Properties of Parallel Lines

What does it mean for two lines to be parallel?

THEY NEVER INTERSECT!

ml //l

m

Parallel Lines cut by a Transversalp

l

m

1 23 4

5 67 8

1) Transversal

2) Identify the Alternate Interior Angles.3) Identify the Corresponding Angles.4) Identify the Same Side Interior Angles.

ALTERNATE INTERIOR ANGLES ARE CONGRUENTCORRESPONDING ANGLES ARE CONGRUENTSame Side interior angles are Supplementary

63 54 62 51 73 84 18053 18064

5) Identify the Alternate Exterior Angles.

81 72

Alternate Exterior Angles are Congruent

Review Theorems:1) Vertical angles are congruent.2) Linear pairs are supplementary

(180°)3) The sum of the measures of the

angles in a triangle is 180°

Review Theorems:1) Vertical angles are congruent.2) Linear pairs are supplementary

(180°)3) The sum of the measures of the

angles in a triangle is 180°

Find m<1 and m<2 and state the theorem(s) that justify your answer.

1)

2)

95°

12

80°

1

2

In the accompanying diagrams, find all the variables.

3)

4)

x

37°

74°

5y 4y

In the accompanying diagrams, find all the variables.

5)

6)

(2x+10)

(3x-20)

(4x+9)

73°

Using the given information, determine if any lines are parallel, and if so, state which lines and theorem used.

z m

x

y

1 2 3 45678

9 10 11 1214 13

7) <7 is supplementary to <6 8) <9 = <11

9) <5 = <9 10) <6 = <12

Proofs Involving Parallel LinesPart 1: Given Parallel Lines

When you know that you are working with parallel lines you can use the theorems we learned yesterdays as reasons within your proof:

A. Alternate interior angles are congruent, when lines are parallel.

B. Corresponding angles are congruent, when lines are parallel.

C. Alternate exterior angles are congruent, when lines are parallel.

D. Same side interior angles are supplementary, when lines are parallel

Examples:

Statements Reasons

1) , and ABDCE // 21 1) Given

2) <1 = <A, <2 = <B2) Alternate Interior angles ,

when lines //.

3) <A = <B 3) Substitution Postulate

Part 2: Proving Lines Parallel

To prove two lines parallel we can use the converse of many of our theorems involving parallel lines.

A. If a pair of alternate interior angles are congruent, then the lines are parallel.

B. If a pair of corresponding angles are congruent, then the lines are parallel.

C. If a pair of same side interior angles are supplementary, then the lines are parallel.

There are two more methods of proving lines are parallel.

D. Two lines parallel to the same line are parallel to each other. (Transitive Property)

lm

p

If and

then

ml //pm //

pl //

If and

, then

ABCD ABEF EFCD //

E. If two lines are perpendicular to the same line, then they are parallel.

1) bisects , and BD ABC CDBC 1) Given

2) <ABD = <CBD2) A bisector divides the < into 2 congruent <‘s

3) <CBD = <CDB 3) If two sides of a Δ are congruent, then <‘s opposite are congruent4) <ABD = <CDB4) Substitution (or Transitive)

5) BACD //5) If a pair of alternate interior angles are congruent, then the lines are parallel.

1) Quad ABCD, and DABC DABC // 1) Given

2) <BCA = <DAC 2) Alternate Interior angles ,

when lines //.

3) AC = AC 3) Reflexive Postulate

4) ΔABC = ΔCDA 4) SAS

5) <BAC = <DCA 5) CPCTC

6) CDAB // 6) If a pair of alternate interior angles are congruent, then the lines are parallel.