Chapter 2.6, Problem 16WE

a.

To determine

To find:angles that must be congruent to $\angle 4$ in the given figure

Answer to Problem 16WE

$\angle 2$ must be congruent to $\angle 4$

Explanation of Solution

Vertical opposite angles are the angles formed on intersection of two lines. Vertical opposite angles are equal.

Here, lines $m$ and $t$ intersect each other.

So,

  $\angle 2\cong \angle 4$ (vertically opposite angles)

b.

To determine

To find: angles that must be congruent to $\angle 4$ if $\angle 4$ and $\angle 5$ are supplementary.

Answer to Problem 16WE

$\angle 6\cong \angle 4$ and $\angle 8\cong \angle 4$

Explanation of Solution

First proof $m\parallel n$

If a transversal intersects two lines such that each pair of interior angles is supplementary then the lines are said to be parallel.

$\angle 4$ and $\angle 5$ are supplementary and $\angle 4,\angle 5$ form a pair of interior angles. Also, take $t$ as a transversal.

So,

  $m\parallel n$

(if a transversal intersects two lines such that each pair of interior angles is supplementary then the lines are said to be parallel)

Now findangles that must be congruent to $\angle 4$ .

Alternate interior angles are equal.

$m\parallel n$ and take $t$ as a transversal

  $\angle 6\cong \angle 4$ (Alternate interior angles)

If a transversal intersects two parallel lines then each pair of corresponding angles are congruent.

$m\parallel n$ and take $t$ as a transversal

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

Therefore,

$\angle 6\cong \angle 4$ and $\angle 8\cong \angle 4$