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 ( mangle4 = 65^{circ} ), then ( mangle1= )
( a. )
( b) ) if the ( mangle7 = 110^{circ} ), then ( mangle2= )
( b. )
( c) ) if the ( mangle5 = 71^{circ} ), then ( mangle4= )
( c. )
( d) ) if the ( mangle3 = 117^{circ} ), then ( mangle5= )
( d. )
( e) ) if the ( mangle8 = 54^{circ} ), then ( mangle1= )
( e. )

Explanation:

Example 3

Step1: Use the property of same - side interior angles

If two parallel lines are cut by a transversal, same - side interior angles are supplementary. So, \(115+(x + 5)=180\).

Step2: Solve the equation

$$
LATEXBLOCK0
$$

Example 4

a)

Step1: Use the property of vertical angles

\(\angle1\) and \(\angle4\) are vertical angles. Vertical angles are congruent. So, \(m\angle1=m\angle4 = 65^{\circ}\). Reason: Vertical Angles Theorem.

b)

Step1: Use the property of corresponding angles and linear pairs

First, \(\angle7\) and \(\angle5\) are vertical angles, so \(m\angle5=m\angle7 = 110^{\circ}\). Then, \(\angle2\) and \(\angle5\) are corresponding angles (if the lines are parallel). So, \(m\angle2=m\angle5=110^{\circ}\). Reason: Vertical Angles Theorem and Corresponding Angles Postulate.

c)

Step1: Use the property of alternate interior angles

\(\angle4\) and \(\angle5\) are alternate interior angles (if the lines are parallel). So, \(m\angle4=m\angle5 = 71^{\circ}\). Reason: Alternate Interior Angles Theorem.

d)

Step1: Use the property of supplementary angles and alternate interior angles

\(\angle3\) and \(\angle4\) are supplementary (\(m\angle3+m\angle4 = 180^{\circ}\)), so \(m\angle4=180 - 117=63^{\circ}\). Then, since \(\angle4\) and \(\angle5\) are alternate interior angles (if the lines are parallel), \(m\angle5=m\angle4 = 63^{\circ}\). Reason: Linear Pair Postulate and Alternate Interior Angles Theorem.

e)

Step1: Use the property of vertical angles and corresponding angles

\(\angle8\) and \(\angle6\) are vertical angles (\(m\angle6=m\angle8 = 54^{\circ}\)). \(\angle2\) and \(\angle6\) are corresponding angles (if the lines are parallel), so \(m\angle2=m\angle6 = 54^{\circ}\). \(\angle1\) and \(\angle2\) are supplementary (\(m\angle1+m\angle2 = 180^{\circ}\)), so \(m\angle1=180 - 54=126^{\circ}\). Reason: Vertical Angles Theorem, Corresponding Angles Postulate, and Linear Pair Postulate.

Answer:

Example 3

\(x = 60\)

Example 4

a) \(65^{\circ}\), Vertical Angles Theorem
b) \(110^{\circ}\), Vertical Angles Theorem and Corresponding Angles Postulate
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 Postulate, and Linear Pair Postulate