Chapter 3.2, Problem 12CE

To determine

To find: the $m\angle 6$ given that $m\angle 4=2m\angle 3$ .

Answer to Problem 12CE

  $m\angle 6=60$

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,

  $\angle 6\cong \angle 3$ (Alternate interior angles)

Thus,

  $m\angle 6=m\angle 3$ ….. (1)

And,

  $m\angle 4+m\angle 6=180$ (Same-side interior angles)

Substitute $m\angle 3=m\angle 6$ and $m\angle 4=2m\angle 3$ in above equation,

  $\begin{array}{c}2m\angle 3+m\angle 3=180\\ 3m\angle 3=180\\ m\angle 3=\frac{180}{3}\\ m\angle 3=60\end{array}$

Then, by equation (1),

  $m\angle 6=m\angle 3=60$