Question

1. ∠a and ∠b are vertical angles. if ( mangle a = 5x - 28 ) and ( mangle b = 3x + 4 ), find the measure of each angle.
2. ∠p and ∠q form a linear pair. if ( mangle p = 7x - 4 ) and ( angle q = 2x - 5 ), find the measure of each angle.

Explanation:

Step1: Use the property of vertical angles

Vertical angles are equal. So, \(5x - 28=3x + 4\).

Step2: Solve the equation for \(x\)

Subtract \(3x\) from both sides: \(5x-3x - 28=3x-3x + 4\), which simplifies to \(2x-28 = 4\).
Add \(28\) to both sides: \(2x-28 + 28=4 + 28\), so \(2x=32\).
Divide both sides by \(2\): \(x=\frac{32}{2}=16\).

Step3: Find \(m\angle A\)

Substitute \(x = 16\) into \(m\angle A=5x - 28\): \(m\angle A=5\times16-28=80 - 28=52\).

Step4: Find \(m\angle B\)

Substitute \(x = 16\) into \(m\angle B=3x + 4\): \(m\angle B=3\times16+4=48 + 4=52\).

Step1: Use the property of linear - pair angles

Linear - pair angles are supplementary, so \(m\angle P+m\angle Q = 180^{\circ}\).
\((7x - 4)+(2x - 5)=180\).

Step2: Simplify the left - hand side of the equation

\(7x-4 + 2x-5=180\), \(9x-9 = 180\).

Step3: Solve the equation for \(x\)

Add \(9\) to both sides: \(9x-9 + 9=180 + 9\), so \(9x=189\).
Divide both sides by \(9\): \(x=\frac{189}{9}=21\).

Step4: Find \(m\angle P\)

Substitute \(x = 21\) into \(m\angle P=7x - 4\): \(m\angle P=7\times21-4=147-4 = 143\).

Step5: Find \(m\angle Q\)

Substitute \(x = 21\) into \(m\angle Q=2x - 5\): \(m\angle Q=2\times21-5=42-5 = 37\).

Answer:

\(m\angle A = 52^{\circ}\), \(m\angle B = 52^{\circ}\)

Now for problem 4: