Chapter 2.3, Problem 20WE

To determine

To prove :  Angles formed by folding two different rectangular sheets of paper along their corners are congruent as shown below.

Explanation of Solution

Given information :  A rectangular sheet of paper is folded from one side. Fold the next corner so that the edges touch each other, as shown below. Measure the angle $\left(\angle c+\angle d\right)$ , formed by the fold lines.

Take another sheet of paper. Fold it from one sideFold the next corner so that the edges touch each other, as shown below. Measure the angle $\left(\angle g+\angle h\right)$ , formed by the fold lines.

It is to be proven that $\left(\angle c+\angle d\right)$ is congruent to $\left(\angle g+\angle h\right)$ .

Formula used :  Angle addition postulate states that if $\angle$ LMN is a straight line and P is a point not on $\overleftrightarrow{LN}$ then $\angle$ LMP + $\angle$ PMN =180 degrees.

Angle bisector theorem states that if $\overrightarrow{BX}$ is the bisector of $\angle$ ABC then $m\angle ABX=\frac{1}{2}\angle ABC$ , and $m\angle XBC=\frac{1}{2}\angle ABC$ .

Proof :  Fold a rectangular paper from one side. Fold the next corner so that the edges touch each other. The fold lines are the angle bisector here. So $\angle a=\angle d$ ; $\angle c=\angle b$ .

By angle addition postulate,

  $\angle a+\angle d+\angle c+\angle b=\text{18}0\text{ degrees}$

Shown earlier $\angle a=\angle d$ and $\angle c=\angle b$ ,

Thus,

  $\begin{array}{l}\angle d+\angle d+\angle c+\angle c=180\text{degrees}\\ 2\angle d+2\angle c=180\text{ degrees}\end{array}$

By division property (from algebra),

  $\angle c+\angle d=90\text{ degrees}$

Fold a rectangular paper from one side. Fold the next corner so that the edges touch each other. The fold lines are the angle bisector here. So $\angle e=\angle h$ ; $\angle g=\angle f$ .

By angle addition postulate,

  $\angle e+\angle h+\angle g+\angle f=180\text{degrees}$

Shown earlier $\angle e=\angle h$ and $\angle g=\angle f$ ,

Thus,

  $\begin{array}{l}\angle h+\angle h+\angle g+\angle g=180\text{degrees}\\ 2\angle g+2\angle h=180\text{degrees}\end{array}$

By division property (from algebra),

  $\angle g+\angle h=90\text{degrees}$

Since both $\angle g+\angle h=90\text{degrees}$ and $\angle c+\angle d=90\text{degrees}$ and for any number of rectangular sheets taken the value comes out to be same.