Question

**#14.) which of the following statements will not prove ( l parallel m )?

1. ( angle 5 cong angle 2 )
2. ( angle 4+angle 5 = 180^{circ} )
3. ( angle 1+angle 7 = 180^{circ} )

1v. ( angle 8 cong angle 6 )
a.) i only
b.) ii only
c.) i and iii
d.) ii and iv
**#15.) spiral of old content: in the figure, ( angle axe ) is a straight angle, and ( angle axc ) is bisected by ( overline{xb} ). if ( mangle bxc=(8x - 30)^{circ} ) and ( mangle axb=(4x + 10)^{circ} ), find ( mangle bxe ).
a.) ( 10^{circ} )
b.) ( 130^{circ} )
c.) ( 50^{circ} )
d.) ( 40^{circ} )

Explanation:

Step1: Analyze each statement for proving \(l\parallel m\)

• Statement i: \(\angle5\cong\angle2\). By the converse of the alternate - exterior angles theorem, if two lines are cut by a transversal and alternate - exterior angles are congruent, then the lines are parallel. So, if \(\angle5\cong\angle2\), then \(l\parallel m\).
• Statement ii: \(\angle4+\angle5 = 180^{\circ}\). By the converse of the same - side interior angles theorem, if two lines are cut by a transversal and the sum of same - side interior angles is \(180^{\circ}\), then the lines are parallel. So, if \(\angle4+\angle5=180^{\circ}\), then \(l\parallel m\).
• Statement iii: \(\angle1+\angle7 = 180^{\circ}\). \(\angle1\) and \(\angle7\) are not a pair of angles (corresponding, alternate - interior, alternate - exterior, or same - side interior) that can be used to prove the parallelism of \(l\) and \(m\) based on the standard parallel line theorems.
• Statement iv: \(\angle8\cong\angle6\). \(\angle8\) and \(\angle6\) are vertical angles. Vertical angles being congruent does not give any information about the parallelism of \(l\) and \(m\).

Answer:

d) ii and iv