Lines and Angles

You will be able to identify relationships between lines and angles formed by transversals.

Relationships between lines

Parallel Lines two lines that are coplanar and do not intersect.

Skew Lines lines that do not intersect and are not coplanar.

Parallel Planes two planes that do not intersect.

Segments and rays can also be parallel if they lie on parallel lines.

The notation for parallel lines: “line m and line n are parallel” m || n

Parallel and Perpendicular Postulates

Parallel Postulate If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.

Perpendicular Postulate If there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line.

Try It Out!

Draw a line (a straight one!) and a point anywhere in relation to that line (above or below).

How many lines can you draw through that one point that are parallel to your line?

How many lines can you draw through that one point hat are perpendicular to your line?

Identifying Angles formed by transversals.

Transversal a line that intersects two or more coplanar lines at different points. Transversal

Coplanar Lines

Corresponding Angles two angles that are in corresponding (“the same”) positions. Angles 1 and 5 are corresponding angles.

1

58

3

2

4

6

7

Alternate Exterior Angles two angles that lie outside the parallel lines and on opposite sides of the transversal. Angles 1 and 8 are alternate exterior angles.

1

58

3

2

4

6

7

Alternate Interior Angles two angles that lie between the parallel lines and on opposite sides of the transversal. Angles 3 and 6 are alternate interior angles.

1

58

3

2

4

6

7

Consecutive Interior Angles two angles that lie between the parallel lines and are on the same side of the transversal. Angles 3 and 5 are consecutive interior angles. These angles are sometimes called “same side interior angles.”

1

58

3

2

4

6

7

Using the diagram, list all pairs of angles that fit the description.

1. Corresponding 2. Alternate exterior3. Alternate interior4. Consecutive interior

1

2

3

4

5

6

8

7

Homework Assignment

Worksheet 3.1 all of it

Parallel Lines and Transversals

GeometrySection 3.2Objective: To identify relationships of angles formed by parallel lines cut by a transversal.

Angles and Parallel Lines Activity

Using a ruler, trace over two of the parallel lines on your paper that are near the middle of the your half piece of paper and about an inch apart.

Draw a transversal that makes clearly acute and clearly obtuse angles near the center of the paper

Label the angles with numbers from 1 to 8

Sketch the parallel lines, transversal, and number labels in your notes. We will use this to record observations.

Angles and Parallel Lines Activity Cut the paper carefully along the lines you

first drew to make six pieces. Try stacking different numbered angles

onto each other and see what you observe. Try placing different numbered angles next

to each other and see what you Observe Mark your observations on the sketch in

your notes

Angles and Parallel Lines Activity

Answer the following questions How many different sizes of angles where

formed? 2

What special relationships exist between the angles Congruent and supplementary

Indicate the two different sizes of angles in your sketch.

Angles and Parallel Lines Activity

How can we use the vocabulary learned yesterday, to describe these relationships?

IF parallel lines are cut by a transversal, THEN corresponding angles are congruent (Postulate in Text) alternate interior angles are congruent (Theorem in Text) alternate exterior angles are congruent (Theorem in Text) Consecutive Interior angles are Supplementary (Theorem in

Text)

Perpendicular Transversal In your notes, trace over two of the parallel lines

about one inch apart. Using a protractor, draw a line perpendicular to one

of the parallel lines. Extend this perpendicular so that it crosses the

other parallel line. Based on your observations in the previous exercise,

what should be true about the new angles formed? Verify this with your protractor.

If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other. (Theorem in Text)

3.3 Proving Lines are Parallel

Standard/Objectives:

Standard 3: Students will learn and apply geometric concepts

Objectives: Prove that two lines are parallel. Use properties of parallel lines to solve

real-life problems, such as proving that prehistoric mounds are parallel.

Properties of parallel lines help you predict.

PropertiesReflexive Property -

General: a =aAngles:Segments: AB = AB

Symmetric Property –General: If a = b then b = a.Angles:Segments: If AB = CD then CD = AB

Transitive Property-General: If a = b and b = c then a = cAngles:Segments If AB = CD and CD = EF then AB = EF

m A m A

then m B = m AIf m A m B

and m then m = mIf m A m B B m C A C

Postulate: Corresponding Angles Converse

If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.

Theorem: Alternate Interior Angles Converse If two lines are cut by a

transversal so that alternate interior angles are congruent, then the lines are parallel.

Theorem: Consecutive Interior Angles Converse If two lines are cut by a

transversal so that consecutive interior angles are supplementary, then the lines are parallel.

Theorem: Alternate Exterior Angles Converse If two lines are cut by a

transversal so that alternate exterior angles are congruent, then the lines are parallel.

Prove the Alternate Interior Angles ConverseGiven: 1 2

Prove: m ║ n

1

2

3m

n

Example 1: Proof of Alternate Interior Converse

Statements:

1. 1 22. 2 33. 1 34. m ║ n

Reasons:

1. Given2. Vertical Angles3. Transitive prop.4. Corresponding

angles converse

Proof of the Consecutive Interior Angles ConverseGiven: 4 and 5 are

supplementary

Prove: g ║ h

6g

h

5

4

Paragraph ProofYou are given that 4 and 5 are

supplementary. By the Linear Pair Postulate, 5 and 6 are also supplementary because they form a linear pair. By the Congruent Supplements Theorem, it follows that 4 6. Therefore, by the Alternate Interior Angles Converse, g and h are parallel.

Find the value of x that makes j ║ k.

Solution:Lines j and k will be parallel if the marked angles are supplementary.

x + 4x = 180 5x = 180 X = 36 4(36) = 144

So, if x = 36, then j ║ k.

x 4x

Using Parallel Converses:Using Corresponding Angles Converse

SAILING. If two boats sail at a 45 angle to the wind as shown, and the wind is constant, will their paths ever cross? Explain

Solution:Because corresponding angles are

congruent, the boats’ paths are parallel. Parallel lines do not intersect, so the boats’ paths will not cross.

Example 5: Identifying parallel linesDecide which rays are parallel.

6261

5958

A B

E H G

DC

A. Is EB parallel to HD?B. Is EA parallel to HC?

Example 5: Identifying parallel linesDecide which rays are parallel.

6158

B

E H G

D

A. Is EB parallel to HD?mBEH = 58m DHG = 61 The angles are

corresponding, but not congruent, so EB and HD are not parallel.

Example 5: Identifying parallel linesDecide which rays are parallel.

120120

A

E H G

C

A. B. Is EA parallel to HC?m AEH = 62 + 58m CHG = 59 + 61AEH and CHG are congruent

corresponding angles, so EA ║HC.

Conclusion:Two lines are cut by a transversal.

How can you prove the lines are parallel?

Show that either a pair of alternate interior angles, or a pair of corresponding angles, or a pair of alternate exterior angles is congruent, or show that a pair of consecutive interior angles is supplementary.