Question

(parallel & perpendicular lines) date: topic 1: parallel lines & planes use the diagram to the right for questions 1 - 5. ① name a plane parallel to plane wxt. ② name two segments parallel to $overline{vu}$. ③ name two segments parallel to $overline{sw}$. 4. name two segments skew to $overline{xy}$. 5. name two segments skew to $overline{vz}$. 6. name each angle - pair as corresponding, alternate interior, alternate exterior, consecutive interior, consecutive exterior, or no relationship. identify the transversal that connects each angle - pair. a) $angle4$ and $angle10$ ; transversal: b) $angle8$ and $angle11$ ; transversal: c) $angle1$ and $angle4$ ; transversal: d) $angle2$ and $angle12$ ; transversal: e) $angle5$ and $angle7$ ; transversal: f) $angle2$ and $angle13$ ; transversal: topic 2: parallel lines & angles 7. if $pparallel q$, $mangle7 = 131^{circ}$, and $mangle16 = 88^{circ}$, give the measure of each angle. a. $mangle1=$ f. $mangle6=$ k. $mangle12=$ b. $mangle2=$ g. $mangle8=$ l. $mangle13=$ c. $mangle3=$ h. $mangle9=$ m. $mangle14=$ d. $mangle4=$ i. $mangle10=$ n. $mangle15=$ e. $mangle5=$ j. $mangle11=$ 8. if $aparallel b$, $mangle2 = 63^{circ}$, and $mangle9 = 105^{circ}$, find the measure of each angle. a. $mangle1=$ e. $mangle6=$ i. $mangle11=$ b. $mangle3=$ f. $mangle7=$ j. $mangle12=$ c. $mangle4=$ g. $mangle8=$ k. $mangle13=$ d. $mangle5=$ h. $mangle10=$ l. $mangle14=$

Explanation:

Step1: Recall parallel - plane concept

In a 3 - D figure, planes that do not intersect are parallel. For the first question, plane $SYZ$ is parallel to plane $WXT$.

Step2: Recall parallel - segment concept

Segments that lie in parallel lines or in non - intersecting lines in the same or parallel planes are parallel. For $\overline{VU}$, segments $\overline{SW}$ and $\overline{TX}$ are parallel. For $\overline{SW}$, segments $\overline{VU}$ and $\overline{TX}$ are parallel.

Step3: Recall skew - segment concept

Skew segments are non - parallel and non - intersecting and do not lie in the same plane. For $\overline{XY}$, segments $\overline{SZ}$ and $\overline{VW}$ are skew. For $\overline{VZ}$, segments $\overline{TX}$ and $\overline{WY}$ are skew.

Step4: Recall angle - pair relationships

a) $\angle4$ and $\angle10$ are corresponding angles with transversal $k$.
b) $\angle8$ and $\angle11$ are alternate interior angles with transversal $j$.
c) $\angle1$ and $\angle4$ are consecutive interior angles with transversal $l$.
d) $\angle2$ and $\angle12$ are no relationship as they are not formed by a single transversal.
e) $\angle5$ and $\angle7$ are vertical angles (not among the given types in a strict sense for this problem context, but if we consider in terms of the basic angle relationships) with no relevant transversal for the given types.
f) $\angle2$ and $\angle13$ are alternate exterior angles with transversal $l$.

Step5: Use angle - measure properties for parallel lines

For the problem with $p\parallel q$, $m\angle7 = 131^{\circ}$, and $m\angle16=88^{\circ}$:

• Corresponding angles are equal, alternate interior angles are equal, and consecutive interior angles are supplementary.
• $m\angle1 = 131^{\circ}$ (corresponding to $\angle7$)
• $m\angle2 = 49^{\circ}$ (supplementary to $\angle1$)
• $m\angle3 = 131^{\circ}$ (vertical to $\angle1$)
• $m\angle4 = 49^{\circ}$ (vertical to $\angle2$)
• $m\angle5 = 131^{\circ}$ (alternate interior to $\angle7$)
• $m\angle6 = 49^{\circ}$ (alternate interior to $\angle2$)
• $m\angle8 = 131^{\circ}$ (corresponding to $\angle5$)
• $m\angle9 = 88^{\circ}$ (corresponding to $\angle16$)
• $m\angle10 = 92^{\circ}$ (supplementary to $\angle9$)
• $m\angle11 = 88^{\circ}$ (vertical to $\angle9$)
• $m\angle12 = 92^{\circ}$ (vertical to $\angle10$)
• $m\angle13 = 88^{\circ}$ (alternate interior to $\angle9$)
• $m\angle14 = 92^{\circ}$ (alternate interior to $\angle10$)
• $m\angle15 = 88^{\circ}$ (corresponding to $\angle13$)

For the problem with $a\parallel b$, $m\angle2 = 63^{\circ}$, and $m\angle9 = 105^{\circ}$:

• $m\angle1 = 117^{\circ}$ (supplementary to $\angle2$)
• $m\angle3 = 63^{\circ}$ (vertical to $\angle2$)
• $m\angle4 = 117^{\circ}$ (vertical to $\angle1$)
• $m\angle5 = 63^{\circ}$ (alternate interior to $\angle2$)
• $m\angle6 = 117^{\circ}$ (alternate interior to $\angle1$)
• $m\angle7 = 105^{\circ}$ (corresponding to $\angle9$)
• $m\angle8 = 75^{\circ}$ (supplementary to $\angle7$)
• $m\angle10 = 75^{\circ}$ (vertical to $\angle8$)
• $m\angle11 = 105^{\circ}$ (vertical to $\angle7$)
• $m\angle12 = 75^{\circ}$ (alternate interior to $\angle8$)
• $m\angle13 = 105^{\circ}$ (alternate interior to $\angle7$)
• $m\angle14 = 75^{\circ}$ (corresponding to $\angle12$)

Answer:

1. Plane $SYZ$
2. $\overline{SW}$, $\overline{TX}$
3. $\overline{VU}$, $\overline{TX}$
4. $\overline{SZ}$, $\overline{VW}$
5. $\overline{TX}$, $\overline{WY}$

6.
a) Corresponding; $k$
b) Alternate interior; $j$
c) Consecutive interior; $l$
d) No relationship; None
e) No relevant transversal for given types; None
f) Alternate exterior; $l$
7.
a. $131^{\circ}$
b. $49^{\circ}$
c. $131^{\circ}$
d. $49^{\circ}$
e. $131^{\circ}$
f. $49^{\circ}$
g. $131^{\circ}$
h. $88^{\circ}$
i. $92^{\circ}$
j. $88^{\circ}$
k. $92^{\circ}$
l. $88^{\circ}$
m. $92^{\circ}$
n. $88^{\circ}$
8.
a. $117^{\circ}$
b. $63^{\circ}$
c. $117^{\circ}$
d. $63^{\circ}$
e. $117^{\circ}$
f. $105^{\circ}$
g. $75^{\circ}$
h. $75^{\circ}$
i. $105^{\circ}$
j. $75^{\circ}$
k. $105^{\circ}$
l. $75^{\circ}$