Question

Exercise 19

Find the values of x and y. Then find the measures of the interior angles of the polygon.

Answer 1

x = 25, y = 5

The interior angles:

<ABC = 75°

<BCD = 50°

<CDA = 105°

<DAC = 130

Explanation:

The question has no attachment. See attachment for complete question.

Polygon ABCD is a cyclic quadrilateral.

For cyclic quadrilaterals, their opposite angles add up to 180°.

Meaning that;

2x + 26y = 180 ---- equation 1

and

3x + 21y = 180 ------ equation 2

These will be solved simultaneously..

Multiply ½ to both sides of equation 1

½(2x + 26y) = ½ * 180

½ * 2x + ½ * 26y = ½ * 180

x + 13y = 90

Make x the subject of formula

x = 90 - 13y

Multiply both sides of equation 2 by ⅓

⅓(3x + 21y) = ⅓ * 180

⅓ * 3x + ⅓ * 21y = ⅓ * 180

x + 7y = 60

Recall that x = 90 - 13y.

So, x + 7y = 60 becomes

90 - 13y + 7y = 60

90 - 6y = 60

Collect like terms

-6y = 60 - 90

-6y = -30

Divide both sides by -6

-6y/-6 = -30/-6

y = 5

Recall that x = 90 - 13y.

So, x = 90 - 13(5)

x = 90 - 65

x = 25

Having calculated x and y, the interior angles can then be calculated.

The interior angles of the polygon are

1. <ABC

2. <BCD

3. <CDA

4. <DAC

Recall that x = 25 and y = 5

<ABC = 3x

<ABC = 3 * 25

<ABC = 75°

<BCD = 2x

<BCD = 2 * 25

<BCD = 50°

<CDA = 21y

<CDA = 21 * 5

<CDA = 105°

<DAC = 26y

<DAC = 26 * 5

<DAC = 130

Hence,

x = 25, y = 5

The interior angles:

<ABC = 75°

<BCD = 50°

<CDA = 105°

<DAC = 130