Geometry : Chapter 3 Sect. 3.1 Lines and Angles

Sect. 3.2 Properties of Parallel Lines and

Sect. 3.3 Proving lines are Parallel

Connections

• Chapter 2

• Inductive reasoning

• Deductive reasoning

• Proofs and Theorems

• Chapter 1:

▫ A point - •B

▫ A line- ↔ CD

▫ A ray- → EF

▫ An angle- M or LMN

▫ A segment- GH

▫ Collinear points

▫ Coplanar points or lines

▫ A plane-

•A

B•

C•

P

•A

•C

• B

Objectives

• Identify relationships between figures in space.

• Identify angles formed by two lines and a transversal

• To prove theorems about parallel lines.

• To use properties of parallel lines to find angle measures.

Essential Question

• How can you prove two lines are parallel?

Vocabulary-

• Transversal Line-

▫ Line that intersects two or more coplanar lines at distinct points

▫ 8 angles formed by line t

▫ Line t intersects line l and m

Alternate Interior angle-

▫ Non-adjacent interior angles

▫ Lie on opposite sides of transversal line

▫ 12

▫ 34

Example #1

• Solution:

• Step 1:Set them equal to each other

• Step 2: Solve for x If 2 = x+15 and 8= 77 What is x?

Vertical Angle Theorem 2-1

• Vertical angles are congruent

Example #2

• Solution:

• Step 1:Set them equal to each other

• Step 2: Solve for x

• If 5 =2x+10 and 7= 156

• What is x?

Corresponding Angles- ▫ Lie on same side of

transversal t in equivalent (the same) positions

12 56 34 78

Example #3

• Solution:

• Step 1:Set them equal to each other

• Step 2: Solve for x • What is the value of x for which a||b, if 6=3x-2 and 2= 55?

Alternate exterior angles- ▫ Nonadjacent exterior angles

▫ That lie on opposite sides of transversal t.

▫ 12

▫ 43

Same-Side Interior- ▫ Interior angles

▫ Lie on same-side of transversal line

Example #4

• In the figure at the right, what is the value of x?

• What is the value of y?

• What are the measures of each angle in the figure?

2x˚ 3y˚

(x-12)˚ (y+20)˚

Adjacent Angles and Linear Pairs

• Since a straight angle contains 180°, these two adjacent angles add to 180.

• They form a linear pair. (Adjacent angles share a vertex, share a side, and do not overlap.)

Question # 1

• What is 2 and 7?

• A. corresponding angles

• B. same-side interior angles

• C. alternate interior angles

• D. alternate exterior angles

Question # 2

• What is 3 and 5?

• A. corresponding angles

• B. same-side interior angles

• C. alternate interior angles

• D. alternate exterior angles

Question # 3

• What is 3 and 6?

• A. corresponding angles

• B. same-side interior angles

• C. alternate interior angles

• D. alternate exterior angles

Question # 4

• What is 6 and 7?

• A. corresponding angles

• B. same-side interior angles

• C. alternate interior angles

• D. alternate exterior angles

• E. vertical angles

Question # 5

• What is 3 and 4?

• A. corresponding angles

• B. same-side interior angles

• C. linear angle pairs (adjacent)

• D. alternate interior angles

• E. alternate exterior angles

Question # 6

• What is 1 and 5?

• A. corresponding angles

• B. same-side interior angles

• C. alternate interior angles

• D. alternate exterior angles

Real World Connections

Recap: Summary

• If two lines are cut by a transversal they form special properties.

▫ Corresponding angles

▫ Alternate interior angles

▫ Alternate exterior angles

Are congruent.

Are parallel

▫ Same-side interior angles are supplementary.

• List all different ways that you can prove that two lines are parallel.

• Show that same-side interior angles are supplementary.

• Show that one of the following pairs are congruent: alternate exterior, alternate interior, corresponding angles.

• Sect. 3-1 Pg. 152-153 #’s 16- 19, 32

• Sect 3-2 pg. 162-163 7, 8, 11,12, 14, 15, 16, 19

• Sect 3-3 Pg 171-172 #’s 5, 6, 8, 9, 10,11, 12

Ticket Out Homework