Question

exit ticket
name: _________________________ period: _
are the triangles similar? if so, write the similarity statement and determine the scale factor.

Explanation:

Step1: Identify Isosceles Triangles

Triangle \( \triangle OPR \): \( OP = OR = 10 \) (marked congruent), so it's isosceles with \( \angle P=\angle R = 70^\circ \).
Triangle \( \triangle UST \): \( US = UT = 5 \) (marked congruent), so it's isosceles with \( \angle S=\angle T \).

Step2: Check Angle-Angle (AA) Similarity

In \( \triangle OPR \), \( \angle P = 70^\circ \). Assume \( \angle T = 70^\circ \) (from the note "Assume \( \angle T = 70^\circ \)"), so \( \angle S=\angle T = 70^\circ \) (isosceles \( \triangle UST \)). Thus, \( \angle P=\angle T = 70^\circ \) and \( \angle R=\angle S = 70^\circ \) (since \( \angle R=\angle P = 70^\circ \) and \( \angle S=\angle T = 70^\circ \)). By AA similarity, \( \triangle OPR \sim \triangle UTS \) (order of angles: \( \angle O \) corresponds to \( \angle U \), \( \angle P \) to \( \angle T \), \( \angle R \) to \( \angle S \)).

Step3: Calculate Scale Factor

Scale factor is the ratio of corresponding sides. Corresponding sides: \( OP = 10 \) and \( UT = 5 \), or \( PR = 3 \) and \( ST = 1.5 \).
\( \text{Scale factor} = \frac{OP}{UT} = \frac{10}{5} = 2 \) (or \( \frac{PR}{ST} = \frac{3}{1.5} = 2 \)).

Answer:

The triangles are similar. Similarity statement: \( \triangle OPR \sim \triangle UTS \). Scale factor: \( 2 \).