Question

name:
mr. laughlin
date:
period:
2.3 parallel lines / transversals thms. app.
given: ( w||x )
( angle c = 3y - 8 )( angle a = 8x )
( angle d = 4z + 16 )
( angle f = 2x + 3 )
( angle g = 4x + 7 )
find: all angle measures
the values of variables ( x ), ( y ), and ( z )

Explanation:

Step1: Find the value of \( x \)

Since \( \angle A = 8x \), \( \angle F=2x + 3 \), and \( \angle G = 4x+7 \), and \( \angle A+\angle F+\angle G = 180^{\circ} \) (sum of angles on a straight - line).

$$8x+(2x + 3)+(4x+7)=180$$

$$8x+2x+3 + 4x+7=180$$

$$14x+10 = 180$$

$$14x=180 - 10$$

$$14x=170$$

$$x = 10$$

Step2: Find the value of \( y \)

Since \( w\parallel x \), \( \angle C=\angle A \) (corresponding angles).
Given \( \angle C = 3y-8 \) and \( \angle A=8x \), substituting \( x = 10 \), \( \angle A=8\times10 = 80^{\circ} \)

$$3y-8=80$$

$$3y=80 + 8$$

$$3y=88$$

$$y = 29.33$$

Step3: Find the value of \( z \)

Since \( w\parallel x \), \( \angle D=\angle A \) (alternate interior angles).
Given \( \angle D = 4z+16 \) and \( \angle A = 80^{\circ} \)

$$4z+16=80$$

$$4z=80 - 16$$

$$4z=64$$

$$z = 16$$

Step4: Calculate angle measures

• \( \angle A=8x=8\times10 = 80^{\circ} \)
• \( \angle F=2x + 3=2\times10+3 = 23^{\circ} \)
• \( \angle G=4x+7=4\times10+7 = 47^{\circ} \)
• \( \angle C=\angle A = 80^{\circ} \)
• \( \angle D=\angle A = 80^{\circ} \)
• \( \angle B = 180-\angle A=100^{\circ} \) (linear - pair with \( \angle A \))
• \( \angle O=\angle A = 80^{\circ} \) (vertical angles)
• \( \angle E=\angle B = 100^{\circ} \) (vertical angles)
• \( \angle I=\angle O = 80^{\circ} \) (vertical angles)
• \( \angle H=\angle B = 100^{\circ} \) (vertical angles)
• \( \angle G = 47^{\circ} \), \( \angle F = 23^{\circ} \), and \( \angle G+\angle F+\angle E=180^{\circ} \) (sum of angles on a straight - line)

Answer:

\( x = 10 \), \( y=\frac{88}{3}\approx29.33 \), \( z = 16 \)
\( \angle A = 80^{\circ} \), \( \angle B = 100^{\circ} \), \( \angle C = 80^{\circ} \), \( \angle D = 80^{\circ} \), \( \angle E = 100^{\circ} \), \( \angle F = 23^{\circ} \), \( \angle G = 47^{\circ} \), \( \angle H = 100^{\circ} \), \( \angle I = 80^{\circ} \), \( \angle O = 80^{\circ} \)