Question

$\overline{\text{OR}}$is a common side of two congruent quadrilaterals.

In your own words explain why each of the following statements must be true.

a. $O$is the midpoint of role="math" localid="1648818703436" $\overline{\mathit{\text{NM}}}$.

b. $\angle \mathit{\text{NOR}}\cong \angle \mathit{\text{MOR}}$.

c.$\overline{\mathit{\text{RO}}}\perp \overline{\mathit{\text{NM}}}$

Short answer

Hence, it is explained in the below steps that

a.$\mathit{O}$ is the midpoint of $\overline{\mathit{\text{NM}}}$.

b. role="math" localid="1648818813775" $\mathbf{\angle }\mathit{\text{NOR}}\mathbf{\cong }\mathbf{\angle }\mathit{\text{MOR}}$.

c.role="math" localid="1648818846151" $\overline{\mathit{\text{RO}}}\mathbf{\perp }\overline{\mathit{\text{NM}}}$

Step-by-step solution

Part a. Step 1. Consider the diagram.

Here, both the quadrilaterals are congruent and$\overline{OR}$ is the common side of the two congruent quadrilaterals.

Part a. Step 2. State the explanation.

If two quadrilaterals are congruent then the following conditions are satisfied:

1. Corresponding sides of the two quadrilaterals are congruent.

2. Corresponding angles of the two quadrilaterals are congruent.

As quadrilateral $NORE$is congruent to quadrilateral $MORA$,

$\overline{NO}\cong \overline{MO}$(Corresponding sides of congruent quadrilaterals are congruent).

As the midpoint of the line segment is a point that divides it into two congruent segments, $O$is the midpoint of $\overline{NM}$.

Part a. Step 3. State the conclusion.

Therefore, $O$ is the midpoint of$\overline{NM}$ is true.

Part b. Step 1. Consider the diagram.

Here, both the quadrilaterals are congruent and $\overline{OR}$is the common side of the two congruent quadrilaterals.

Part b. Step 2. State the explanation.

If two quadrilaterals are congruent then the following conditions are satisfied:

1. Corresponding sides of the two quadrilaterals are congruent.

2. Corresponding angles of the two quadrilaterals are congruent.

As quadrilateral $NORE$is congruent to quadrilateral $MORA$,

$\angle NOR\cong \angle MOR$(Corresponding angles of congruent quadrilaterals are congruent).

Part b. Step 3. State the conclusion

Therefore, $\angle NOR\cong \angle MOR$is true.

Part c. Step 1. Consider the diagram.

Here, both the quadrilaterals are congruent and $\overline{OR}$is the common side of the two congruent quadrilaterals.

Part c. Step 2. State the explanation.

From part (b),

$\angle NOR\cong \angle MOR$

So,

$m\angle NOR=m\angle MOR$(Congruent angles are equal to each other).

$m\angle NOR+m\angle MOR={180}^{\circ }$( $\angle NOR,\angle MOR$form a linear pair and measure of linear pair of angles is ${180}^{\circ }$).

So,

$m\angle NOR+m\angle MOR={180}^{\circ }$

Since $\angle NOR=\angle MOR$, then,

$\begin{array}{c}2\angle NOR={180}^{\circ }\\ \angle NOR={90}^{\circ }\end{array}$

So,

$\overline{RO}\perp \overline{NM}$ (Two lines segments are perpendicular if they form a right angle).

Part c. Step 3. State the conclusion.

Therefore, $\overline{RO}\perp \overline{NM}$is true.