Question

name: date: period: mr. laughlin 2.3 parallel lines / transversals thms. app. 2. given: \\( \angle a = 5 y - 2 \\) \\( \angle b = 15 x - 7 \\) \\( \angle c = 5 x - 13 \\) find: \\( \angle a, \angle b, \angle c \\) the values of variables \\( x \\) and \\( y \\)

Explanation:

Step1: Use the property of parallel lines and transversals

Since \( \angle B\) and \( \angle C\) are supplementary (consecutive - interior angles), \( \angle B+\angle C = 180^{\circ}\).
Substitute \( \angle B=15x - 7\) and \( \angle C=5x - 13\) into the equation:
\((15x - 7)+(5x - 13)=180\)

Step2: Solve the equation for \(x\)

Combine like - terms:
\(15x+5x-7 - 13=180\)
\(20x-20 = 180\)
Add 20 to both sides:
\(20x=180 + 20\)
\(20x=200\)
Divide both sides by 20:
\(x=\frac{200}{20}=10\)

Step3: Find \( \angle B\) and \( \angle C\)

Substitute \(x = 10\) into the expressions for \( \angle B\) and \( \angle C\):
\(\angle B=15x - 7=15\times10-7=150 - 7 = 143^{\circ}\)
\(\angle C=5x - 13=5\times10-13=50 - 13 = 37^{\circ}\)

Step4: Use the property of parallel lines to find \( \angle A\)

Since \( \angle A\) and \( \angle C\) are alternate - interior angles, \( \angle A=\angle C\)
Also, we can use the fact that \( \angle A+\angle B = 180^{\circ}\) (consecutive - interior angles).
If \( \angle A = 5y-2\) and \( \angle A=\angle C = 37^{\circ}\)
Set \(5y-2 = 37\)
Add 2 to both sides: \(5y=37 + 2=39\)
Divide both sides by 5: \(y=\frac{39}{5}=7.8\)

Answer:

\(x = 10\), \(y = 7.8\), \( \angle A=37^{\circ}\), \( \angle B=143^{\circ}\), \( \angle C=37^{\circ}\)