Question

given: ∠a = 6x + 2
<b = 4x - 2
and ∠a and ∠b are complementary
find: the measure of each angle

given: ∠a = 2x + 4
<b = 3x - 16
and ∠a and ∠b are corresponding angles
find: the measure of each angle

given: ∠a = 4x + 5
<b = x - 10
and ∠a and ∠b are consecutive interior angles
find: the measure of each angle

Explanation:

Step1: Use the property of complementary angles

Complementary angles sum to \(90^{\circ}\). So, \(\angle A+\angle B = 90^{\circ}\). Substitute \(\angle A = 6x + 2\) and \(\angle B=4x - 2\) into the equation:
\((6x + 2)+(4x - 2)=90\)
Simplify the left - hand side:
\(6x+2 + 4x-2=90\)
\(10x=90\)
Solve for \(x\):
\(x = 9\)
Then \(\angle A=6x + 2=6\times9+2=56^{\circ}\) and \(\angle B=4x - 2=4\times9-2 = 34^{\circ}\)

Step2: Use the property of corresponding angles

Corresponding angles are equal. So, \(\angle A=\angle B\). Substitute \(\angle A = 2x + 4\) and \(\angle B=3x - 16\) into the equation:
\(2x + 4=3x - 16\)
Solve for \(x\):
\(3x-2x=4 + 16\)
\(x = 20\)
Then \(\angle A=2x + 4=2\times20+4=44^{\circ}\) and \(\angle B=3x - 16=3\times20-16 = 44^{\circ}\)

Step3: Use the property of consecutive interior angles

Consecutive interior angles sum to \(180^{\circ}\). So, \(\angle A+\angle B = 180^{\circ}\). Substitute \(\angle A = 4x + 5\) and \(\angle B=x - 10\) into the equation:
\((4x + 5)+(x - 10)=180\)
Simplify the left - hand side:
\(4x+5+x - 10=180\)
\(5x-5 = 180\)
\(5x=180 + 5\)
\(5x=185\)
Solve for \(x\):
\(x = 37\)
Then \(\angle A=4x + 5=4\times37+5=153^{\circ}\) and \(\angle B=x - 10=37-10 = 27^{\circ}\)

Answer:

For the first pair of angles: \(\angle A = 56^{\circ}\), \(\angle B=34^{\circ}\)
For the second pair of angles: \(\angle A = 44^{\circ}\), \(\angle B=44^{\circ}\)
For the third pair of angles: \(\angle A = 153^{\circ}\), \(\angle B=27^{\circ}\)