two triangles are equal, the triangles are similar.
For \(\triangle PRT\) and \(\triangle XWT\):
\(\frac{PR}{XW}=\frac{15}{10} = 1.5\), \(\frac{PT}{XT}=\frac{12}{8}=1.5\), \(\frac{RT}{WT}=\frac{18}{12} = 1.5\)
Since \(\frac{PR}{XW}=\frac{PT}{XT}=\frac{RT}{WT}\), by the SSS similarity criterion, \(\triangle PRT\sim\triangle XWT\)

Answer:

a. Similar, \(\triangle ABC\sim\triangle MLK\), Similarity condition: \(AA\)
b. Similar, \(\triangle PRT\sim\triangle XWT\), Similarity condition: \(SSS\)