Question

1. solve for ( x ) in each figure. explain your reasoning by classifying the relationship between the pairs of angles.

a.
b.

Explanation:

Part a

Step1: Identify angle relationship

The angles \(72^\circ\) and \((7x + 24)^\circ\) are same - side interior angles. Same - side interior angles are supplementary, meaning their sum is \(180^\circ\). So we set up the equation:
\(7x + 24+72 = 180\)

Step2: Simplify the left - hand side

Combine like terms: \(7x+(24 + 72)=180\), so \(7x+96 = 180\)

Step3: Solve for x

Subtract 96 from both sides: \(7x=180 - 96\)
\(7x = 84\)
Divide both sides by 7: \(x=\frac{84}{7}=12\)

Part b

Step1: Identify angle relationship

The angles \(135^\circ\) and \((2x + 15)^\circ\) are corresponding angles. Corresponding angles are congruent (equal) when two parallel lines are cut by a transversal. So we set up the equation:
\(2x+15 = 135\)

Step2: Solve for x

Subtract 15 from both sides: \(2x=135 - 15\)
\(2x = 120\)
Divide both sides by 2: \(x=\frac{120}{2}=60\)

Answer:

a. \(x = 12\) (The angles are same - side interior angles, so they are supplementary. Solving \(7x + 24+72=180\) gives \(x = 12\))
b. \(x = 60\) (The angles are corresponding angles, so they are congruent. Solving \(2x + 15=135\) gives \(x = 60\))