Chapter 3.2, Problem 36PPS

(a)

To determine

Find the relation between the measures of the odd-numbered angles.

Answer to Problem 36PPS

  $\angle 1\cong \angle 3\cong \angle 5\cong \angle 7\cong \angle 9\cong \angle 11\cong \angle 13\cong \angle 15$

Explanation of Solution

Given:

  

The two levels of the bridge are parallel.

Formula Used:

Alternate Interior Angle Theorem : If two lines are parallel and are cut by a transversal , then the pair of alternate interior angles is congruent.

Corresponding Angle Postulate: If two lines are parallel and are cut by a transversal , then the pair of corresponding angles is congruent.

Calculation:

The following pairs are alternate interior angles and are congruent by Alternate Interior Angles Theorem :

  $\begin{array}{l}\angle 1\cong \angle 3\\ \angle 5\cong \angle 7\\ \angle 9\cong \angle 11\\ \angle 13\cong \angle 15\end{array}$

The following pairs are corresponding angles and are congruent by Corresponding Angle Postulate :

  $\begin{array}{l}\angle 1\cong \angle 5\cong \angle 9\cong \angle 13\\ \angle 3\cong \angle 7\cong \angle 11\cong \angle 15\end{array}$

So,

  $\angle 1\cong \angle 3\cong \angle 5\cong \angle 7\cong \angle 9\cong \angle 11\cong \angle 13\cong \angle 15$ (By Transitive Property)

(b)

To determine

Find the relation between the measures of the even-numbered angles.

Answer to Problem 36PPS

  $\angle 2\cong \angle 4\cong \angle 6\cong \angle 8\cong \angle 10\cong \angle 12\cong \angle 14\cong \angle 16$

Explanation of Solution

Given:

  

The two levels of the bridge are parallel.

Formula Used:

Alternate Interior Angle Theorem : If two lines are parallel and are cut by a transversal , then the pair of alternate interior angles is congruent.

Calculation:

The following pairs are alternate interior angles and are congruent by Alternate Interior Angles Theorem :

  $\begin{array}{l}\angle 2\cong \angle 4\\ \angle 6\cong \angle 8\\ \angle 10\cong \angle 12\\ \angle 14\cong \angle 16\end{array}$

The following pairs are alternate interior angles and are congruent by Alternate Interior Angles Theorem :

  $\begin{array}{l}\angle 4\cong \angle 6\\ \angle 8\cong \angle 10\\ \angle 12\cong \angle 14\end{array}$

So,

  $\angle 2\cong \angle 4\cong \angle 6\cong \angle 8\cong \angle 10\cong \angle 12\cong \angle 14\cong \angle 16$ (By Transitive Property)

(c)

To determine

Find the relation between the measures of the odd-even numbered angles.

Answer to Problem 36PPS

All the pairs consisting one odd-even angle are complementary.

Explanation of Solution

Given:

  

The two levels of the bridge are parallel.

Formula Used:

Consecutive Interior Angle Theorem : If two lines are parallel and are cut by a transversal , then the pair of consecutive interior angles is congruent.

Calculation:

From part (a) and (b) :

  $\angle 1\cong \angle 3\cong \angle 5\cong \angle 7\cong \angle 9\cong \angle 11\cong \angle 13\cong \angle 15$

  $\angle 2\cong \angle 4\cong \angle 6\cong \angle 8\cong \angle 10\cong \angle 12\cong \angle 14\cong \angle 16$

Also,

  $m\angle 1+m\angle 2=90°$ , since, they are consecutive interior angles .

Hence,

  $\begin{array}{l}m\angle 3+m\angle 4=90°\\ m\angle 5+m\angle 6=90°\\ m\angle 7+m\angle 8=90°\\ m\angle 9+m\angle 10=90°\\ m\angle 11+m\angle 12=90°\\ m\angle 13+m\angle 14=90°\\ m\angle 15+m\angle 16=90°\end{array}$

So, all the pairs consisting one odd-even angle are complementary.

(d)

To determine

Find the geometric term(s) that can be used to relate the two roadways contained by the bridge.

Answer to Problem 36PPS

Parallel

Explanation of Solution

Given:

  

The two levels of the bridge are parallel.

Calculation:

From part (a) , (b) and (c) , we can conclude that :

The planes of the two roadways contained by the bridge are parallel.