Question

1. match the correct answer.

triangle mon with o to n and m to o marked congruent, and triangle prq with r to q and p to r marked congruent.

\\(\overline{on} \cong \overline{rq}, \overline{mo} \cong \overline{pr}, m\angle o > m\angle r\\)

according to the hinge theorem:

dropdown 1 > dropdown 2

options:
a. \\(pq\\)
b. \\(mn\\)
c. \\(m\angle m\\)
d. \\(m\angle n\\)

Explanation:

Step1: Recall Hinge Theorem

The Hinge Theorem (SAS Inequality Theorem) states that if two sides of one triangle are congruent to two sides of another triangle, but the included angle of the first triangle is larger than the included angle of the second, then the third side of the first triangle is longer than the third side of the second.

Step2: Identify Triangles and Sides/Angles

• Triangles: $\triangle MON$ and $\triangle PRQ$ (from the diagram and given congruences $\overline{ON} \cong \overline{RQ}$, $\overline{MO} \cong \overline{PR}$).
• Included angles: $\angle O$ (in $\triangle MON$) and $\angle R$ (in $\triangle PRQ$), with $m\angle O > m\angle R$.
• Third sides: $MN$ (in $\triangle MON$) and $PQ$ (in $\triangle PRQ$).

Step3: Apply Hinge Theorem

By Hinge Theorem, since two sides ($\overline{MO} \cong \overline{PR}$, $\overline{ON} \cong \overline{RQ}$) are congruent and $m\angle O > m\angle R$ (included angle), the third side $MN$ (opposite $\angle O$) is longer than $PQ$ (opposite $\angle R$). So $MN > PQ$.

Answer:

First dropdown: b. \( MN \)
Second dropdown: a. \( PQ \)