ANGLES AND PARALLEL LINES WORKSHEET

1. In the figure shown below, m∠2 = 75°. Find the measures of the remaining angles.

2. In the figure shown below, m∠3 = 102°. Find the measure of each angle. Tell which postulate(s) or theorem(s) you used.

a. ∠5       b. ∠6       c. ∠11       d. ∠7       e. ∠15       f. ∠14

3. In the figure shown below, m∠2 = 92° and m∠12 = 74°. Find the measure of each angle. Tell which postulate(s) or theorem(s) you used.

a. ∠10       b. ∠8       c. ∠9       d. ∠5       e. ∠11       f. ∠13

4. If m∠1 = 3x + 15, m∠2 = 4x - 5, and m∠3 = 5y, find the values of x and y.

5. In the figure shown below, find the values of x and y.

6. In the figure shown below, find the values of x, y and z.

7. In the figure shown below, find the values of x and y.

8. Using a 3rd parallel Line – Auxiliary Line, find the value of x. 

9. Using a 3rd parallel Line – Auxiliary Line, find the value of x. 

10. A diagonal brace strengthens the wire fence and prevents it from sagging. The brace makes a 50° angle with the wire as shown. Find the value of the variable.

Answers

1. Answer :

∠1 and ∠2 form a linear pair and they are supplementary.

m∠1 + m∠2 = 180°

m∠1 + 75° = 180°

m∠1 = 105°

∠1 and ∠3 are vertically opposite angles and they are equal.

m∠3 = m∠1

m∠3 = 105°

∠2 and ∠4 are vertically opposite angles and they are equal.

m∠4 = m∠2

m∠4 = 75°

∠1 and ∠5 are corresponding angles and they are equal.

m∠5 = m∠1

m∠5 = 105°

∠2 and ∠6 are corresponding angles and they are equal.

m∠6 = m∠2

m∠6 = 75°

∠4 and ∠8 are corresponding angles and they are equal.

m∠8 = m∠4

m∠8 = 75°

∠3 and ∠7 are corresponding angles and they are equal.

m∠7 = m∠3

m∠7 = 105°

2. Answer :

(a) :

m∠5 = m∠3

m∠5 = 102°

(Alternate Interior Angles Theorem)

(b) :

m∠3 + m∠6 = 180°

102° + m∠6 = 180°

m∠6 = 78°

(Interior Angles on the Same Side of the Transversal Theorem)

(c) :

m∠11 = m∠3

m∠11 = 102°

(Corresponding Angles Postulate)

(d) :

m∠7 = m∠3

m∠7 = 102°

(Corresponding Angles Postulate)

(e) :

m∠15 = m∠7

m∠15 = 102°

(Corresponding Angles Postulate)

(f) :

m∠14 = m∠6

m∠14 = 78°

(Corresponding Angles Postulate)

3. Answer :

(a) :

m∠10 = m∠2

m∠10 = 92°

(Corresponding Angles Postulate)

(b) :

m∠8 = m∠2

m∠8 = 92°

(Vertical Angles Theorem)

(c) :

m∠9 + m∠10 = 180°

m∠9 + 92° = 180°

m∠9 = 88°

(Linear Pair Postulate)

(d) :

m∠5 + m∠12 = 180°

m∠5 + 74° = 180°

m∠5 = 106°

(Interior Angles on the Same Side of the Transversal Theorem)

(e) :

m∠11 = m∠5

m∠11 = 106°

(Alternate Interior Angles Theorem)

(f) :

m∠13 = m∠11

m∠13 = 106°

(Vertical Angles Theorem)

4. Answer :

∠1 and ∠2 are corresponding angles and they are equal.

m∠1 = m∠2

3x + 15 = 4x - 5

Subtract 3x from each side.

15 = x - 5

Add 5 to each side.

20 = x

∠2 and ∠3 are corresponding angles and they are equal.

m∠2 = m∠3

4x - 5 = 5y

Substitute x = 20.

4(20) - 5 = 5y

80 - 5 = 5y

75 = 5y

Divide each side by 5.

15 = y

Therefore,

x = 20 and y = 15

5. Answer :

(3y + 18)° and 90° are interior angles on the same side of the transversal and they are supplementary.

(3y + 18)° + 90° = 180°

3y + 18 + 90 = 180

3y + 108 = 180

Subtract 108 from each side.

3y = 72

Divide each side by 3.

y = 24

10x° and (15x + 30)° are interior angles on the same side of the transversal and they are supplementary.

10x° + (15x + 30)° = 180°

10x + 15x + 30 = 180

25x + 30 = 180

Subtract 180 from each side.

25x = 150

Divide each side by 25.

x = 6

Therefore,

x = 6 and y = 24

6. Answer :

2x°, 90° and x° together form a straight angle.

2x° + 90° + x° = 180°

3x + 90 = 180

Subtract 90 from each side.

3x = 90

Divide each side by 3.

x = 30

x° and 2y° are alternate interior angles and they are equal.

2y° = x°

2y = x

Substitute x = 30.

2y = 30

Divide each side by 2.

y = 15

2y° and z° form a linear pair, they are supplementary.

2y° + z° = 180°

2y + z = 180

Substitute y = 15.

2(15) + z = 180

30 + z = 180

Subtract 30 from each side.

z = 150

Therefore,

x = 30, y = 15 and z = 150

7. Answer :

Mark a new angle a°.

a° and (5y - 4)° form a linear pair.

a° + (5y - 4)° = 180°

a° and 3y° are corresponding angles, then  a° = 3y°.

3y° + (5y - 4)° = 180°

3y + 5y - 4 = 180

8y - 4 = 180

Add 4 to each side.

8y = 184

Divide each side by 8.

y = 23

3y° and (2x + 13)° are corresponding angles and they are equal.

(2x + 13)° = 3

2x + 13 = 3y

Substitute y = 23.

2x + 13 = 3(23)

2x + 13 = 69

Subtract 13 from each side.

2x = 56

Divide each side by 2.

x = 28

8. Answer :

In the figure above, a° and 50° are corresponding angles and they are equal.

a° = 50°

b° and 100° are interior angles on the same side of the transversal and they are supplementary.

b° + 100° = 180°

Subtract 100° from each side.

b° = 80°

In the above figure,

x = a + b

50 + 80

= 130

9. Answer :

In the figure above, a° and 62° are alternate interior angles and they are equal.

a° = 62°

b° and 144° are interior angles on the same side of the transversal and they are supplementary.

b° + 144° = 180°

Subtract 144° from each side.

b° = 36°

In the above figure,

x = a + b

= 62 + 36

= 98

10. Answer :

Mark a new angle x°.

x° and y° form a linear pair.

x° + y° = 180°

x° and 50° are corresponding angles, then  x° = 50°.

50° + y° = 180°

50 + y = 180

Subtract 50 from each side.

y = 130

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