Lesson 2-8

Proving Angle Relationships

5-Minute Check on Lesson 2-75-Minute Check on Lesson 2-75-Minute Check on Lesson 2-75-Minute Check on Lesson 2-7 Transparency 2-8

Justify each statement with a property of equality or aproperty of congruence.1. If AB CD and CD EF, then AB EF.2. RS RS3. If H is between G and I, then GH + HI = GI.

State a conclusion that can be drawn from the statementsgiven using the property indicated.4. W is between X and Z; Segment Addition Postulate

5. LM NO and NO PQ; Transitive Property of Congruence

6. Which statement is true, given that K is between J and L?

Standardized Test Practice:

A

C

B

D

JK + KL = JL JL + LK = JK

LJ + JK = LK JK KL

5-Minute Check on Lesson 2-75-Minute Check on Lesson 2-75-Minute Check on Lesson 2-75-Minute Check on Lesson 2-7 Transparency 2-8

Justify each statement with a property of equality or aproperty of congruence.1. If AB CD and CD EF, then AB EF. Transitive Property2. RS RS Reflexive Property3. If H is between G and I, then GH + HI = GI.

Segment Addition PostulateState a conclusion that can be drawn from the statementsgiven using the property indicated.4. W is between X and Z; Segment Addition Postulate

XW + WZ = XZ5. LM NO and NO PQ; Transitive Property of Congruence

LM PQ6. Which statement is true, given that K

is between J and L?

Standardized Test Practice:

A

C

B

D

JK + KL = JL JL + LK = JK

LJ + JK = LK JK KL

Objectives

• Write proofs involving supplementary and complementary angles

• Write proofs involving congruent and right angles

Vocabulary

• No new vocabulary

Postulate 2.10, Protractor Postulate: Given ray AB and a number r between 0 and 180, there is exactly one ray with endpoint A, extending on either side of ray AB, such that the angle formed measures r°.

Postulate 2.11, Angle Addition Postulate: If R is in the interior of PQS, then mPQR + mRQS = mPQS and if mPQR + mRQS = mPQS, then R is in the interior of PQS

Postulates

Theorem 2.3, Supplement Theorem: if two angles form a linear pair, then they are supplementary angles.

Theorem 2.4, Complement Theorem: if the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles.

Theorem 2.5, Angles supplementary to the same angle or to congruent angles are congruent.

Theorem 2.6, Angles complementary to the same angle or to congruent angles are congruent.

Theorem 2.7, Vertical Angles Theorem: If two angles are vertical angles, then they are congruent.

Theorems

Theorem 2.9, Perpendicular lines intersect to form four right angles.

Theorem 2.10, All right angles are congruent.

Theorem 2.11, Perpendicular lines form congruent adjacent angles.

Theorem 2.12, If two angles are congruent and supplementary, then each angle is a right angle.

Theorem 2.13, If two congruent angles form a linear pair, then they are right angles.

Theorems

Given: 1 and 3 are vertical anglesm1 = 3x + 5, m3 = 2x + 8

Prove: m1 = 14

Angle Proof

Statement Reason

1 and 3 are vertical angles Given

m1 m3 Vertical Angles Theorem

m1 = m3 Congruence Definition

m1 = 3x + 5 Given

m3 = 2x + 8 Given

3x + 5 = 2x + 8 Substitution

3x – 2x + 5 = 2x – 2x + 8 Subtraction

x + 5 = 8 Substitution

x + 5 – 5 = 8 – 5 Subtraction

x = 3 Substitution

m1 = 3 (3) + 5 = 14 Substitution (twice)

2 3

1 4

TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands?

Solution: If the second hand stops where the angle is bisected, then the angle between the minute and second hands is one-half the measure of the angle formed by the hour and minute hands, or ½(120º) = 60º.

By the Angle Addition Postulate, the sum of the two angles is 120º, so the angle between the second and hour hands is also 60º.

Answer: They are both 60º by the definition of angle bisector and the Angle Addition Postulate.

Answer: 50

QUILTING The diagram below shows one square for a particular quilt pattern. If mBAC = mDAE = 20, and BAE is a right angle, find mCAD.

m1 + m2 = 180 Supplement Theorem

m1 + 166 – 166 = 180 – 166 Subtraction Property

m1 = 14 Substitution

Answer: 14

If 1 and 2 form a linear pair, and m2 = 166, find m1.

Solution:

m2 = 166 Given

m1 + 166 = 180 Substitution

Answer: 28

If 1 and 2 are complementary angles and m1 = 62, find m2.

Given: 1 and 4 form a linear pair m3 + m1 = 180.

Prove: 3 4

1. Given1.

3. Definition of supplementary angles

3.

4. Subtraction Property4.

5. Substitution5.

6. Definition of congruent angles

6.

Proof:

Statements Reasons

2. Linear pairs are supplementary.

2.

Given: NYR and RYA form a linear pair,AXY and AXZ form a linear pair, RYA AXZ.

Prove: RYN AXY

Proof:

Statements Reasons

1. Given

2. If two s form a linear pair, then they are suppl. s.

3. Given

4.

1.

2.

3.

4.

linear pairs.

d – 32 = 175 – 2d Substitution

3d – 32 = 175 Add 2d to each side.

3d = 207 Add 32 to each side.

d = 69 Divide each side by 3.

If 1 and 2 are vertical angles and m1 = d - 32 and m2 = 175 – 2d, find m1 and m2.

1 2 Vertical Angles Theorem

m1 = m2 Definition of congruent angles

Answer: m1 = 37 and m2 = 37

Answer: mA = 52; mZ = 52

If A and Z are vertical angles and mA = 3b -23 and mZ = 152 – 4b, find mA and mZ.

Summary & Homework

• Summary:– Properties of equality and congruence can be

applied to angle relationships

• Homework: – pg 112-3: 16-23, 27-32, 41