Question

if m∠1 = 65°, find each measure. give your reasoning.
a. m∠2=
b. m∠3=
c. m∠4=
d. m∠5=
e. m∠6=
f. m∠7=
g. m∠8=
if m∠6 = 142°, find each measure. give your reasoning.
a. m∠1=
b. m∠2=
c. m∠3=
d. m∠4=
e. m∠5=
f. m∠7=
g. m∠8=

Explanation:

Step1: Identify vertical - angle relationship

Vertical angles are equal. $\angle1$ and $\angle3$ are vertical angles, $\angle2$ and $\angle4$ are vertical angles, $\angle5$ and $\angle7$ are vertical angles, $\angle6$ and $\angle8$ are vertical angles.

Step2: Identify supplementary - angle relationship

Adjacent angles that form a straight - line are supplementary (sum to 180°). For example, $\angle1+\angle2 = 180^{\circ}$.

Step3: Identify corresponding - angle relationship

If the lines are parallel, corresponding angles are equal.

Case 1: When $m\angle1 = 65^{\circ}$

• a. $m\angle2=180 - 65=115^{\circ}$ (Supplementary to $\angle1$)
• b. $m\angle3 = m\angle1=65^{\circ}$ (Vertical angles)
• c. $m\angle4 = m\angle2 = 115^{\circ}$ (Vertical angles)
• d. If the lines are parallel, $m\angle5 = m\angle1=65^{\circ}$ (Corresponding angles)
• e. $m\angle6 = m\angle2 = 115^{\circ}$ (Corresponding angles)
• f. $m\angle7 = m\angle5 = 65^{\circ}$ (Vertical angles)
• g. $m\angle8 = m\angle6 = 115^{\circ}$ (Vertical angles)

Case 2: When $m\angle6 = 142^{\circ}$

• a. If the lines are parallel, $m\angle1 = m\angle6 = 142^{\circ}$ (Corresponding angles)
• b. $m\angle2=180 - 142 = 38^{\circ}$ (Supplementary to $\angle1$)
• c. $m\angle3 = m\angle1 = 142^{\circ}$ (Vertical angles)
• d. $m\angle4 = m\angle2 = 38^{\circ}$ (Vertical angles)
• e. $m\angle5=180 - 142 = 38^{\circ}$ (Supplementary to $\angle6$)
• f. $m\angle7 = m\angle5 = 38^{\circ}$ (Vertical angles)
• g. $m\angle8 = m\angle6 = 142^{\circ}$ (Vertical angles)

Answer:

When $m\angle1 = 65^{\circ}$

a. $m\angle2 = 115^{\circ}$
b. $m\angle3 = 65^{\circ}$
c. $m\angle4 = 115^{\circ}$
d. $m\angle5 = 65^{\circ}$
e. $m\angle6 = 115^{\circ}$
f. $m\angle7 = 65^{\circ}$
g. $m\angle8 = 115^{\circ}$

When $m\angle6 = 142^{\circ}$

a. $m\angle1 = 142^{\circ}$
b. $m\angle2 = 38^{\circ}$
c. $m\angle3 = 142^{\circ}$
d. $m\angle4 = 38^{\circ}$
e. $m\angle5 = 38^{\circ}$
f. $m\angle7 = 38^{\circ}$
g. $m\angle8 = 142^{\circ}$