Chapter 11.1, Problem 37HP

To determine

To find: Describe how the following angles are related.

Answer to Problem 37HP

  $\angle 5+\angle 6={180}^0$

  $\angle 3+\angle 4={180}^0$

  $\angle 2+\angle 1={180}^0$

Explanation of Solution

Given information:

Lines $l$ and $m$ are parallel and are cut by transversals $j$ and $k$

Since $\angle 1$ and $\angle 4$ are alternate interior angles, they are congruent.

So,

  $\angle 1=\angle 4$

Since $\angle 2$ and $\angle 4$ forms straight line, so they are supplementary angles

Sum of their measures is ${180}^0$

Using that

We get

  $\angle 2+\angle 4={180}^0$

Substituting $\angle 4=\angle 1$

We get

  $\angle 2+\angle 1={180}^0$

So, we can say that sum of measures of $\angle 2$ and $\angle 1$ is ${180}^0$

Since $\angle 2$ and $\angle 3$ are alternate interior angles, they are congruent.

So,

  $\angle 2=\angle 3$

We got

  $\angle 2+\angle 4={180}^0$

Substituting $\angle 3=\angle 2$

We get

  $\angle 3+\angle 4={180}^0$

So, we can say that sum of measures of $\angle 3$ and $\angle 4$ is ${180}^0$

We know that

The interior angles on the same side of a transversal are supplementary

So,

  $\angle 5+\angle 6={180}^0$

Therefore,

  $\angle 5+\angle 6={180}^0$

  $\angle 3+\angle 4={180}^0$

  $\angle 2+\angle 1={180}^0$

Are related