Lesson 4.4Angle Properties

pp. 135-141

Lesson 4.4Angle Properties

pp. 135-141

Objectives:1. To identify linear pairs and

vertical, complementary, and supplementary angles.

2. To prove theorems on related angles.

Objectives:1. To identify linear pairs and

vertical, complementary, and supplementary angles.

2. To prove theorems on related angles.

A linear pair is a pair of adjacent angles whose noncommon sides form a straight angle (are opposite rays).

A linear pair is a pair of adjacent angles whose noncommon sides form a straight angle (are opposite rays).

DefinitionDefinitionDefinitionDefinition

AA BB CC

DD

Vertical angles are angles adjacent to the same angle and forming linear pairs with it.

Vertical angles are angles adjacent to the same angle and forming linear pairs with it.

DefinitionDefinitionDefinitionDefinition

AA BB CC

DD

EE

Two angles are complementary if the sum of their measures is 90°.

Two angles are supplementary if the sum of their measures is 180°.

Two angles are complementary if the sum of their measures is 90°.

Two angles are supplementary if the sum of their measures is 180°.

DefinitionDefinitionDefinitionDefinition

23°23°67°67°

TT FF XX

YY

CC

CFY and YFX are complementaryCFY and YFX are complementary

23°23°157°157°

TT FF XX

YY

CC

TFY and YFX are supplementaryTFY and YFX are supplementary

Theorem 4.1All right angles are congruent.

Theorem 4.1All right angles are congruent.

STATEMENTS REASONS

A and B are Givenright angles

12. mA = 90° 12. _______________mB = 90°

13. mA = mB 13. _______________

14.A B 14. _______________

STATEMENTS REASONS

A and B are Givenright angles

12. mA = 90° 12. _______________mB = 90°

13. mA = mB 13. _______________

14.A B 14. _______________

Def. of rt. angleDef. of rt. angle

SubstitutionSubstitution

Def. of anglesDef. of angles

Theorem 4.2If two angles are adjacent and supplementary, then they form a linear pair.

Theorem 4.2If two angles are adjacent and supplementary, then they form a linear pair.

Theorem 4.3Angles that form a linear pair are supplementary.

Theorem 4.3Angles that form a linear pair are supplementary.

Theorem 4.4If one angle of a linear pair is a right angle, then the other angle is also a right angle.

Theorem 4.4If one angle of a linear pair is a right angle, then the other angle is also a right angle.

Theorem 4.5Vertical Angle Theorem. Vertical angles are congruent.

Theorem 4.5Vertical Angle Theorem. Vertical angles are congruent.

Theorem 4.6Congruent supplementary angles are right angles.

Theorem 4.6Congruent supplementary angles are right angles.

Theorem 4.7Angle Bisector Theorem. If

AB bisects CAD, then mCAB = ½mCAD.

Theorem 4.7Angle Bisector Theorem. If

AB bisects CAD, then mCAB = ½mCAD.

Practice: If the mA = 58°, find the measure of the supplement of A.

Practice: If the mA = 58°, find the measure of the supplement of A.

Practice: If the mA = 58°, find the measure of the complement of A.

Practice: If the mA = 58°, find the measure of the complement of A.

Practice: If the mA = 58°, find the measure of an angle that makes a vertical angle with A.

Practice: If the mA = 58°, find the measure of an angle that makes a vertical angle with A.

Practice: If the mA = 58°, find the measure of an angle that makes a linear pair with A.

Practice: If the mA = 58°, find the measure of an angle that makes a linear pair with A.

Practice: If the mA = 58°, find the measures of the angles formed when A is bisected.

Practice: If the mA = 58°, find the measures of the angles formed when A is bisected.

Homeworkpp. 137-141Homeworkpp. 137-141

►A. ExercisesmAGF = 40°; mBGC = 50°; mAGE = 90°; mEGD = 90°.7. Name two pairs

of supplementaryangles.

►A. ExercisesmAGF = 40°; mBGC = 50°; mAGE = 90°; mEGD = 90°.7. Name two pairs

of supplementaryangles.

A

G

B

CD

E

F

►A. ExercisesmAGF = 40°; mBGC = 50°; mAGE = 90°; mEGD = 90°.9. What is mFGE?

►A. ExercisesmAGF = 40°; mBGC = 50°; mAGE = 90°; mEGD = 90°.9. What is mFGE?

A

G

B

CD

E

F

►B. ExercisesGive the reason for each step in the proofs below.18-22. Theorem 4.3Angles that form a linear pair are supplementary.Given: PAB and BAQ form a linear pairProve: PAD and BAQ are supplementary

►B. ExercisesGive the reason for each step in the proofs below.18-22. Theorem 4.3Angles that form a linear pair are supplementary.Given: PAB and BAQ form a linear pairProve: PAD and BAQ are supplementary

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?41. Addition property of

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?41. Addition property of

■ Cumulative Review Review properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?42. Multiplication property of

■ Cumulative Review Review properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?42. Multiplication property of

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?43. Reflexive property of

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?43. Reflexive property of

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?44. Transitive property of

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?44. Transitive property of

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?45. Why is not an equivalence

relation?

■ Cumulative ReviewReview properties of equality and inequality (Sections 3.1, 4.1). What would each property of inequality below be?45. Why is not an equivalence

relation?