﻿ Parallel Lines and Transversals Worksheet | Problems & Solutions

# Parallel Lines and Transversals Worksheet

Parallel Lines and Transversals Worksheet
• Page 1
1.
The lines $j$ and $k$ are parallel. Find the values of $x$ and $y$ in the figure shown.

 a. $x$ = 36 and $y$ = 72 b. $x$ = 36 and $y$ = 108 c. $x$ = 72 and $y$ = 36 d. $x$ = 72 and $y$ = 108

#### Solution:

2x° + 3x° = 180°

5x° = 180°

x° = 180° / 5 = 36°

If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
[Corresponding angles postulate .]

y = (2x)
[From step 4.]

y = 2(36) = 72

So, x = 36 and y = 72

2.
The lines $l$ and $m$ are parallel. Find the values of $x$ and $y$ in the figure shown.

 a. $x$ = 10 and $y$ = 5 b. $x$ = 5 and $y$ = 20 c. $x$ = 10 and $y$ = 20 d. $x$ = 5 and $y$ = 10

#### Solution:

If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
[Corresponding angles postulate.]

13y° = (20x + 3y
[Corresponding angles.]

10y = 20x y = 2x
[Simplify.]

13y° + (8x + y)° = 180°

14y + (8x) = 180°
[Simplify.]

14(2x) + 8x = 180 36x = 180
[Substitute y = 2x and simplify.]

x = 180 / 36 x = 5
[Simplify.]

y = 2(5) = 10
[From step 3.]

So, the values of x and y are 5 and 10.

3.
If $p$ and $q$ are parallel and $m$$\angle$$A$ = 100, find the $m$$\angle$1 and $m$$\angle$2.

 a. $m$$\angle$1 = 80 and $m$$\angle$2 = 100 b. $m$$\angle$1 = 100 and $m$$\angle$2 = 80 c. $m$$\angle$1 = 200 and $m$$\angle$2 = 50 d. $m$$\angle$1 = 260 and $m$$\angle$2 = 100

#### Solution:

If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
[Corresponding angles postulate.]

1 = 100°
[Corresponding angles.]

1 + 2 = 180°

100° + 2 = 180°

2 = 180°- 100°

2 = 80°

Hence m1 = 100 and m2 = 80

4.
Find the values of $\angle$1 and $\angle$2 in the figure.

 a. $\angle$1 = 60° and $\angle$2 = 100° b. $\angle$1 = 60° and $\angle$2 = 80° c. $\angle$1 = 100° and $\angle$2 = 60° d. $\angle$1 = 80° and $\angle$2 = 100°

#### Solution:

MNO + 60° = 180°

MNO = 180° - 60° = 120°
[Simplify.]

If two parallel lines are cut by a transversal, then corresponding angles are congruent.
[Alternate interior angle theorem.]

2 = 60°
[MN and OP are parallel lines cut by a transversal NO.]

If two parallel lines are cut by a transversal, then the pairs of Same-side interior angles are supplementary.
[Same-side interior angles theorem.]

1 + 80° = 180°
[MN and OP are parallel lines cut by a transversal MP.]

1 = 180° - 80° = 100°
[Simplify.]

Hence 1 = 60° and 2 = 100°

5.
Find the values of $m$$\angle$1 and $m$$\angle$2 in the figure.

 a. $m$$\angle$1 = 120 and $m$$\angle$2 = 60 b. $m$$\angle$1 = 80 and $m$$\angle$2 = 100 c. $m$$\angle$1 = 80 and $m$$\angle$2 = 80 d. $m$$\angle$1 = 100 and $m$$\angle$2 = 100

#### Solution:

If two parallel lines are cut by a transversal, then the pairs of Same-side interior angles are supplementary.
[Same-side interior angles theorem.]

1 + 100° = 180°
[AB and DC are parallel lines cut by a transversal AD.]

1 = 180° - 100° = 80°
[Simplify.]

m1 + m2 = 180
[Alternate interior angles.]

80 + m2 = 180
[From step 3.]

m2 = 180 - 80 = 100
[Simplify.]

So, m1 = 80 and m2 = 100

6.
The lines $p$ and $q$ are parallel. Find the values of $x$ and $y$.

 a. $x$ = 16 and $y$ = 20 b. $x$ = 32 and $y$ = 20 c. $x$ = 16 and $y$ = 80 d. $x$ = 20 and $y$ = 60

#### Solution:

4x° + (6x + y)° = 180°

10x + y = 180 ---------- (1)
[Simplify.]

If two parallel lines are cut by a transversal, then the corresponding angles are congruent.
[Corresponding angles postulate.]

(6x + y)° = (x + 5y
[Corresponding angles.]

5x - 4y = 0 ---------- (2)
[Simplify.]

Solving equations (1) and (2), 45x = 720
[From steps (2) and (5).]

x = 16

5(16) - 4y = 0
[Substitute the value of x in equation (2).]

y = 20

So, the values of x and y are 16 and 20.

7.
The lines $a$ and $b$ are parallel. Find the $m$$\angle$M and $m$$\angle$P.

 a. $m$$\angle$M = 135 and $m$$\angle$N = 1150 b. $m$$\angle$M = 120 and $m$$\angle$N = 100 c. $m$$\angle$M = 100 and $m$$\angle$N = 80 d. $m$$\angle$M = 130 and $m$$\angle$N = 110

#### Solution:

If two parallel lines are cut by a transversal, then the pairs of Same-side interior angles are supplementary.
[Same-side interior angles theorem.]

a° + (a - 20)° = 180°
[Same-side interior angles.]

2a = 200 a = 100
[Simplify.]

M = a° = 100°
[From step 3.]

P = (a - 20)° = (100 - 20)° = 80°
[From the figure.]

So, mM = 100 and mP = 80.

8.
Find the measures of $\angle$A and $\angle$B if the lines AB and CD are parallel to each other.

 a. $m$ $\angle$A = 115 and $m$ $\angle$B = 65 b. $m$ $\angle$A = 55 and $m$ $\angle$B = 35 c. $m$ $\angle$A = 70 and $m$ $\angle$B = 35 d. $m$ $\angle$A = 15 and $m$ $\angle$B = 165

#### Solution:

If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
[Alternate interior angle theorem.]

B = D = x°
[Alternate interior angles.]

COD = AOB =90°
[Vertically opposite angles.]

(2x - 15)° + 90° + x = 180
[Sum of the angles in a triangle is 180°.]

3x = 105 x = 35
[Simplify.]

A = (2x - 15)
[From the figure.]

2(35) - 15 = 55°

mB = x = 35

So, the mA = 55 and mB = 35

9.
Find the value of $x$.

 a. 60 b. 50 c. 45 d. 40

#### Solution:

If two parallel lines are cut by a transversal, then corresponding angles are congruent.
[Corresponding angles postulate.]

QPR = SRT = 40°
[Corresponding angles.]

40° + 90° + x° = 180°
[Sum of the angles in a triangle is 180°.]

x° = 50°
[Simplify.]

So, the value of x is 50.

10.
If $m$$\angle$2 = $m$$\angle$7 and line $l$ is parallel to line $m$, then which of the following is true?
1. Line $n$ is perpendicular to lines $l$ and $m$.
2. All the angles 1 to 8 are equal.

 a. 1 only b. 1 or 2 c. 1 and 2 d. 2 only

#### Solution:

2 = 6
[l || m, corresponding angles are equal.]

m6 + m7 = 180

m2 = m7
[As per the question.]

m6 = m7 = 90
[Steps 1 and 2.]

Line n is perpendicular to line m.
[Step 4.]

m5 = m8 = 90 = m6 = m7
[Step 5.]

Line n is perpendicular to line l.
[Line l and m are parallel.]

m1 = m2 = 90 = m3 = m4
[Step 7.]

So, both the statements are correct.