Question

in the diagram below, ( mangle a = 55^{circ} ) and ( mangle e = 35^{circ} ). which best explains the relationship between triangle acb and triangle dce? the triangles are not similar because only one pair of corresponding angles is congruent. the triangles are similar because all right triangles can be mapped to each other using a series of transformations. the triangles are not similar because they share a common segment and vertex. the triangles are similar because all pairs of corresponding angles are congruent.

Explanation:

Step1: Find angle in triangle \(ACB\)

In right - triangle \(ACB\), \(\angle ACB = 90^{\circ}\), \(\angle A=55^{\circ}\). Using the angle - sum property of a triangle (\(\angle A+\angle B+\angle ACB = 180^{\circ}\)), we find \(\angle B=180^{\circ}-\angle A - \angle ACB=180^{\circ}-55^{\circ}-90^{\circ}=35^{\circ}\).

Step2: Find angle in triangle \(DCE\)

In right - triangle \(DCE\), \(\angle DCE = 90^{\circ}\), \(\angle E = 35^{\circ}\). Using the angle - sum property of a triangle (\(\angle DCE+\angle E+\angle CDE=180^{\circ}\)), we find \(\angle CDE=180^{\circ}-\angle DCE-\angle E=180^{\circ}-90^{\circ}-35^{\circ}=55^{\circ}\).

Step3: Check angle congruence

For \(\triangle ACB\) and \(\triangle DCE\): \(\angle ACB=\angle DCE = 90^{\circ}\), \(\angle A=\angle CDE = 55^{\circ}\), \(\angle B=\angle E = 35^{\circ}\). By the AA (angle - angle) similarity criterion (if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar), \(\triangle ACB\sim\triangle DCE\) since all pairs of corresponding angles are congruent.

The first option is wrong because there are two pairs of congruent non - right angles (in addition to the right angles). The second option is wrong because not all right - triangles are similar (right - triangles need at least one more pair of congruent non - right angles to be similar). The third option is wrong because sharing a common segment and vertex is not a valid reason for non - similarity (similarity is based on angle measures and side - length ratios).

Answer:

The triangles are similar because all pairs of corresponding angles are congruent.