The line BD bisects the angle ABC solve for x then find the

The line BD bisects the angle ABC, solve for x then find the degree measure of the 2 angles.

1 a)

Step 1: Given the line BD bisects angle ABC

Step 2: This means Angle ABD = Angle DBC

Step 3: Given Angle ABD = 4x-5 and Angle DBC = 2x+9

Step 4: From Step 2, since Angle ABD = Angle DBC

This implies, 4x-5 = 2x+9

Solving this equation for x

4x-2x = 5+9

2x = 14

x=7

Step 5: Angle ABD = 4x-5 = 4*7-5 = 23

Step 6: Angle DBC = 2x+9 = 2*7+9 = 23

Step 7: Final Answer is x=7, Angle ABD = 23 and Angle DBC = 23

1 b)

Step 1: Given the line BD bisects angle ABC

Step 2: This means Angle ABD = Angle DBC

Step 3: Given Angle ABD = 27-4x and Angle DBC = x² + 2x

Step 4: From Step 2, since Angle ABD = Angle DBC

This implies, 27-4x = x² + 2x

x² + 2x + 4x -27 = 0

x² +6x - 27 = 0

x² + 9x - 3x -27 = 0

x(x+9) -3(x + 9) = 0

(x+9)(x-3) = 0

x = -9 or x = 3

Step 5: For x = -9, Solving for Angle ABD = 27 - 4x = 27 - 4(-9) = 63

Step 6: For x=-9, Solving for Angle DBC = x² + 2x = (-9)² + 2(-9) = 63

Step 7: For x=3, Solving for Angle ABD = 27-4x = 27 -4(3) = 15

Step 8: For x=3, Solving for Angle DBC = x² + 2x = 3²⁺ 2*3 = 15

Step 9: Final answer

For x = -9, Angle ABD = 63, Angle DBC = 63

For x=3, Angle ABD = 15, Angle DBC = 15