Question

1. if angle 10 is 120 degrees and angle 13 is 65 degrees, find all other angles.

Explanation:

Step1: Vertical angles

Vertical angles are equal.

• $\angle1=\angle3$, $\angle2=\angle4$, $\angle5=\angle7$, $\angle6=\angle8$, $\angle9=\angle11$, $\angle10=\angle12$, $\angle13=\angle15$, $\angle14=\angle16$
• Given $\angle10 = 120^{\circ}$, so $\angle12=120^{\circ}$
• Given $\angle13 = 65^{\circ}$, so $\angle15=65^{\circ}$

Step2: Linear - pair angles

Linear - pair angles sum to $180^{\circ}$.

• $\angle1+\angle2 = 180^{\circ}$, $\angle3+\angle4 = 180^{\circ}$, $\angle5+\angle6 = 180^{\circ}$, $\angle7+\angle8 = 180^{\circ}$, $\angle9+\angle10 = 180^{\circ}$, $\angle11+\angle12 = 180^{\circ}$, $\angle13+\angle14 = 180^{\circ}$, $\angle15+\angle16 = 180^{\circ}$
• Since $\angle10 = 120^{\circ}$, then $\angle9=180 - 120=60^{\circ}$, and $\angle11 = 60^{\circ}$
• Since $\angle13 = 65^{\circ}$, then $\angle14=180 - 65 = 115^{\circ}$, and $\angle16 = 115^{\circ}$

Step3: Corresponding angles, alternate - interior angles and alternate - exterior angles

Assume the lines are parallel.

• Corresponding angles: $\angle1=\angle5$, $\angle2=\angle6$, $\angle3=\angle7$, $\angle4=\angle8$, $\angle9=\angle13$, $\angle10=\angle14$, $\angle11=\angle15$, $\angle12=\angle16$
• Alternate - interior angles: $\angle3=\angle5$, $\angle4=\angle6$, $\angle12=\angle16$, $\angle11=\angle13$
• Alternate - exterior angles: $\angle1=\angle7$, $\angle2=\angle8$, $\angle9=\angle15$, $\angle10=\angle16$
• $\angle1=\angle5=\angle7=\angle9 = 60^{\circ}$, $\angle2=\angle6=\angle8=\angle10=\angle14=\angle16 = 120^{\circ}$, $\angle3=\angle11=\angle13=\angle15 = 65^{\circ}$, $\angle4=\angle12 = 120^{\circ}$

Answer:

$\angle1 = 60^{\circ}$, $\angle2 = 120^{\circ}$, $\angle3 = 65^{\circ}$, $\angle4 = 120^{\circ}$, $\angle5 = 60^{\circ}$, $\angle6 = 120^{\circ}$, $\angle7 = 60^{\circ}$, $\angle8 = 120^{\circ}$, $\angle9 = 60^{\circ}$, $\angle11 = 60^{\circ}$, $\angle14 = 115^{\circ}$, $\angle16 = 115^{\circ}$