﻿ Parallel and Perpendicular Lines Worksheet | Problems & Solutions Parallel and Perpendicular Lines Worksheet

Parallel and Perpendicular Lines Worksheet
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1.
$\angle$A and $\angle$B are congruent and supplementary. Prove that $\angle$A and $\angle$B are right angles.

Solution:

A two-column proof can be given as shown.

Given: A and B are congruent and supplementary angles. Prove: A and B are right angles.

2.
What conclusion follows from the following pair of statements?
1. If two non vertical lines are perpendicular, then the product of their slopes is - 1.
2. The product of the slopes of non vertical lines $l$ and $n$ is not - 1. a. $l$ is not perpendicular to $n$ b. $l$ || $n$ c. $l$ $\perp$ $n$

Solution:

Had l been perpendicular to n, product of their slopes would have been -1.

As product of slopes of l and n is not - 1, it follows that l is not perpendicular to n.

3.
$\stackrel{‾}{\mathrm{AB}}$ $\perp$ $\stackrel{‾}{\mathrm{CD}}$ as shown. Prove: $\angle$AOC $\cong$ $\angle$BOC

Solution:

A two-column proof can be given as shown.

Given: AB CD as shown. Prove: AOC BOC

4.
It is given that $m$$\angle$A = $m$$\angle$B. Which of the following is the reason for the statement: $\angle$A $\cong$ $\angle$B? a. transitive Property of congruent b. reflexive Property of congruent c. definition of congruent angles d. substitution Property

Solution:

Two angles are congruent if they have the same measure.
[Definition of congruent angles.]

5.
$\angle$1 and $\angle$2 are complementary angles. Which of the following statements can be used to say: $\angle$1 + $\angle$2 = 90o ? a. definition of complementary angles b. angle addition postulate c. definition of right angle d. definition of supplementary angles

Solution:

Two angles are complementary if the sum of their measures is 90o.
[Definition of complementary angles.]

6.
It is given that $m$$\angle$A = 90 and $m$ $\angle$B = 90. Which of the following is the reason for the statement: $m$$\angle$A = $m$$\angle$B? a. transitive Property b. definition of congruent angles c. symmetric Property d. associative Property

Solution:

If a = b and b = c, then a = c.
[Transitive Property.]

7.
Write the first step of an indirect proof of the following statement:
$\begin{array}{c}\stackrel{↔}{\text{AB}}\end{array}$ and $\begin{array}{c}\stackrel{↔}{\text{CD}}\end{array}$ are perpendicular lines. a. assume $\begin{array}{c}\stackrel{↔}{\text{AB}}\end{array}$ and $\begin{array}{c}\stackrel{↔}{\text{CD}}\end{array}$ are parallel lines b. assume $\begin{array}{c}\stackrel{↔}{\text{AB}}\end{array}$ and $\begin{array}{c}\stackrel{↔}{\text{CD}}\end{array}$ are not perpendicular lines c. assume $\begin{array}{c}\stackrel{↔}{\text{AB}}\end{array}$ and $\begin{array}{c}\stackrel{↔}{\text{CD}}\end{array}$ are skew lines d. assume $\begin{array}{c}\stackrel{↔}{\text{AB}}\end{array}$ and $\begin{array}{c}\stackrel{↔}{\text{CD}}\end{array}$ are perpendicular lines

Solution:

In the first step of an indirect proof, we always assume the opposite of what is to be proved.

So, Assume AB and CD are not perpendicular lines.
[Step 1.]

8.
Find the value of $x$.  a. 15 b. 14 c. 12 d. 10

Solution:

4 xo + 5 xo = 90o

9xo = 90o
[Simplify.]

x = 10
[Solve for x.]

9.
Find the value of $x$.  a. 27 b. 28 c. 26 d. 29

Solution:

(2x + 36)o + 90o = 180o
[Definition of a linear pair.]

(2x + 36)o = 90o

x = 27
[Solve for x.]

10.
Write the first step of an indirect proof of the following statement:
$\angle$A is not a right angle. a. assume $\angle$A is an acute angle b. assume $\angle$A is an obtuse angle c. assume $\angle$A is not a right angle d. assume $\angle$A is a right angle

Solution:

In the first step of an indirect proof, we always assume the opposite of what is to be proved.

So, Assume A is a right angle.
[Step 1.]