Question

1. in the figure below, a || b, m∠1 = (9x - 4)° and m∠2 = (13x - 32)°. what is m∠3? a. 54° b. 59° c. 121° d. 126°

Explanation:

Step1: Identify Angle Relationship

Since \(a \parallel b\), \(\angle 1\) and \(\angle 2\) are alternate interior angles, so \(m\angle 1 = m\angle 2\).
Set up the equation: \(9x - 4 = 13x - 32\).

Step2: Solve for \(x\)

Subtract \(9x\) from both sides: \(-4 = 4x - 32\).
Add 32 to both sides: \(28 = 4x\).
Divide by 4: \(x = 7\).

Step3: Find \(m\angle 1\) (or \(m\angle 2\))

Substitute \(x = 7\) into \(m\angle 1 = 9x - 4\): \(m\angle 1 = 9(7) - 4 = 63 - 4 = 59^\circ\).

Step4: Find \(m\angle 3\)

\(\angle 1\) and \(\angle 3\) are supplementary (linear pair), so \(m\angle 3 = 180^\circ - m\angle 1 = 180 - 59 = 121^\circ\)? Wait, no—wait, \(\angle 2\) and \(\angle 3\) are supplementary? Wait, no, let's re - check. Wait, when \(a\parallel b\), \(\angle 1=\angle 2\) (alternate interior). Then \(\angle 2\) and \(\angle 3\) are supplementary? Wait, no, in the diagram, \(\angle 2\) and \(\angle 3\) are adjacent and form a linear pair? Wait, no, maybe I made a mistake. Wait, let's recalculate \(m\angle 2\): \(13x - 32\) with \(x = 7\) is \(13(7)-32=91 - 32 = 59^\circ\). Then \(\angle 2\) and \(\angle 3\) are supplementary? Wait, no, if \(\angle 1=\angle 2 = 59^\circ\), and \(\angle 3\) is supplementary to \(\angle 2\)? Wait, no, maybe the relationship is that \(\angle 1\) and \(\angle 3\) are same - side interior angles? Wait, no, let's look at the diagram again. The transversal cuts \(a\) and \(b\), \(\angle 1\) and \(\angle 2\) are alternate interior, so they are equal. Then \(\angle 2\) and \(\angle 3\) are vertical angles? No, no. Wait, maybe I messed up the angle relationship. Wait, let's start over.

Given \(a\parallel b\), \(\angle 1\) and \(\angle 2\) are alternate interior angles, so \(9x - 4=13x - 32\). Solving: \(13x-9x=32 - 4\), \(4x = 28\), \(x = 7\). Then \(m\angle 1=9(7)-4 = 59^\circ\), \(m\angle 2=13(7)-32 = 59^\circ\). Now, \(\angle 2\) and \(\angle 3\) are supplementary (since they form a linear pair), so \(m\angle 3=180 - 59=121^\circ\)? But wait, the options have 121 as option C. Wait, but earlier I thought I made a mistake, but let's check the calculation again.

Wait, \(9x - 4=13x - 32\)

\(13x-9x = 32 - 4\)

\(4x=28\)

\(x = 7\)

\(m\angle 1=9\times7 - 4=63 - 4 = 59\)

\(m\angle 3=180 - 59 = 121\)

Yes, that's correct.

Answer:

C. \(121^\circ\)