Question

**#8.) based on the diagram below, which of the following makes q and p parallel? (select all that apply)
a.) ( mangle4 + mangle6 = 180 )
b.) ( mangle2 + mangle8 = 180 )
c.) ( angle1congangle4 )
d.) ( angle3congangle6 )
**#9.) in the diagram below, ( pparallel q ), the ( mangle6=(5x + 3)^{circ} ) and the ( mangle1=(11x + 33)^{circ} ). determine the ( mangle5 ).
a.) ( 48^{circ} )
b.) ( 132^{circ} )
c.) ( 9^{circ} )
d.) ( 33^{circ} )
*#10.) given ( overrightarrow{ac}paralleloverrightarrow{dg} ), ( mangle ibc = 63^{circ} ), ( mangle kfg = 44^{circ} ) what is the ( mangle fbe )?
a.) ( 73^{circ} )
b.) ( 63^{circ} )
c.) ( 107^{circ} )
d.) ( 21^{circ} )

Explanation:

Step1: Use the property of parallel lines and transversals

When two parallel lines are cut by a transversal, consecutive interior angles are supplementary. Also, alternate interior angles are congruent.
For problem 8:

• For option a: \(m\angle4 + m\angle6=180^{\circ}\). \(\angle4\) and \(\angle6\) are consecutive interior angles. If consecutive interior angles are supplementary, then the lines \(q\) and \(p\) are parallel (by the consecutive - interior - angles converse).
• For option b: \(m\angle2 + m\angle8 = 180^{\circ}\). \(\angle2\) and \(\angle8\) are not related by any of the parallel - line converse theorems (they are not corresponding, alternate interior or consecutive interior angles).
• For option c: \(\angle1\cong\angle4\). \(\angle1\) and \(\angle4\) are vertical angles. Vertical - angle congruence does not imply parallel lines.
• For option d: \(\angle3\cong\angle6\). \(\angle3\) and \(\angle6\) are alternate interior angles. If alternate interior angles are congruent, then the lines \(q\) and \(p\) are parallel (by the alternate - interior - angles converse).

For problem 9:

• Since \(p\parallel q\), \(\angle1\) and \(\angle6\) are supplementary (consecutive interior angles). So, \((11x + 33)+(5x+3)=180\).
• Combine like terms: \(11x+5x+33 + 3=180\), \(16x+36 = 180\).
• Subtract 36 from both sides: \(16x=180 - 36=144\).
• Divide by 16: \(x=\frac{144}{16}=9\).
• Then \(m\angle6=(5x + 3)=(5\times9+3)=48^{\circ}\).
• \(\angle5\) and \(\angle6\) are supplementary (linear pair). So \(m\angle5=180 - 48=132^{\circ}\).

For problem 10:

• Since \(AC\parallel DG\), \(\angle IBC\) and \(\angle BED\) are congruent (corresponding angles), so \(m\angle BED = 63^{\circ}\).
• \(\angle FBE\) and \(\angle BED\) and \(\angle KFG\) are related by the fact that the sum of angles around a point on a straight line. But using the exterior - angle property (or the fact that \(\angle FBE\) is supplementary to the angle formed by the sum of \(\angle BED\) and \(\angle KFG\) in a non - direct way, we know that \(\angle FBE=180-(63 + 44)=73^{\circ}\) (using the property of angles on a straight line and parallel - line angle relationships).

Answer:

1. a) \(m\angle4 + m\angle6 = 180\), d) \(\angle3\cong\angle6\)
2. b) \(132^{\circ}\)
3. a) \(73^{\circ}\)