PARALLEL LINES CUT BY A TRANSVERSAL

DEFINITIONS

• PARALLEL• TRANSVERSAL• ANGLE• VERTICAL ANGLE• CORRESPONDING ANGLE• ALTERNATE INTERIOR ANGLE• ALTERNATE EXTERIOR ANGLE

DEFINITIONS

• SUPPLEMENTARY ANGLE• COMPLEMENTARY ANGLE• CONGRUENT

Parallel lines cut by a transversal

123 4

567 8

Parallel lines cut by a transversal

123 4

567 8

< 1 and < 2 are called SUPPLEMENTARY ANGLES

DEFINITION:They form a straight angle measuring 180 degrees.

Parallel lines cut by a transversal

123 4

567 8

Name other supplementary pairs:

< 2 and < 3< 3 and < 4< 4 and < 1< 5 and < 6< 6 and < 7< 7 and < 8< 8 and < 5

Parallel lines cut by a transversal

123 4

567 8

< 1 and < 3 are called VERTICAL ANGLES

They are congruent m<1 = m<3

DEFINITION: The angles formed from two lines are crossing.

Parallel lines cut by a transversal

123 4

567 8

Name other vertical pairs:

< 2 and < 4

< 6 and < 8

< 5 and < 7

Parallel lines cut by a transversal

123 4

567 8

< 1 and < 5 are called CORRESPONDING ANGLES

They are congruent m<1 = m<5

DEFINITION: Corresponding angles occupy the same position on the top and bottom parallel lines.

Parallel lines cut by a transversal

123 4

567 8

Name other corresponding pairs:

< 2 and < 6

< 3 and < 7

< 4 and < 8

Parallel lines cut by a transversal

123 4

567 8

< 4 and < 6 are called ALTERNATE INTERIOR ANGLES

They are congruent m<4 = m<6

DEFINITION:Alternate Interior on the inside of the two parallel lines and on opposite sides of the transversal.

Parallel lines cut by a transversal

123 4

567 8

Name other alternate interior pairs:

< 3 and < 5

Parallel lines cut by a transversal

123 4

567 8

< 1 and < 7 are called ALTERNATE EXTERIOR ANGLES

They are congruent m<1 = m<7

Alternate Exterior on the outside of the two parallel lines and on opposite sides of the transversal.

Parallel lines cut by a transversal

123 4

567 8

Name other alternate exterior pairs:

< 2 and < 8

< 1 and < 7

Parallel lines cut by a transversal

123 4

567 8

< 4 and < 5 are called CONSECUTIVE INTERIOR ANGLES

The sum is 180. m<4 = m<5

DEFINITION: Consecutive Interior on the inside of the two parallel lines and on same side of the transversal. Sum = 180

TRY IT OUT

123 4

567 8

The m < 1 is 60 degrees.

What is the m<2 ?

120 degrees

TRY IT OUT

123 4

567 8

The m < 1 is 60 degrees.

What is the m<5 ?

60 degrees

TRY IT OUT

123 4

567 8

The m < 1 is 60 degrees.

What is the m<3 ?

60 degrees

TRY IT OUT

6012060 120

6012060 120

TRY IT OUT

x + 102x + 20

What do you know about the angles?

Write the equation.

Solve for x.

SUPPLEMENTARY

2x + 20 + x + 10 = 180

3x + 30 = 1803x = 150

x = 50

TRY IT OUT

2x - 60

3x - 120

What do you know about the angles?

Write the equation.

Solve for x.

ALTERNATE INTERIOR

3x - 120 = 2x - 60

x = 60Subtract 2x from both sides

Add 120 to both sides

Holt Geometry

3-2 Angles Formed by Parallel Lines and Transversals

Warm UpIdentify each angle pair.

1. 1 and 3

2. 3 and 6

3. 4 and 5

4. 6 and 7 same-side int s

corr. s

alt. int. s

alt. ext. s

Holt Geometry

3-2 Angles Formed by Parallel Lines and Transversals

Find each angle measure.

Example 1: Using the Corresponding Angles Postulate

A. mECF

x = 70

B. mDCE

mECF = 70°

Corr. s Post.

5x = 4x + 22 Corr. s Post.

x = 22 Subtract 4x from both sides.

mDCE = 5x

= 5(22) Substitute 22 for x.

= 110°

Holt Geometry

3-2 Angles Formed by Parallel Lines and Transversals

Check It Out! Example 1

Find mQRS.

mQRS = 180° – x

x = 118

mQRS + x = 180°

Corr. s Post.

= 180° – 118°

= 62°

Subtract x from both sides.

Substitute 118° for x.

Def. of Linear Pair

WEBSITES FOR PRACTICE

Ask Dr. Math: Corresponding /Alternate Angles

Project Interactive: Parallel Lines cut by Transversal

Triangle Sum Theorem

The sum of the angle measures in a triangle equal 180°

32

1

m<1 + m<2 + m<3 = 180°

Corollary•A corollary to a theorem is a statement that follows directly from that theorem

TRIANGLE ANGLE SUM THEOREM COROLLARIES

• If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent

• The acute angles of a right triangle are complementary

• The measure of each angle of an equiangular triangle is 60o

• A triangle can have at most 1 right or 1 obtuse angle

Exterior Angle Theorem(your new best friend)

The measure of an exterior angle in a triangle is the sum of the measures of the 2 remote interior angles

3

2

1 4

exterior angle

remote interior

angles

m<4 = m<1 + m<2

REMOTE INTERIOR ANGLE• In any polygon, a remote interior angle is an

interior angle that is not adjacent to a given exterior angle A and B are remote to angle 1

Exterior Angle Theorem Exterior Angle Theorem

1

2 3 4

P

Q R

In the triangle below, recall that 1, 2, and 3 are _______ angles ofΔPQR.

interior

Angle 4 is called an _______ angle of ΔPQR.exterior

An exterior angle of a triangle is an angle that forms a _________ with one ofthe angles of the triangle.

linear pair

In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3.

____________________ of a triangle are the two angles that do not forma linear pair with the exterior angle.Remote interior angles

In ΔPQR, 1, and 2 are the remote interior angles with respect to 4.

Exterior Angle Theorem Exterior Angle Theorem

1

2

3 4 5

In the figure below, 2 and 3 are remote interior angles with respect towhat angle? 5

an example with numbers

x

82°

30° y

find x & y

x = 68°

y = 112°

40x 10x2

30x find all the angle measures

80°, 60°, 40°

Do you hear the sirens?????

• Determine the measure of <4,

• If <3 = 50, <2 = 70

Exterior Angle Theorem Exterior Angle Theorem

Holt Geometry

4-2 Angle Relationships in Triangles

Find mB.

Example 3: Applying the Exterior Angle Theorem

mA + mB = mBCD Ext. Thm.

15 + 2x + 3 = 5x – 60 Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD.

2x + 18 = 5x – 60 Simplify.

78 = 3xSubtract 2x and add 60 to both sides.

26 = x Divide by 3.

mB = 2x + 3 = 2(26) + 3 = 55°

Holt Geometry

4-2 Angle Relationships in Triangles

There are several ways to prove certain triangles are similar. The following postulate, as well as the SSS and SAS Similarity Theorems, will be used in proofs just as SSS, SAS, ASA, HL, and AAS were used to prove triangles congruent.

Example 1: Using the AA Similarity Postulate

Explain why the triangles are similar and write a similarity statement.

Since , B E by the Alternate Interior Angles Theorem. Also, A D by the Right Angle Congruence Theorem. Therefore ∆ABC ~ ∆DEC by AA~.

Check It Out! Example 1

Explain why the trianglesare similar and write asimilarity statement.

By the Triangle Sum Theorem, mC = 47°, so C F. B E by the Right Angle Congruence Theorem. Therefore, ∆ABC ~ ∆DEF by AA ~.