Question

unit 3 test: properties & proportions

1. find the values of the variables:

(a)

Explanation:

Step1: Find \(a\)

Use the property of supplementary angles (\(a + 45^{\circ}=180^{\circ}\)).
\(a=180^{\circ}- 45^{\circ}=135^{\circ}\)

Step2: Find \(d\)

\(d\) and \(45^{\circ}\) are vertical angles. So \(d = 45^{\circ}\)

Step3: Find \(e\)

In the triangle with angles \(80^{\circ}\), \(45^{\circ}\), and \(e\) (using the angle - sum property of a triangle: \(80^{\circ}+45^{\circ}+e = 180^{\circ}\)).
\(e=180^{\circ}-(80^{\circ}+45^{\circ})=55^{\circ}\)

Step4: Find \(c\)

Since the two lines are parallel, \(c = e\) (alternate - interior angles). So \(c = 55^{\circ}\)

Step5: Find \(f\)

In the triangle with angles \(60^{\circ}\), \(c = 55^{\circ}\), and \(f\) (using the angle - sum property of a triangle: \(60^{\circ}+55^{\circ}+f=180^{\circ}\)).
\(f=180^{\circ}-(60^{\circ}+55^{\circ})=65^{\circ}\)

Step6: Find \(b\)

Since the two lines are parallel, \(b = f\) (alternate - interior angles). So \(b = 65^{\circ}\)

Answer:

\(a = 135^{\circ}\), \(b = 65^{\circ}\), \(c = 55^{\circ}\), \(d = 45^{\circ}\), \(f = 65^{\circ}\)