Question

example 1 - identifying angles
given ( mangle 1 = 105^circ ), find ( mangle 4 ), ( mangle 5 ), and ( mangle 8 ). tell which theorem you use in each case.
solution
( mangle 4 = 105^circ ) by the vertical angles congruence theorem
( mangle 5 = 105^circ ) by the corresponding angles theorem
( mangle 8 = 105^circ ) by the alternate exterior angles theorem

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try this video for extra example 1 - use the properties and theorems of parallel lines to complete the problems below.

1. the measures of three of the numbered angles are ( 75^circ ). identify the angles. tell which theorem you use in each case.

explain 1b using properties of parallel lines
visit bim.easyaccessmaterials.com, read integrated mathematics 1 lesson 10.2, then read the section belo
teacher voice - when parallel lines are cut by a transversal, you can apply the theorems about parallel li
example 2 - using properties of parallel lines
find the value of ( x ).
solution
by the alternate interior angles theorem, the two given angles are congruent.
( (10x - 55)^circ = 45^circ ) alternate interior angles theorem
( 10x - 55 + 55 = 45 + 55 ) addition property of equality
( 10x = 100 ) simplify.
( x = 10 ) division property of equality
so, the value of ( x = 10 ).

Explanation:

Step1: Identify congruent vertical angles

$m\angle 1 = m\angle 4 = 75^\circ$ (Vertical Angles Congruence Theorem)

Step2: Identify congruent corresponding angles

$m\angle 4 = m\angle 8 = 75^\circ$ (Corresponding Angles Theorem)

Step3: Identify congruent alternate interior angles

$m\angle 4 = m\angle 5 = 75^\circ$ (Alternate Interior Angles Theorem)

Step4: Identify supplementary linear pairs

$m\angle 1 + m\angle 2 = 180^\circ$
$m\angle 2 = 180^\circ - 75^\circ = 105^\circ$ (Linear Pair Postulate)

Step5: Identify congruent vertical angles

$m\angle 2 = m\angle 3 = 105^\circ$ (Vertical Angles Congruence Theorem)

Step6: Identify congruent corresponding angles

$m\angle 2 = m\angle 6 = 105^\circ$ (Corresponding Angles Theorem)

Step7: Identify congruent alternate interior angles

$m\angle 3 = m\angle 6 = 105^\circ$ (Alternate Interior Angles Theorem)

Step8: Identify congruent vertical angles

$m\angle 6 = m\angle 7 = 105^\circ$ (Vertical Angles Congruence Theorem)

Answer:

• $m\angle 1 = 75^\circ$ (given)
• $m\angle 2 = 105^\circ$ (Linear Pair Postulate with $\angle 1$)
• $m\angle 3 = 105^\circ$ (Vertical Angles Congruence Theorem with $\angle 2$)
• $m\angle 4 = 75^\circ$ (Vertical Angles Congruence Theorem with $\angle 1$)
• $m\angle 5 = 75^\circ$ (Alternate Interior Angles Theorem with $\angle 4$)
• $m\angle 6 = 105^\circ$ (Corresponding Angles Theorem with $\angle 2$)
• $m\angle 7 = 105^\circ$ (Vertical Angles Congruence Theorem with $\angle 6$)
• $m\angle 8 = 75^\circ$ (Corresponding Angles Theorem with $\angle 4$)

The three $75^\circ$ angles are $\angle 1$, $\angle 4$, $\angle 5$, $\angle 8$ (any three of these match the problem's description, with justifications from the theorems above).