Question

use the diagram to the right to answer the questions below:
a. if ( mangle 1 = 84^circ ), what must be ( mangle 5 ) in order for ( a parallel b )?

b. if ( mangle 2 = 109^circ ), what must be ( mangle 9 ) in order for ( a parallel c )?

c. if ( mangle 5 = 68^circ ), what must be ( mangle 11 ) in order for ( b parallel c )?

d. if ( mangle 3 = mangle 9 ), what converse proves ( a parallel c )?

e. if ( mangle 8 = mangle 12 ), what converse proves ( b parallel c )?

f. if ( mangle 2 + mangle 5 = 180^circ ), what converse proves ( a parallel b )?

Explanation:

Part a

Step1: Identify Angle Relationship

For \(a \parallel b\), \(\angle 1\) and \(\angle 5\) are corresponding angles. Corresponding angles are equal when lines are parallel.

Step2: Determine \(m\angle 5\)

Since \(\angle 1\) and \(\angle 5\) are corresponding angles, \(m\angle 5 = m\angle 1\). Given \(m\angle 1 = 84^\circ\), so \(m\angle 5 = 84^\circ\).

Part b

Step1: Identify Angle Relationship

For \(a \parallel c\), \(\angle 2\) and \(\angle 9\) are same - side exterior angles? Wait, no. Wait, \(\angle 2\) and \(\angle 9\): Let's re - examine. Actually, \(\angle 2\) and \(\angle 9\) are same - side interior angles? Wait, no, if we consider the transversal, \(\angle 2\) and \(\angle 9\) should be supplementary for \(a\parallel c\)? Wait, no, let's think again. Wait, \(\angle 2\) and \(\angle 9\): If \(a\parallel c\), then \(\angle 2\) and \(\angle 9\) are same - side interior angles? Wait, no, maybe alternate exterior? Wait, no, the correct relationship: If \(a\parallel c\), and the transversal is the line that cuts \(a\) and \(c\), then \(\angle 2\) and \(\angle 9\) are supplementary? Wait, no, let's use the converse of same - side interior angles. Wait, no, the measure of \(\angle 2 = 109^\circ\), and for \(a\parallel c\), \(\angle 2\) and \(\angle 9\) should be supplementary? Wait, no, I think I made a mistake. Wait, \(\angle 2\) and \(\angle 9\): Let's look at the diagram. If \(a\parallel c\), then \(\angle 2\) and \(\angle 9\) are same - side interior angles? Wait, no, the sum of same - side interior angles is \(180^\circ\). Wait, no, \(\angle 2\) and \(\angle 9\): Wait, maybe \(\angle 2\) and \(\angle 9\) are corresponding? No, that can't be. Wait, the correct approach: The sum of \(\angle 2\) and \(\angle 9\) should be \(180^\circ\) if they are same - side interior angles. Wait, no, let's calculate. If \(m\angle 2=109^\circ\), then \(m\angle 9 = 180^\circ - 109^\circ=71^\circ\)? Wait, no, maybe I got the angle relationship wrong. Wait, actually, for \(a\parallel c\), \(\angle 2\) and \(\angle 9\) are alternate exterior angles? No, alternate exterior angles are equal. Wait, I think I messed up. Let's start over. The correct relationship: If two lines are parallel, same - side interior angles are supplementary. If \(a\parallel c\), then \(\angle 2\) and \(\angle 9\) are same - side interior angles, so \(m\angle 2 + m\angle 9=180^\circ\). So \(m\angle 9 = 180^\circ - m\angle 2\). Given \(m\angle 2 = 109^\circ\), then \(m\angle 9=180 - 109 = 71^\circ\).

Part c

Step1: Identify Angle Relationship

For \(b \parallel c\), \(\angle 5\) and \(\angle 11\) are corresponding angles. Corresponding angles are equal when lines are parallel.

Step2: Determine \(m\angle 11\)

Since \(\angle 5\) and \(\angle 11\) are corresponding angles, \(m\angle 11=m\angle 5\). Given \(m\angle 5 = 68^\circ\), so \(m\angle 11 = 68^\circ\).

Part d

Answer:

Step1: Identify Angle Relationship

\(\angle 2\) and \(\angle 5\) are same - side interior angles. The converse of the same - side interior angles theorem states that if same - side interior angles are supplementary (their sum is \(180^\circ\)), then the two lines are parallel. So if \(m\angle 2 + m\angle 5=180^\circ\), by the converse of same - side interior angles theorem, \(a\parallel b\).

Part a Answer: \(84^\circ\)

Part b Answer: \(71^\circ\)

Part c Answer: \(68^\circ\)

Part d Answer: Converse of Alternate Interior Angles Theorem

Part e Answer: Converse of Corresponding Angles Theorem

Part f Answer: Converse of Same - Side Interior Angles Theorem