Question

example 3) find the value of x.
example 4) use the diagram below to find the angle measure. tell which postulate or theorem you use.
angle measure
reason
a) if the ( mangle 4 = 65^{circ} ), then ( mangle 1 = )
a.

b) if the ( mangle 7 = 110^{circ} ), then ( mangle 2 = )
b.

c) if the ( mangle 5 = 71^{circ} ), then ( mangle 4 = )
c.

d) if the ( mangle 3 = 117^{circ} ), then ( mangle 5 = )
d.

e) if the ( mangle 8 = 54^{circ} ), then ( mangle 1 = )
e.

Explanation:

Step1: Find \(m\angle1\) when \(m\angle4 = 65^{\circ}\)

Vertical angles are equal. \(\angle1\) and \(\angle4\) are vertical angles. So \(m\angle1=m\angle4 = 65^{\circ}\).

Step2: Find \(m\angle2\) when \(m\angle7 = 110^{\circ}\)

First, \(m\angle5=180^{\circ}-m\angle7 = 180^{\circ}- 110^{\circ}=70^{\circ}\) (linear - pair). Then, since \(\angle2\) and \(\angle5\) are corresponding angles (if we assume the two horizontal lines are parallel), \(m\angle2=m\angle5 = 70^{\circ}\).

Step3: Find \(m\angle4\) when \(m\angle5 = 71^{\circ}\)

\(\angle4\) and \(\angle5\) are alternate - interior angles (if the two horizontal lines are parallel). So \(m\angle4=m\angle5 = 71^{\circ}\).

Step4: Find \(m\angle5\) when \(m\angle3 = 117^{\circ}\)

\(\angle3\) and \(\angle4\) are a linear pair, so \(m\angle4=180^{\circ}-m\angle3=180^{\circ}-117^{\circ} = 63^{\circ}\). Then, since \(\angle4\) and \(\angle5\) are alternate - interior angles (if the two horizontal lines are parallel), \(m\angle5=m\angle4 = 63^{\circ}\).

Step5: Find \(m\angle1\) when \(m\angle8 = 54^{\circ}\)

\(m\angle6=m\angle8 = 54^{\circ}\) (vertical angles). Then, \(m\angle2=m\angle6 = 54^{\circ}\) (corresponding angles). Since \(\angle1\) and \(\angle2\) are a linear pair, \(m\angle1=180^{\circ}-m\angle2=180^{\circ}-54^{\circ}=126^{\circ}\).

Answer:

a) \(65^{\circ}\), Vertical Angles Theorem
b) \(70^{\circ}\), Linear - Pair Postulate and Corresponding Angles Theorem
c) \(71^{\circ}\), Alternate - Interior Angles Theorem
d) \(63^{\circ}\), Linear - Pair Postulate and Alternate - Interior Angles Theorem
e) \(126^{\circ}\), Vertical Angles Theorem, Corresponding Angles Theorem and Linear - Pair Postulate