Question

triangles below are congruent and their corresponding parts are marked. name all the corresponding congruent angles and sides. then, complete the triangle congruence statement. (a) ∠a ≅ ∠□ (b) $overline{ab}$ ≅ $overline{□}$ ∠b ≅ ∠□ $overline{ac}$ ≅ $overline{□}$ ∠c ≅ ∠□ $overline{bc}$ ≅ $overline{□}$ (c) △cab ≅ △□

Explanation:

Step1: Identify Corresponding Angles

In congruent triangles, corresponding angles are marked with the same number of arcs. ∠A in △CAB has one arc, so it corresponds to ∠Z in △XYZ (which also has one arc). ∠B has two arcs, corresponding to ∠X (two arcs). ∠C has three arcs, corresponding to ∠Y (three arcs). So:

• ∠A ≅ ∠Z
• ∠B ≅ ∠X
• ∠C ≅ ∠Y

Step2: Identify Corresponding Sides

Corresponding sides have the same number of tick marks. \( \overline{AB} \) has one tick, corresponding to \( \overline{XZ} \) (one tick). \( \overline{AC} \) has two ticks, corresponding to \( \overline{YZ} \) (two ticks). \( \overline{BC} \) has three ticks, corresponding to \( \overline{XY} \) (three ticks). So:

• \( \overline{AB} \cong \overline{XZ} \)
• \( \overline{AC} \cong \overline{YZ} \)
• \( \overline{BC} \cong \overline{XY} \)

Step3: Triangle Congruence Statement

Match the vertices: C corresponds to Y, A corresponds to Z, B corresponds to X. So \( \triangle CAB \cong \triangle YZX \).

Answer:

(a) \( \angle A \cong \angle Z \), \( \angle B \cong \angle X \), \( \angle C \cong \angle Y \)
(b) \( \overline{AB} \cong \overline{XZ} \), \( \overline{AC} \cong \overline{YZ} \), \( \overline{BC} \cong \overline{XY} \)
(c) \( \triangle CAB \cong \triangle YZX \)