5.3 Congruent Angles Associated With Parallel Lines

By:Kristi Polizzano

Objective

• Apply the parallel postulate• Identify the pairs of angles formed by a

transversal cutting parallel lines• Apply six theorems about parallel lines

The Parallel Postulate

• Through a point not on a line there is exactly one parallel to the given line.

• Although this postulate may seem reasonable, mathematicians have argued its truth. For our purposes, we will assume the postulate is true.

• <-------p------> a<--------------->

Angles Formed When Parallel Lines Are Cut by a Transversal

• We know when alternate interior angles are congruent, lines are parallel. With this section, we will prove that the opposite is true as well.

• If we start with parallel lines, then we can conclude that alternate interior angles are congruent.

Theorem 37

• If two parallel lines are cut by a transversal, each pair of alternate interior angles are congruent. Short form: ll lines alt. int.<s

• Given: Lines a and b are parallel Prove: <1 <2

1

2

Theorem 38If two parallel lines are cut by a transversal, then any pair of the angles formed are either congruent of supplementary. To prove this, study the diagram below:

x x

x *Vertical angles are congruent.

xx

(180-x)(180-x)

*A straight angle = 180

xx

(180-x)(180-x)

x (180-x)

* ll lines Alt. int. angles congruent

xx

(180-x)(180-x)

x (180-x)(180-x) x

*vertical angles are congruent.

Theorem 39If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. Alt. ext. )(ll lines

Given: a ll bProve: 1 2 1

8

a

b

Theorem 40If two parallel lines are cut by a transversal, each pair of corresponding angles are congruent. ( ll lines Corr.s )

Given: a ll bProve: 1 5

1

5

Theorem 41If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary.

Given: a ll bProve: 3 supp. 5 a

b

3

5

Theorem 42If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the transversal are supplementary.

Given: a ll bProve: 1 supp. 7

a

b

1

7

Theorem 43In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other.

Given: a ll b c a

Prove: c b

a

b

c

Theorem 44If two lines are parallel to a third lines, they are parallel to each other. (Transitive Property of Parallel Lines).Given: a ll b, b ll cProve: a ll c

ac

b

Theorem 44 (cont’d).By using “Parallel lines implies alternate interior angles congruent” and “alternate interior angles congruent implies parallel lines,” you can prove that Theorem 44 is true when all three lines lie in the same plane. It also can be shown that the theorem holds for lines in three- dimensional space.

In summary, if two parallel lines are cut by a transversal, then…

• Each pair of alternate interior angles are congruent.•Each pair of alternate exterior angles are congruent. •Each pair of corresponding angles are congruent. •Each pair of interior angles on the same side of the transversal are supplementary.•Each pair of exterior angles on the same side of the transversal are supplementary.

!Just a note…!

*These theorems are extremely important in this chapter, along with all the upcoming chapters. It is very crucial that you learn each meaning and recognize the associations between parallel lines and congruent angles whenever you are stuck in a proof.-Check out this video to get a Close look on how to applyThe parallel lines and congruentAngles theorems. http://youtube.com/watch?v=XQUbFCAv_U4

Sample ProblemsIf k ll p, find m 1

k

p

1(6x+10)

(4x+20)

•Since alt. int. angles are congruent, we can see that 6x+10 is going to equal 4x+20. So put it into the equation:

6x+10=4x+202x=10

X=5Now you know x=5. Since you know from the theorem “vertical angles are congruent”, angle 1 must be congruent to 6x+10. (Also from the new theorem of parallel lines implies corresponding congruent angles, you can see that angles 1 and 4x+20 are congruent.) So, plug in 5 for the x value and solve the equation.

6(5)+10=40.Thus, since vertical angles are congruent, the measure of angle 1 is 40.

Given: m ll oProve: 1 supp. 2

m

o

1

23

1. m ll o

2. 2 Supp. 3

3. 31

4. 1 Supp. 2

1. Given2. If two angles form a straight

angles, they are supplementary.

3. ll lines Alt. ext. s4. Substitution

The Crook Problem•Crook problems test your ability to apply all you know about parallel lines and congruent angles to a problem that involves multiple steps with this concept.•They aren’t as difficult as they look. Just take your time and take the problem apart step by step.

100

1 c

40

1. If a ll b. Find the m1a

b2.To solve this problem, think of the theorems and postulates you just learned. According to the parallel postulate, through a point not on a line there is exactly one parallel to the given line. Using this, draw line MC , so it is parallel to a.

Step 1:

Step 2:

a

b

100

1

40

cm

Continued...Step 3:

100

140

a

b

m c

3. Now, use what you know about parallel lines and congruent angles. You know that parallel lines for alt. int. angles congruent, so what angles can be established from this theorem?

You can label the one side of angle 1, 40 because alt. int. angles shows that this angle is congruent to 40.

You can also see from alt. int. angles congruent, that part of angle c is 100. You know MC is a straight line. Straight lines have a measure of 180. So, subtract 180-100 and you get 80. This is the measure for the upper half of the angle 1.

1008040

Step 4:The problem asked you to find angle 1. To find this, just simply add the two measurements you found. This equals 120. This is the measure of angle 1.

Problems Answers on separate slides.

1) Given: 54

RS ll MP

Prove: Triangle KMP is isosceles.

R sK

M P

4 5

2) Given: a ll bFind: m 1

(2x+5)

1

(3x-13)

a

b

3) Are K and P parallel?

k

p

(2x+10)

(3x+30)

(5x-20)

4) *Crook ProblemIf V ll P, find m 2

70

2

80

v

p

Answers1)

1. 54

2. RS ll MP3. KPM

4.

5

4 KMP

5. M P

6. KMP is isos.

1. Given2. Given3. Alt. int. angles implies congruent

angles4. Same as 35. Substitution6. If angles then triangle is isos.

2) 41

3)No, because when you solve for x, it doesn’t fit into all equations.4)150

Rhoad, Richard, and George Milauskas. Geometry for Enjoyment and Challenge. Evanston: McDougal Littell, 1997.

"Watch video on parallel lines-geometry help." Youtube. Youtube. 30 May 2008 <http://youtube.com/watch?v=XQUbFCAv_U4>.