# ALGEBRA AND ANGLE MEASURES

Algebra can be used to find unknown values in angles using appropriate theorems and postulates in geometry.

Example 1 :

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

Solution :

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

m∠1  =  m∠2

3x + 15  =  4x - 5

Subtract 3x from each side.

15  =  x - 5

20  =  x

m∠2 and m∠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

Example 2 :

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

Solution :

(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

Example 3 :

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

Solution :

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

Example 4 :

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

Solution :

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

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

Example 5 :

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

Solution :

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

Example 6 :

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

Solution :

3x° and (5x - 20)° are alternate interior angles and they are equal.

3x°  =  (5x - 20)°

3x  =  5x - 20

Subtract 3x from each side.

0  =  2x - 20

20  =  2x

Divide each side by 2.

10  =  x

By Triangle Angle Sum Theorem,

(5x - 20)° + 2y° + 4y°  =  180°

5x - 20 + 2y + 4y  =  180

5x - 20 + 6y  =  180

Substitute x = 10.

5(10) - 20 + 6y  =  180

50 - 20 + 6y  =  180

30 + 6y  =  180

Subtract 30 from each side.

6y  =  150

Divide each side by 6.

y  =  25

Therefore,

x  =  10  and  y  =  25

Example 7 :

In the figure shown below, find the value of x.

Solution :

Since the inscribed angle ∠WXY intercepts the diameter, it is a right angle.

m∠WXY  =  90°

(13x - 1)°  =  90°

13x - 1  =  90

13x  =  91

Divide each side by 13.

x  =  7

Example 8 :

In the figure shown below, find the value of x.

Solution :

The measure of an inscribed angle is equal to half of the measure of its intercepted arc.

m∠ONM  =  (1/2) ⋅ m∠arc OLM

m∠ONM  =  (1/2) ⋅ (91° + 135°)

m∠ONM  =  (1/2) ⋅ 226°

m∠ONM  =  113°

Since the quadrilateral LMNO is inscribed in a circle, its opposite angles are supplementary.

m∠OLM + m∠ONM  =  180°

Substitute.

(15x - 23)° + 113°  =  180°

15x - 23 + 113  =  180

15x + 90  =  180

Subtract 90 from each side.

15x  =  90

Divide each side by 15.

x  =  6

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