Question

In ABCD, if BC BD, m/B = (13x-35), m2C = (5x-19), and m2D = (2x+14)", find x and the

measure
of each angle.

Ignore the writing I need 12,13, and 14 but if you want you can just do one

Answer 1

The values of x, the measures of the angles and the side lengths are;

12. x = 26

m∠P = 116°

m∠Q = 21°

m∠R = 43°

13. m∠J = 45°

m∠K = 83°

m∠L = 52°

14. x = 11

m∠B = 108°

m∠C = 36°

m∠D = 36°

15. x = 4

WX = 25

XY = 17

WY = 25

The steps used to find the above value of x, measures of the angles, and side lengths are presented as follows;

12. m∠P = 5·x - 14, m∠Q = x - 5, and m∠R = 2·x - 9, we get;

5·x - 14 + (x - 5) + (2·x - 9) = 180° (Angle sum property in a triangle)

5·x - 14 + (x - 5) + (2·x - 9) = 8·x - 14

Therefore; 8·x - 28 = 180°

8·x= 180 + 28

x = (180 + 28)/8

x = 26

m∠P is; 5 × 26 - 14 = 116°

m∠Q is; 26 - 5 = 21°

m∠R is 2 × 26 - 9 = 43°

13. m∠J = m∠L - 7

m∠K = 2 × m∠L - 21

Therefore; m∠L + m∠L - 7 + 2 × m∠L - 21 = 180°

4 × m∠L - 28 = 180°

m∠L = (180 + 28)/4°

m∠L = 52°

m∠J is; (52 - 7) = 45°

m∠K is; 2 × 52 - 21 = 83°

14. Triangle ΔBCD has a pair of congruent sides,
`\overline{BC}` and
`\overline{BD}`, therefore, ΔBCD is an isosceles triangle, and ∠C ≅ ∠D

m∠C = m∠D (Definition of congruent angles)

5·x - 19 = 2·x + 14

5·x - 2·x = 14 + 19

3·x = 33

x = 33/3

33/3 = 11

x = 11

m∠B is; 13 × 11 - 35 = 108°

m∠C is; 5 × 11 - 19 = 36°

m∠D is; 2 × 11 + 14 = 36°

15. Triangle ΔWXY has a pair of congruent interior angles, therefore, triangle ΔWXY is an isosceles triangle (Definition of isosceles triangles)

`\overline{WX}` ≅
`\overline{WY}` (legs of an isosceles triangle)

WX = WY (Definition of congruent segments)

9·x - 11 = 7·x - 3

9·x - 7·x = 11 - 3

2·x = 8

x = 8/2

x = 4

The measure of WX is 9 × 4 - 11 = 25

The measure of WY is 7 × 4 - 3 = 25

The measure of XY is; 4 × 4 + 1 = 17