Question

the 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) $\angle a \cong \angle \square$ (b) $\overline{ab} \cong \overline{\square}$
$\angle b \cong \angle \square$ $\overline{ac} \cong \overline{\square}$
$\angle c \cong \angle \square$ $\overline{bc} \cong \overline{\square}$
(c) $\triangle cab \cong \triangle \square$

Explanation:

To solve this, we analyze the congruent triangles by matching the marked angles and sides. The markings (tick marks for sides, arc marks for angles) help identify correspondences.

Step 1: Identify Corresponding Angles

• ∠A has the same arc marking as ∠Z, so ∠A ≅ ∠Z.
• ∠B is a single arc (right angle - like ∠Y), so ∠B ≅ ∠Y.
• ∠C has the same arc marking as ∠X, so ∠C ≅ ∠X.

Step 2: Identify Corresponding Sides

• \( \overline{AB} \): 2 tick marks, matches \( \overline{ZY} \) (2 tick marks). So \( \overline{AB} \cong \overline{ZY} \).
• \( \overline{AC} \): 1 tick mark, matches \( \overline{ZX} \) (1 tick mark). So \( \overline{AC} \cong \overline{ZX} \).
• \( \overline{BC} \): 2 tick marks, matches \( \overline{XY} \) (2 tick marks). So \( \overline{BC} \cong \overline{XY} \).

Step 3: Triangle Congruence Statement

For \( \triangle CAB \cong \triangle \), we match vertices: C ↔ X, A ↔ Z, B ↔ Y. So \( \triangle CAB \cong \triangle XZY \).

Answer:

s:
(a)
\( \angle A \cong \angle Z \)
\( \angle B \cong \angle Y \)
\( \angle C \cong \angle X \)

(b)
\( \overline{AB} \cong \overline{ZY} \)
\( \overline{AC} \cong \overline{ZX} \)
\( \overline{BC} \cong \overline{XY} \)

(c)
\( \triangle CAB \cong \triangle XZY \)