Question

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

Explanation:

Step1: Identify right angles

Both $\triangle ACB$ and $\triangle DCE$ have right angles ($\angle ACB = \angle DCE = 90^\circ$), so one pair of corresponding angles is congruent.

Step2: Calculate third angles

For $\triangle ACB$: $m\angle B = 180^\circ - 90^\circ - 55^\circ = 35^\circ$.
For $\triangle DCE$: $m\angle D = 180^\circ - 90^\circ - 35^\circ = 55^\circ$.
Thus, $\angle A \cong \angle D$ ($55^\circ$) and $\angle B \cong \angle E$ ($35^\circ$).

Step3: Check similarity condition

All three pairs of corresponding angles are congruent, so triangles are similar by AA similarity.

Answer:

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