Chapter 3.2, Problem 41PPS

(a)

To determine

Draw five pair of parallel lines , m and n , a and b , r and s , j and k , and x and y , cut by a transversal t .

Explanation of Solution

Given:

  $\begin{array}{l}m\parallel n\\ a\parallel b\\ r\parallel s\\ j\parallel k\\ x\parallel y\end{array}$

Calculation:

Draw the lines :

  

(b)

To determine

Record the data in a table of the measures of each angle.

Explanation of Solution

Calculation:

From part (a), we measure the angles of 1,2,3,4 in each pair of parallel lines :

$m\angle 1$	$m\angle 2$	$m\angle 3$	$m\angle 4$
88	92	88	92
60	120	60	120
90	90	90	90
80	100	80	100
30	150	30	150

(c)

To determine

Make a conjucture about the relationship between the pair of angles formed on the exterior of parallel lines and on the same side of transversal.

Answer to Problem 41PPS

The pair is supplementary.

Explanation of Solution

Calculation:

From part (a) and (b), $\angle 1$ and $\angle 4$ are a pair of exterior angles on the same side of transversal.

$m\angle 1$	$m\angle 2$	$m\angle 3$	$m\angle 4$
88	92	88	92
60	120	60	120
90	90	90	90
80	100	80	100
30	150	30	150

  $\begin{array}{l}m\angle 1+m\angle 4=88+92=180\\ m\angle 1+m\angle 4=60+120=180\\ m\angle 1+m\angle 4=90+90=180\\ m\angle 1+m\angle 4=80+100=180\\ m\angle 1+m\angle 4=30+150=180\end{array}$

So, a pair of exterior angles on the same side of transversal is supplementary.

(d)

To determine

Find the type of reasoning did you use to form the conjucture in part (c).

Answer to Problem 41PPS

Inductive

Explanation of Solution

Calculation:

Since the conjucture was deduced by using the reasoning of certain pattern in all 5 pairs of parallel lines, hence the conjucture was based on inductive reasoning.

(e)

To determine

Write a proof of your conjucture.

Explanation of Solution

Calculation:

To Prove : A pair of exterior angles on the same side of transversal is supplementary.

Proof:

  $\begin{array}{l}m\angle 1+m\angle 2=\mathrm{180.................}[\text{Supplementary theorem]}\\ m\angle 2=m\angle \mathrm{4.........................}[\text{corresponding angles are congruent]}\\ \Rightarrow m\angle 1+m\angle 4=\mathrm{180............}[substitute]\end{array}$

Hence , $\angle 1\text{ and }\angle 4$ are supplementary.

Hence proved.