Geometry Similarity Proofs and Calculations: Triangles ABC, DBF, LMN, XYZ, PFE, GFQ, STU, GTJ

Published on July 20, 2026

Geometry

This problem involves proving triangle similarity using AA and SAS criteria and calculating unknown side lengths using proportions derived from similarity.

Question

Answer

Based on the provided image, here are the step-by-step solutions for the math problems.

Part (a)

Problem: Prove that

is similar to

. Find

and

Proof:

In and , we are given that (indicated by the arrows on the lines).
Since , the corresponding angles are equal:
Both triangles share .
Therefore, by the AA (Angle-Angle) similarity criterion.

Finding and :
Since the triangles are similar, their corresponding sides are proportional:

From the figure:

, and

Find :

Simplify to :
Find :

Part (b)

Problem: Prove that

is similar to

. If

, and

, find

Proof:

In , and .
The third angle .
In , and .
Since and , by the AA similarity criterion.

Finding :
The corresponding sides are proportional:

.
Given:

, and

Part ©

Problem: Prove that

is similar to

. If

, and

, find the length of

Proof:

(vertically opposite angles).
Assuming (typical for this trapezoid-like figure), then (alternate interior angles).
Therefore, by the AA similarity criterion.