# ANGLES AND PARALLEL LINES WORKSHEET

Problem 1 :

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

Problem 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

Problem 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

Problem 4 :

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

Problem 5 :

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

Problem 6 :

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

Problem 7 :

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

Problem 8 :

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

Problem 9 :

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

Problem 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.

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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