Chapter 3.2, Problem 11CE

To determine

To find: the measure of other numbered angles given that $m\angle 1=x$ .

Answer to Problem 11CE

  $m\angle 2=180-x$

  $m\angle 3=180-x$

  $m\angle 4=x$

  $m\angle 5=x$

  $m\angle 6=180-x$

  $m\angle 7=180-x$

  $m\angle 8=x$

Explanation of Solution

Given:

The diagram:

Concept Used:

Postulate 10: If two parallel lines are cut by a transversal then the corresponding angles are congruent

Theorem 3-2: If two parallel lines are cut by a transversal then the alternate interior angles are congruent.

Theorem 3-3: If two parallel lines are cut by a transversal then the same-side interior angles are supplementary.

Vertically opposite angles are congruent.

Consider the given figure:

In this figure,

  $\angle 1\cong \angle 4$ (Vertically opposite angles)

Thus,

  $m\angle 4=m\angle 1=x$

  $\angle 4\cong \angle 8$ (Corresponding angles)

Thus,

  $m\angle 8=m\angle 4=x$

  $\angle 8\cong \angle 5$ (Vertically opposite angles)

Thus,

  $m\angle 5=m\angle 8=x$

And,

  $m\angle 1+m\angle 2=180$ (Same-side interior angles)

Substitute $m\angle 1=x$

  $\begin{array}{c}x+m\angle 2=180\\ m\angle 2=180-x\end{array}$

  $\angle 2\cong \angle 3$ (Vertically opposite angles)

Thus,

  $m\angle 3=m\angle 2=180-x$

  $\angle 3\cong \angle 7$ (Corresponding angles)

Thus,

  $m\angle 7=m\angle 3=180-x$

  $\angle 7\cong \angle 6$ (Vertically opposite angles)

Thus,

  $m\angle 6=m\angle 7=180-x$