Question

in the diagram, $\triangle efg \cong \triangle opq$. complete the statement.

1. $\overline{ef} \cong \underline{quad}$
2. $\angle f \cong \underline{quad}$
3. $\angle o \cong \underline{quad}$
4. $m\angle o = \underline{quad}$
5. $pq = \underline{quad}$
6. $m\angle efg = \underline{quad}$

write a congruence statement for any figure that can be proved congruent. explain your reasoning.

8.

9.

Explanation:

Step1: Match congruent sides/angles

Since $\triangle EFG \cong \triangle OPQ$, corresponding parts are congruent.

1. Corresponding side of $\overline{EF}$: $\overline{OP}$
2. Corresponding angle of $\angle F$: $\angle P$
3. Corresponding angle of $\angle G$: $\angle Q$
4. Corresponding angle of $\angle O$: $\angle E$
5. Corresponding side of $\overline{QO}$: $\overline{GE}$ (or $\overline{EG}$)
6. Corresponding triangle of $\triangle GFE$: $\triangle QPO$

Step2: Analyze problem 8

Check congruent figures: $\triangle EGI$ and $\triangle QOM$.

• $\angle E = \angle Q$, $\angle G = \angle M$, $\overline{EG} = \overline{QM}$. By AAS congruence, $\triangle EGI \cong \triangle QOM$.

Step3: Analyze problem 9

Check congruent figures: $\triangle RTV$ and $\triangle YXZ$.

• $\overline{RT} = \overline{YX}$, $\overline{TV} = \overline{XZ}$, $\angle T = \angle X$. By SAS congruence, $\triangle RTV \cong \triangle YXZ$.

Answer:

1. $\overline{OP}$
2. $\angle P$
3. $\angle Q$
4. $\angle E$
5. $\overline{GE}$ (or $\overline{EG}$)
6. $\triangle QPO$

1. Congruence statement: $\triangle EGI \cong \triangle QOM$

Reason: Two pairs of corresponding angles are congruent ($\angle E \cong \angle Q$, $\angle G \cong \angle M$) and a pair of non-included corresponding sides are congruent ($\overline{EG} \cong \overline{QM}$), satisfying the AAS (Angle-Angle-Side) congruence criterion.

1. Congruence statement: $\triangle RTV \cong \triangle YXZ$

Reason: Two pairs of corresponding sides are congruent ($\overline{RT} \cong \overline{YX}$, $\overline{TV} \cong \overline{XZ}$) and the included corresponding angles are congruent ($\angle T \cong \angle X$), satisfying the SAS (Side-Angle-Side) congruence criterion.