Parallel Lines and Transversals-Geometry-Solved Examples

Solved Examples and Worksheet for Parallel Lines and Transversals

Q1Find the value of

$x$ , if

$y$ = 35° and

$b$ = 70°.

A. 75°
B. 205°
C. 145°
D. 110°

Solutions

Step: 1

$a$ + 35° = 70°

[Alternate Interior Angles Theorem.]

Step: 2

$a$ = 35°

[Simplify.]

Step: 3

$a$ +

$x$ = 180°

[Same-Side Interior Angles Theorem.]

Step: 4

35° +

$x$ = 180°

[Substitute.]

Step: 5

$x$ = 145°

[Simplify.]

Correct Answer is : 145°

$m$

$∠$ ECB = 140°, find

$m$

$∠$ ABC.

A. 40°
B. 140°
C. 50°
D. 320°

Solutions

Step: 1

$m$

$∠$ ECD = 180°

[

$∠$ ECD is a straight angle.]

Step: 2

$m$

$∠$ ECB +

$m$

$∠$ BCD = 180°

[Angle Addition Postulate.]

Step: 3

140° +

$m$

$∠$ BCD = 180°

[Substitute.]

Step: 4

$m$

$∠$ BCD = 40°

[Simplify.]

Step: 5

$m$

$∠$ ABC =

$m$

$∠$ BCD

[Alternate Interior Angles Theorem.]

Step: 6

$m$

$∠$ ABC = 40°

[Substitute.]

Correct Answer is : 40°

Q3In the following figure, two parallel lines are cut by a transversal. Find

$x$ .

A. 13
B. 52
C. 106
D. 117
E. 128

Solutions

Step: 1

The two labeled angles are supplementary angles and hence their sum is equal to 180^o.

Step: 2

$x$ + 9

$x$ + 11 = 180

Step: 3

$x$ + 11 = 180

Step: 4

$x$ = 169

$⇒$

$x$ = 13

Correct Answer is : 13

Q4If

$p$ and

$q$ are parallel and

$m$

$∠$

$A$ = 125, find the

$m$

$∠$ 1 and

$m$

$∠$ 2.

$m$

$∠$ 1 = 250 and

$m$

$∠$ 2 = 62
B.

$m$

$∠$ 1 = 55 and

$m$

$∠$ 2 = 125
C.

$m$

$∠$ 1 = 125 and

$m$

$∠$ 2 = 55
D.

$m$

$∠$ 1 = 235 and

$m$

$∠$ 2 = 125

Solutions

Step: 1

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

[Corresponding angles postulate.]

Step: 2

$∠$ 1 = 125°

[Corresponding angles.]

Step: 3

$∠$ 1 +

$∠$ 2 = 180°

[Linear Pair.]

Step: 4

125° +

$∠$ 2 = 180°

Step: 5

$∠$ 2 = 180°- 125°

Step: 6

$∠$ 2 = 55°

Step: 7

Hence

$m$

$∠$ 1 = 125 and

$m$

$∠$ 2 = 55

Correct Answer is :

$m$

$∠$ 1 = 125 and

$m$

$∠$ 2 = 55

Q5Find the value of

$x$ .

A. 140
B. 120
C. 110
D. 60
E. 100

Solutions

Step: 1

From the figure,

$x$ ° = 2

$y$ °

$⇒$

$12$

$x$ ° =

$y$ °

[Alternate angles.]

Step: 2

From the figure,

$x$ ° +

$y$ ° = 180°

[Supplementary angles.]

Step: 3

$x$ ° +

$y$ ° = 180°

$⇒$

$x$ ° +

$12$

$x$ ° = 180°

[From step 1 and step 2.]

Step: 4

$⇒$

$x$ ° = 120°.

[Solve for

$x$ .]

Correct Answer is : 120

Q6Two parallel lines L₁ and L₂ are cut by a transversal T. Use the figure to find how are

$∠$ 4 and

$∠$ 5 related.

A. complementary.
B. congruent.
C. supplementary.
D. none of the above

Solutions

Step: 1

$∠$ 4 +

$∠$ 5 = 180^o

[Same-Side Interior Angles Theorem.]

Step: 2

$∠$ 4 and

$∠$ 5 are Supplementary.

[Step 1.]

Correct Answer is : supplementary.

Q7Find the measures of

$∠$ 1 and

$∠$ 2, if

$m$

$∠$ A equals 135.

$m$

$∠$ 1 = 45 and

$m$

$∠$ 2 = 135
B.

$m$

$∠$ 1 = 270 and

$m$

$∠$ 2 = 67
C.

$m$

$∠$ 1 = 225 and

$m$

$∠$ 2 = 135
D.

$m$

$∠$ 1 = 135 and

$m$

$∠$ 2 = 45

Solutions

Step: 1

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

[Corresponding angles postulate.]

Step: 2

$∠$ 1 = 135°

[Corresponding angles.]

Step: 3

$∠$ 1 +

$∠$ 2 = 180°

[Adjacent angles.]

Step: 4

135° +

$∠$ 2 = 180°

Step: 5

$∠$ 2 = 180°- 135°

Step: 6

$∠$ 2 = 45°

Step: 7

Hence

$m$

$∠$ 1 = 135 and

$m$

$∠$ 2 = 45

Correct Answer is :

$m$

$∠$ 1 = 135 and

$m$

$∠$ 2 = 45

$m$

$∠$ ECB = 130, find

$m$

$∠$ ABC.

A. 310
B. 130
C. 50
D. 40

Solutions

Step: 1

$m$

$∠$ ECD = 180

[

$∠$ ECD is a straight angle.]

Step: 2

$m$

$∠$ ECB +

$m$

$∠$ BCD = 180

[Angle Addition Postulate.]

Step: 3

130 +

$m$

$∠$ BCD = 180

[Substitute.]

Step: 4

$m$

$∠$ BCD = 50

[Simplify.]

Step: 5

$m$

$∠$ ABC =

$m$

$∠$ BCD

[Alternate Interior Angles Theorem.]

Step: 6

$m$

$∠$ ABC = 50

[Substitute.]

Correct Answer is : 50

Q9Find the value of

$x$ .

A. 90°
B. 30°
C. 60°
D. 45°

Solutions

Step: 1

In a parallelogram adjacent angles are supplementary .

Step: 2

$x$ + 3

$x$ = 180°

[Adjacent angles.]

Step: 3

$x$ = 180°

$⇒$

$x$ = 45°

[Simplify.]

Correct Answer is : 45°

Q10

$m$

$∠$

$E$

$C$

$B$ = 160, find

$m$

$∠$

$A$

$B$

$C$ .

A. 340
B. 160
C. 20
D. 115

Solutions

Step: 1

$m$

$∠$ ECD = 180

[

$∠$ ECD is a straight angle.]

Step: 2

$m$

$∠$ ECB +

$m$

$∠$ BCD = 180

[Angle Addition Postulate.]

Step: 3

160 +

$m$

$∠$ BCD = 180

[Substitute.]

Step: 4

$m$

$∠$ BCD = 20

[Simplify.]

Step: 5

$m$

$∠$ ABC =

$m$

$∠$ BCD

[Alternate Interior Angles Theorem.]

Step: 6

$m$

$∠$ ABC = 20

[Substitute.]

Correct Answer is : 20

Q11Find the

$m$

$∠$ 1 and

$m$

$∠$ 2 in the figure shown.

$m$ ∠1 = 60 and

$m$ ∠2 = 120
B.

$m$ ∠1 = 100 and

$m$ ∠2 = 60
C.

$m$ ∠1 = 120 and

$m$ ∠2 = 60
D.

$m$ ∠1 = 120 and

$m$ ∠2 = 80

Solutions

Step: 1

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

[

$AB¯$ and

$DC¯$ are parallel lines cut by a transversal

$AD¯$ .]

Step: 2

∠1 + 120° = 180°

[Same - side interior angles theorm.]

Step: 3

∠1 = 180° - 120° = 60°

[Simplify.]

Step: 4

$m$ ∠1 +

$m$ ∠2 = 180

[Same - side interior angles theorm.]

Step: 5

60 +

$m$ ∠2 = 180

[From step 3.]

Step: 6

$m$ ∠2 = 180 - 60 = 120

[Simplify.]

Step: 7

So,

$m$ ∠1 = 60 and

$m$ ∠2 = 120

Correct Answer is :

$m$ ∠1 = 60 and

$m$ ∠2 = 120

Q12The lines

$j$ and

$k$ are parallel. Find the values of

$x$ and

$y$ in the figure shown. [

$a$ = 10,

$b$ = 2.]

$x$ = 15 and

$y$ = 17
B.

$x$ = 15 and

$y$ = 150
C.

$x$ = 15 and

$y$ = 30
D.

$x$ = 7 and

$y$ = 14

Solutions

Step: 1

$x$ ° + 2

$x$ ° = 180°

[Adjacent angles.]

Step: 2

$x$ ° = 180°

Step: 3

$x$ ° =

$180°12$ = 15°

Step: 4

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

[Corresponding angles postulate .]

Step: 5

$y$ = (2

$x$ )

[From step 4.]

Step: 6

$y$ = 2

$15$ = 30

Step: 7

So,

$x$ = 15 and

$y$ = 30

Correct Answer is :

$x$ = 15 and

$y$ = 30

Q13The lines

$l$ and

$m$ are parallel. Find the values of

$x$ and

$y$ in the figure shown.

$x$ = 6 and

$y$ = 18
B.

$x$ = 12 and

$y$ = 4
C.

$x$ = 4 and

$y$ = 8
D.

$x$ = 4 and

$y$ = 12

Solutions

Step: 1

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

[Corresponding angles postulate.]

Step: 2

$y$ ° = (21

$x$ + 5

$y$ )°

[Corresponding angles.]

Step: 3

$y$ = 21

$x$

$⇒$

$y$ = 3

$x$

[Simplify.]

Step: 4

$y$ ° + (6

$x$ +

$y$ )° = 180°

[Adjacent angles.]

Step: 5

$y$ + (6

$x$ ) = 180°

[Simplify.]

Step: 6

13(3

$x$ ) + 6

$x$ = 180

$⇒$ 45

$x$ = 180

[Substitute

$y$ = 3

$x$ and simplify.]

Step: 7

$x$ =

$18045$

$⇒$

$x$ = 4

[Simplify.]

Step: 8

$y$ = 3

$4$ = 12

[From step 3.]

Step: 9

So, the values of

$x$ and

$y$ are 4 and 12.

Correct Answer is :

$x$ = 4 and

$y$ = 12

Q14The lines

$p$ and

$q$ are parallel. Find the values of

$x$ .

$x$ = 40
B.

$x$ = 60
C.

$x$ = 50
D.

$x$ = 30

Solutions

Step: 1

From the data, one of the large angles is (3

$x$ +

$y$ )°.

Step: 2

From the data, two of the small angles are 40° and (3

$x$ -

$y$ )°.

Step: 3

If a transversal intersects two or more parallel lines, all small angles are equal.
Therefore, 3

$x$ -

$y$ = 40 ----

$1$

Step: 4

If a transversal intersects two or more parallel lines, then sum of small angle and large angle equals straight angle.
Therefore, 3

$x$ +

$y$ + 40 = 180
3

$x$ +

$y$ = 140 ----

$2$

Step: 5

Now, add equation

$1$ and equation

$2$ and solve for

$x$ .
6

$x$ = 180

$x$ = 30

Correct Answer is :

$x$ = 30

Q15From the given figure If

$∠$

$E$

$C$

$B$ = 150, then

$∠$

$A$

$B$

$C$ = _________ .

A. 30
B. 150
C. 330
D. 105

Solutions

Step: 1

$∠$ ECD = 180

[

$∠$ ECD is a straight angle.]

Step: 2

$∠$ ECB + m

$∠$ BCD = 180

[Angle Addition Postulate.]

Step: 3

150 + m

$∠$ BCD = 180

[Substitute.]

Step: 4

$∠$ BCD = 30

[Simplify.]

Step: 5

$∠$ ABC = m

$∠$ BCD

[Alternate Interior Angles Theorem.]

Step: 6

$∠$ ABC = 30

[Substitute.]

Correct Answer is : 30

Solved Examples and Worksheet for Parallel Lines and Transversals

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