Question

is quadrilateral defg congruent to quadrilateral mnop? explain
choose the correct answer below.
a. yes. △def ≅ mno by aas. (overline{df} cong overline{mo}) by cpctc, and △dfg ≅ △mop by asa. since defg and mnop are composed of corresponding congruent triangles, defg ≅ mnop.
b. yes. △def ≅ mno by sas, and △dfg ≅ △mop by aas. since defg and mnop are composed of corresponding congruent triangles, defg ≅ mnop.
c. no. △def ≅ mno by aas, but △dfg and △mop cannot be shown to be congruent because only two pairs of congruent angles are known. therefore, defg and mnop are not composed of corresponding congruent triangles
d. no. △def ≅ mno by sas, but △dfg and △mop cannot be shown to be congruent because only two pairs of congruent angles are known. therefore, defg and mnop are not composed of corresponding congruent triangles

Explanation:

To determine if quadrilaterals DEFG and MNOP are congruent, we analyze the triangles they are composed of. First, check $\triangle DEF$ and $\triangle MNO$:

• In $\triangle DEF$ and $\triangle MNO$, we have two angles (e.g., $\angle D \cong \angle M$, $\angle E \cong \angle N$) and a side (e.g., $EF \cong NO$) which fits AAS (Angle - Angle - Side) congruence, so $\triangle DEF \cong \triangle MNO$ by AAS.
• By CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $DF \cong MO$.
• For $\triangle DFG$ and $\triangle MOP$, we know $\angle D \cong \angle M$ (from before), $DF \cong MO$ (from CPCTC), and $\angle DFG \cong \angle MOP$ (marked angles), so $\triangle DFG \cong \triangle MOP$ by ASA (Angle - Side - Angle).

Since DEFG is made of $\triangle DEF$ and $\triangle DFG$, and MNOP is made of $\triangle MNO$ and $\triangle MOP$, and the corresponding triangles are congruent, the quadrilaterals are congruent. This matches option A.

Answer:

A. Yes. $\triangle DEF \cong MNO$ by AAS. $\overline{DF} \cong \overline{MO}$ by CPCTC, and $\triangle DFG \cong \triangle MOP$ by ASA. Since DEFG and MNOP are composed of corresponding congruent triangles, $DEFG \cong MNOP$.