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

Explanation:

Step1: Match congruent angles

Look at the angle markings:

• $\angle A$ (marked with single arc) corresponds to $\angle Z$ (single arc)
• $\angle B$ (marked with double arc) corresponds to $\angle Y$ (double arc)
• $\angle C$ (unmarked, remaining angle) corresponds to $\angle X$ (unmarked, remaining angle)

Expressions:
$\angle A \cong \angle Z$
$\angle B \cong \angle Y$
$\angle C \cong \angle X$

Step2: Match congruent sides

Look at the side markings:

• $\overline{AB}$ (single tick) corresponds to $\overline{ZY}$ (single tick)
• $\overline{AC}$ (double tick) corresponds to $\overline{ZX}$ (double tick)
• $\overline{BC}$ (triple tick) corresponds to $\overline{YX}$ (triple tick)

Expressions:
$\overline{AB} \cong \overline{ZY}$
$\overline{AC} \cong \overline{ZX}$
$\overline{BC} \cong \overline{YX}$

Step3: Write congruence statement

Map vertices from $\triangle BCA$ using corresponding parts: $B \to Y$, $C \to X$, $A \to Z$
Expression:
$\triangle BCA \cong \triangle YXZ$

Answer:

(a) $\angle A \cong \angle Z$, $\angle B \cong \angle Y$, $\angle C \cong \angle X$
(b) $\overline{AB} \cong \overline{ZY}$, $\overline{AC} \cong \overline{ZX}$, $\overline{BC} \cong \overline{YX}$
(c) $\triangle BCA \cong \triangle YXZ$