Question

1. here are triangles abc and dec.

decide whether triangles abc and dec are similar. explain.
angle are congruent

Explanation:

Step1: Check Vertical Angles

Angles at \( C \) ( \( \angle ACB \) and \( \angle DCE \)) are vertical angles, so \( \angle ACB = \angle DCE = 123^\circ \).

Step2: Check Angle \( A \) and \( D \)

\( \angle A = 36^\circ \) and \( \angle D = 36^\circ \), so \( \angle A \cong \angle D \).

Step3: Check Angle \( B \) and \( E \)

In \( \triangle DEC \), sum of angles is \( 180^\circ \). So \( \angle E = 180^\circ - 123^\circ - 36^\circ = 21^\circ \). \( \angle B = 21^\circ \), so \( \angle B \cong \angle E \).

Step4: Apply AA Similarity

Since two pairs of corresponding angles are congruent (AA criterion), \( \triangle ABC \sim \triangle DEC \).

Answer:

Triangles \( ABC \) and \( DEC \) are similar by the AA (Angle - Angle) similarity criterion. Two pairs of corresponding angles are congruent: \( \angle A=\angle D = 36^\circ \), \( \angle B=\angle E = 21^\circ \), and \( \angle ACB=\angle DCE = 123^\circ \) (vertical angles), so the triangles have the same shape (similar).