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: Identify corresponding angles

In congruent triangles, corresponding angles are marked with the same number of arcs or have the same angle markings. Looking at the triangles, $\angle A$ (with one arc) corresponds to $\angle X$ (with one arc), $\angle B$ (unmarked angle) corresponds to $\angle Z$ (unmarked angle), and $\angle C$ (with two arcs) corresponds to $\angle Y$ (with two arcs). So:

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

Step2: Identify corresponding sides

Corresponding sides are marked with the same number of tick marks. Side $AB$ (two ticks) corresponds to side $ZX$ (two ticks), side $AC$ (one tick) corresponds to side $XY$ (one tick), and side $BC$ (two ticks? Wait, no: Wait, triangle $BCA$: $BC$ has two ticks? Wait, looking at the first triangle: $AB$ has two ticks, $BC$ has two ticks? Wait no, first triangle: $AB$ (from $B$ to $A$) has two ticks, $BC$ (from $B$ to $C$) has two ticks? Wait no, the first triangle: $AB$ (two ticks), $AC$ (one tick), $BC$ (two ticks? Wait, no, the first triangle: $B$ to $A$: two ticks, $A$ to $C$: one tick, $C$ to $B$: two ticks? Wait the second triangle: $Y$ to $Z$: two ticks, $Z$ to $X$: two ticks? Wait no, second triangle: $Y$ to $Z$: two ticks, $Z$ to $X$: two ticks? Wait no, the second triangle: $Y$ to $Z$ (two ticks), $Z$ to $X$ (two ticks), $X$ to $Y$ (one tick). Wait, no, let's re-examine:

First triangle: vertices $B$, $A$, $C$. Sides: $AB$ (two ticks), $AC$ (one tick), $BC$ (two ticks? Wait, no, the first triangle: $B$ to $A$: two ticks, $A$ to $C$: one tick, $C$ to $B$: two ticks? Wait the second triangle: $Y$, $Z$, $X$. Sides: $Y$ to $Z$: two ticks, $Z$ to $X$: two ticks, $X$ to $Y$: one tick. Wait, no, the markings: first triangle: $AB$ (two ticks), $BC$ (two ticks)? No, the first triangle: $AB$ (two ticks), $AC$ (one tick), $BC$ (two ticks? Wait, the first triangle: angle at $A$ (one arc), angle at $C$ (two arcs), angle at $B$ (unmarked). Second triangle: angle at $X$ (one arc), angle at $Y$ (two arcs), angle at $Z$ (unmarked). So sides: $AB$ (two ticks) corresponds to $ZX$ (two ticks), $AC$ (one tick) corresponds to $XY$ (one tick), $BC$ (two ticks? Wait, no, $BC$: in first triangle, $B$ to $C$: two ticks? Wait, the first triangle: $B$ to $A$: two ticks, $A$ to $C$: one tick, $C$ to $B$: two ticks? Wait the second triangle: $Z$ to $X$: two ticks, $X$ to $Y$: one tick, $Y$ to $Z$: two ticks. So:

• $AB$ (two ticks) $\cong ZX$ (two ticks)
• $AC$ (one tick) $\cong XY$ (one tick)
• $BC$ (two ticks) $\cong YZ$ (two ticks)

Step3: Triangle congruence statement

The order of the vertices in the congruence statement should match the corresponding vertices. So $\triangle BCA$: vertices $B$, $C$, $A$. Corresponding vertices in the second triangle: $Z$ (corresponds to $B$), $Y$ (corresponds to $C$), $X$ (corresponds to $A$). So $\triangle BCA \cong \triangle ZYX$.

Answer:

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