Section 2: Angles and Parallel Lines
Postulate 3.1: If two parallel lines are cut by a transveral, then each pair of corresponding angles are congruent.
In the image above, the measure of three of the numbered angles is 120.
In the figure, m<8= 74. Find the measure of each angle. Tell which postulate (or theorems) you used.
A. <4 ; Angle 4 is congruent to angle 8.
<4 ≅ <8 Corresponding angles postulate
m<4 = m<8 Definition of congruent angles
m<4 = 74 Substitution
B. <1
<1 ≅ <4 Vertical angles theorem
<4 ≅ <8 Corresponding angles postulate
<1 ≅ <8 Transitive property of congruence
m<1 = m<8 Definition of congruent angles
m<1 = 74 Substitution
A. <4 ; Angle 4 is congruent to angle 8.
<4 ≅ <8 Corresponding angles postulate
m<4 = m<8 Definition of congruent angles
m<4 = 74 Substitution
B. <1
<1 ≅ <4 Vertical angles theorem
<4 ≅ <8 Corresponding angles postulate
<1 ≅ <8 Transitive property of congruence
m<1 = m<8 Definition of congruent angles
m<1 = 74 Substitution
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then each pair of alternate interior angles is congruent
So in the figure, if line K is parallel to line L, then <2 is congruent to
<8 and <3 is congruent to <5
Consecutive Interior Angles Theorem: If two parallel lines are cut by a transversal, then each pair of consecutive interior angles is supplementary
In the figure, angles 3 and 5 are consecutive interior angles
because they are supplementary, along with angles 4 and 6
Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then each pair of alternate exterior angles is congruent
In the figure, if line K is parallel to line L, then <1 is congruent to <7 and <4 is congruent to <6
Keep in mind that postulates are accepted without proof
Example:
Given: Angle K || L and according to the corresponding angles postulate, angle 4 is congruent to angle 8. By definition of congruent angles, <4 is congruent to <8. Since <5 and <8 form a linear pair, they are supplementary, so <5+<8= 180. When substituting m<4 for m<8, we get m<4+m<5=180.
Perpendicular Transversal Theorem: In a plane, is a line is perpendicular to one of two parallel lines, then it is perpendicular to the other
If line K || line L and line K is perpendicular to line T, then line
L is perpendicular to line T
Real - World Problem
On the train train, if m<10= 2x-5 and m< 1= 67
<6 ≅ <1 Vertical Angles Theorem
m<6 and m<1 Definition of congruent
m<6 = 67 Substitution
since line s and line r are parallel, <6 and < 10 are supplementary by the Consecutive Interior Angles Theorem.
m<6 + m<10= 180 definition of supplementary angles
67+ 2x-5= 180 Substitution
2x+62=180 Simplify
2x=118 Subtract 62 from each side.
x= 59 divide by 2
<6 ≅ <1 Vertical Angles Theorem
m<6 and m<1 Definition of congruent
m<6 = 67 Substitution
since line s and line r are parallel, <6 and < 10 are supplementary by the Consecutive Interior Angles Theorem.
m<6 + m<10= 180 definition of supplementary angles
67+ 2x-5= 180 Substitution
2x+62=180 Simplify
2x=118 Subtract 62 from each side.
x= 59 divide by 2