Question

question 8 of 10 based only on the information given in the diagram, it is guaranteed that △rst ~ △uvw. a. true b. false

Explanation:

Step1: Analyze △RST

In right - triangle \( \triangle RST \), \( \angle T = 90^{\circ} \), \( \angle R=72^{\circ} \). Using the angle - sum property of a triangle (\( \angle R+\angle S+\angle T = 180^{\circ} \)), we can find \( \angle S \) as follows:
\( \angle S=180^{\circ}-\angle R - \angle T=180^{\circ}-72^{\circ}-90^{\circ}=18^{\circ} \)

Step2: Analyze △UVW

In right - triangle \( \triangle UVW \), \( \angle W = 90^{\circ} \), \( \angle V = 18^{\circ} \). Using the angle - sum property of a triangle (\( \angle V+\angle U+\angle W = 180^{\circ} \)), we can find \( \angle U \) as follows:
\( \angle U=180^{\circ}-\angle V-\angle W = 180^{\circ}-18^{\circ}-90^{\circ}=72^{\circ} \)

Step3: Check for similarity

For two triangles to be similar, the corresponding angles must be equal.
In \( \triangle RST \) and \( \triangle UVW \):

• \( \angle R = 72^{\circ}=\angle U \)
• \( \angle S=18^{\circ}=\angle V \)
• \( \angle T = 90^{\circ}=\angle W \)

By the AA (Angle - Angle) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the triangles are similar), \( \triangle RST\sim\triangle UVW \).

Answer:

A. True