Question

show if the triangles below are similar or not.
when two polygons are similar, we can write a similarity statement using the symbol “∼”.

1. are the following rectangles similar?
2. are the following triangles similar?

Explanation:

Part 1: Triangles Similarity (Top Problem)

Step1: Analyze Angles of Triangles

Triangle \( ACB \): \( \angle C = 90^\circ \), \( \angle A = 35^\circ \), so \( \angle B = 180^\circ - 90^\circ - 35^\circ = 55^\circ \).
Triangle \( XZY \): \( \angle Z = 90^\circ \), \( \angle X = 35^\circ \), so \( \angle Y = 180^\circ - 90^\circ - 35^\circ = 55^\circ \).
All corresponding angles are equal (\( 90^\circ, 35^\circ, 55^\circ \)).

Step2: Check Side Ratios

For \( \triangle ACB \) and \( \triangle XZY \):
\( \frac{AC}{XZ} = \frac{24}{15} = \frac{8}{5} \), \( \frac{CB}{ZY} = \frac{32}{20} = \frac{8}{5} \), \( \frac{AB}{XY} = \frac{40}{25} = \frac{8}{5} \).
All side ratios are equal.

Step1: Recall Rectangle Properties

Rectangles have all angles \( 90^\circ \), so corresponding angles are equal. Check side ratios.

Step2: Calculate Side Ratios

Rectangle \( ABCD \): length \( 36 \), width \( 18 \).
Rectangle \( WXYZ \): length \( 45 \), width \( 25 \).
Ratio of lengths: \( \frac{36}{45} = \frac{4}{5} \).
Ratio of widths: \( \frac{18}{25} \).
\( \frac{4}{5}
eq \frac{18}{25} \) (since \( 4 \times 25 = 100 \), \( 5 \times 18 = 90 \)).

Step1: Analyze Angles

\( \triangle PTN \): \( \angle P = 90^\circ \), \( \angle N = 30^\circ \), \( \angle T = 60^\circ \).
\( \triangle RKG \): \( \angle K = 90^\circ \), \( \angle G = 60^\circ \), so \( \angle R = 30^\circ \).
Corresponding angles: \( 90^\circ, 30^\circ, 60^\circ \) (equal).

Step2: Check Side Ratios

\( \triangle PTN \): sides \( 16, 30, 34 \).
\( \triangle RKG \): sides \( 8, 15, 17 \).
Ratios: \( \frac{16}{8} = 2 \), \( \frac{30}{15} = 2 \), \( \frac{34}{17} = 2 \). All ratios equal.

Answer:

The triangles are similar. Similarity statement: \( \triangle ACB \sim \triangle XZY \)

Part 1 (Sub-Question 1: Rectangles Similarity)