Chapter 4.1, Problem 21WE

a.

To determine

To state: The reason why the statement ‘O is the midpoint of $\overline{NM}$ ’ is true.

Explanation of Solution

Given information:Quadrilateral NORE is congruent to quadrilateral MORA.

  

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 $\overline{NM}$ .

b.

To determine

To state: The reason why the statement ‘ $\angle NOR\cong \angle MOR$ ’ is true.

Explanation of Solution

Given information: Quadrilateral NORE is congruent to quadrilateral MORA

  

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)

c.

To determine

To state: The reason why the statement ‘ $\overline{RO}\perp \overline{NM}$ ’ is true.

Explanation of Solution

Given information: Quadrilateral NORE is congruent to quadrilateral MORA.

  

From part b.,

  $\angle NOR\cong \angle MOR$

So,

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

  $m\angle NOR+m\angle MOR=180°$ ( $\angle NOR,\angle MOR$ form a linear pair and measure of linear pair of angles is $180°$ )

So,

  $m\angle NOR+m\angle NOR=180°\left\{\because \angle NOR=\angle MOR\right\}$

  $\begin{array}{l}\Rightarrow 2\angle NOR=180°\\ \Rightarrow \angle NOR=\frac{180°}{2}\\ =90°\end{array}$

So,

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