Question

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unit 3: parallel & perpendicular lines
homework 2: parallel lines cut by a transv
this is a 2 - page document!
(1) if (mangle8 = 23^{circ}), find each measure. give your reasoning.
a. (mangle1=)
b. (mangle2=)
c. (mangle3=)
d. (mangle4=)
e. (mangle5=)
f. (mangle6=)
g. (mangle7=)
(2) if (mangle9 = 97^{circ}) and (mangle12 = 114^{circ}), find each measure

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle 1$ and $\angle 4$ are vertical angles, $\angle 2$ and $\angle 3$ are vertical angles, $\angle 5$ and $\angle 8$ are vertical angles, $\angle 6$ and $\angle 7$ are vertical angles.

Step2: Identify corresponding - angle relationship

Corresponding angles are equal when lines are parallel. $\angle 1$ and $\angle 5$, $\angle 2$ and $\angle 6$, $\angle 3$ and $\angle 7$, $\angle 4$ and $\angle 8$ are corresponding angles.

Step3: Identify alternate - interior and alternate - exterior angle relationships

Alternate - interior angles are equal when lines are parallel ($\angle 3$ and $\angle 5$, $\angle 4$ and $\angle 6$), and alternate - exterior angles are equal when lines are parallel ($\angle 1$ and $\angle 7$, $\angle 2$ and $\angle 8$).
Given $m\angle8 = 23^{\circ}$:
a. $m\angle1=23^{\circ}$ (alternate - exterior angles)
b. $m\angle2 = 157^{\circ}$ ($\angle2$ and $\angle8$ are supplementary, $180 - 23=157$)
c. $m\angle3 = 23^{\circ}$ (corresponding to $\angle7$, and $\angle7$ and $\angle8$ are supplementary, so $\angle3$ and $\angle8$ are alternate - interior)
d. $m\angle4 = 157^{\circ}$ ($\angle4$ and $\angle8$ are supplementary)
e. $m\angle5 = 23^{\circ}$ (vertical to $\angle8$)
f. $m\angle6 = 157^{\circ}$ ($\angle6$ and $\angle5$ are supplementary)
g. $m\angle7 = 23^{\circ}$ (vertical to $\angle5$)

Answer:

a. $23^{\circ}$
b. $157^{\circ}$
c. $23^{\circ}$
d. $157^{\circ}$
e. $23^{\circ}$
f. $157^{\circ}$
g. $23^{\circ}$