Chapter 6.3, Problem 29PPS

To determine

To prove:

The coordinate proof for the following:

‘If a parallelogram has one right angle, it has four right angles.’

Explanation of Solution

Given information:

  $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram

  $\angle \mathrm{A}$ is a right angleProof:

First draw a parallelogram $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$

as below:

  

Now, the slope of $\overline{\mathrm{B}\mathrm{C}}$ is calculated as:

  $\overline{\mathrm{B}\mathrm{C}}=\frac{\mathrm{b}-\mathrm{b}}{\mathrm{a}-0}=0$

The slope of $\overline{\mathrm{B}\mathrm{C}}$ is $0$ .

The slope of $\overline{\mathrm{C}\mathrm{D}}$ is undefined.

The slope of $\overline{\mathrm{A}\mathrm{D}}$ is calculated as:

  $\overline{\mathrm{A}\mathrm{D}}=\frac{0-0}{\mathrm{a}-0}=0$

The slope of $\overline{\mathrm{A}\mathrm{B}}$ is $0$ .

Therefore, $\overline{\mathrm{B}\mathrm{C}}\perp \overline{\mathrm{C}\mathrm{D}},\overline{\mathrm{C}\mathrm{D}}\perp \overline{\mathrm{A}\mathrm{D}}$ and $\overline{\mathrm{A}\mathrm{B}}\parallel \overline{\mathrm{B}\mathrm{C}}$ .

So,

  $\angle \mathrm{B},\angle \mathrm{C},\angle \mathrm{D}$ are right angles