Question

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) $\triangle cba \cong \triangle \square$

Explanation:

Step1: Analyze congruent angles

In congruent triangles, corresponding angles are equal. Looking at the triangles, $\angle A$ corresponds to $\angle Z$, $\angle B$ corresponds to $\angle X$, and $\angle C$ corresponds to $\angle Y$. So:

• $\angle A \cong \angle Z$
• $\angle B \cong \angle X$
• $\angle C \cong \angle Y$

Step2: Analyze congruent sides

Corresponding sides in congruent triangles are equal. The markings on the sides show:

• $\overline{AB}$ corresponds to $\overline{ZX}$
• $\overline{AC}$ corresponds to $\overline{ZY}$
• $\overline{BC}$ corresponds to $\overline{XY}$

Step3: Triangle congruence statement

From the correspondence of angles and sides, $\triangle CBA$ (which is $\triangle ABC$) is congruent to $\triangle XYZ$ (or $\triangle ZXY$? Wait, let's check the order. The vertices of the first triangle are $C, B, A$, so we need to match the corresponding vertices. From angles: $C \to Y$, $B \to X$, $A \to Z$. So $\triangle CBA \cong \triangle XYZ$. Wait, no, let's re - check. The first triangle: vertices $A, B, C$ with $AB$ and $AC$? Wait, the first triangle (top) has $A$ at the top, $B$ at the bottom right, $C$ at bottom left. The second triangle (bottom) has $X$ at bottom left, $Y$ at top, $Z$ at bottom right. Wait, maybe the correct correspondence is: $\angle A$ (single arc) corresponds to $\angle Z$ (single arc), $\angle B$ (double arc) corresponds to $\angle X$ (double arc), $\angle C$ (no arc? Wait, the first triangle: $\angle C$ is a single arc? Wait, the first triangle: $\angle A$ has one arc, $\angle B$ has two arcs, $\angle C$ has one arc? Wait, no, the first triangle: $\angle A$ (top) has one arc, $\angle B$ (bottom right) has two arcs, $\angle C$ (bottom left) has one arc. The second triangle: $\angle X$ (bottom left) has two arcs, $\angle Y$ (top) has one arc, $\angle Z$ (bottom right) has one arc. Wait, maybe I mixed up. Let's do it by side markings. The first triangle: $AC$ has two marks, $BC$ has two marks, $AB$ has one mark? Wait, no, the first triangle: the sides from $A$ to $B$: one mark, $A$ to $C$: two marks, $B$ to $C$: two marks? Wait, the first triangle (top) has $AC$ with two marks, $BC$ with two marks, $AB$ with one mark. The second triangle (bottom) has $XY$ with two marks, $ZY$ with two marks, $ZX$ with one mark. So $AC$ (two marks) $\cong ZY$ (two marks), $BC$ (two marks) $\cong XY$ (two marks), $AB$ (one mark) $\cong ZX$ (one mark). Angles: $\angle B$ (two arcs) $\cong \angle X$ (two arcs), $\angle A$ (one arc) $\cong \angle Z$ (one arc), $\angle C$ (one arc) $\cong \angle Y$ (one arc). So the correspondence of vertices: $B \to X$, $C \to Y$, $A \to Z$. So $\triangle CBA$ (vertices $C, B, A$) corresponds to $\triangle YXZ$? No, let's write the congruence statement. If $\angle A \cong \angle Z$, $\angle B \cong \angle X$, $\angle C \cong \angle Y$, then the order of the triangles should be $C$ (corresponds to $Y$), $B$ (corresponds to $X$), $A$ (corresponds to $Z$). So $\triangle CBA \cong \triangle YXZ$? Wait, maybe the correct answer is:

(a) $\angle A \cong \angle Z$, $\angle B \cong \angle X$, $\angle C \cong \angle Y$

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

(c) $\triangle CBA \cong \triangle XYZ$ (Wait, maybe I made a mistake in vertex order, but following the angle and side correspondences, the most probable is $\triangle CBA \cong \triangle XYZ$)

Answer:

(a) $\angle A \cong \boldsymbol{\angle Z}$, $\angle B \cong \boldsymbol{\angle X}$, $\angle C \cong \boldsymbol{\angle Y}$

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

(c) $\triangle CBA \cong \boldsymbol{\triangle XYZ}$