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Splash Screen. Five-Minute Check (over Lesson 2–7) CCSS Then/Now Postulate 2.10: Protractor Postulate Postulate 2.11: Angle Addition Postulate Example 1: Use the Angle Addition Postulate Theorems 2.3 and 2.4 Example 2: Real-World Example: Use Supplement or Complement

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lesson menu

Five-Minute Check (over Lesson 2–7)

CCSS

Then/Now

Postulate 2.10: Protractor Postulate

Postulate 2.11: Angle Addition Postulate

Example 1: Use the Angle Addition Postulate

Theorems 2.3 and 2.4

Example 2: Real-World Example: Use Supplement or Complement

Theorem 2.5: Properties of Angle Congruence

Proof: Symmetric Property of Congruence

Theorems 2.6 and 2.7

Proof: One Case of the Congruent Supplements Theorem

Example 3: Proofs Using Congruent Comp. or Suppl. Theorems

Theorem 2.8: Vertical Angles Theorem

Example 4: Use Vertical Angles

Theorems 2.9–2.13: Right Angle Theorems

Lesson Menu
5 minute check 1

Justify the statement with a property of equality or a property of congruence.

A. Transitive Property

B. Symmetric Property

C. Reflexive Property

D. Segment Addition Postulate

5-Minute Check 1
5 minute check 2

Justify the statement with a property of equality or a property of congruence.

A. Transitive Property

B. Symmetric Property

C. Reflexive Property

D. Segment Addition Postulate

5-Minute Check 2
5 minute check 3

Justify the statement with a property of equality or a property of congruence.If H is between G and I, then GH + HI = GI.

A. Transitive Property

B. Symmetric Property

C. Reflexive Property

D. Segment Addition Postulate

5-Minute Check 3
5 minute check 4

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

A.WX > WZ

B.XW + WZ = XZ

C.XW + XZ = WZ

D.WZ – XZ = XW

5-Minute Check 4
5 minute check 5

State a conclusion that can be drawn from the statements given using the property indicated.

___

___

LMNO

A.

B.

C.

D.

5-Minute Check 5
5 minute check 6

___

Given B is the midpoint of AC, which of the following is true?

A.AB + BC = AC

B.AB + AC = BC

C.AB = 2AC

D.BC = 2AB

5-Minute Check 6
slide9

Content Standards

G.CO.9 Prove theorems about lines and angles.

Mathematical Practices

3 Construct viable arguments and critique the reasoning of others.

6 Attend to precision.

CCSS
then now

You identified and used special pairs of angles.

  • Write proofs involving supplementary and complementary angles.
  • Write proofs involving congruent and right angles.
Then/Now
example 1

Use the Angle Addition Postulate

CONSTRUCTIONUsing a protractor, a construction worker measures that the angle a beam makes with a ceiling is 42°. What is the measure of the angle the beam makes with the wall?

The ceiling and the wall make a 90 angle. Let 1 be the angle between the beam and the ceiling. Let 2 be the angle between the beam and the wall.

m1 + m2 = 90 Angle Addition Postulate

42 + m2 = 90 m1 = 42

42 – 42 + m2 = 90 – 42 Subtraction Property of Equality

m2 = 48 Substitution

Example 1
example 11

Use the Angle Addition Postulate

Answer: The beam makes a 48° angle with the wall.

Example 1
example 2

Use Supplement or Complement

TIMEAt 4 o’clock, the angle between the hour and minute hands of a clock is 120º. When the second hand 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?

UnderstandMake a sketch of the situation. The time is 4 o’clock and the second hand bisects the angle between the hour and minute hands.

Example 2
example 21

60 + 60 = 120

Use Supplement or Complement

PlanUse the Angle Addition Postulate and the definition of angle bisector.

Solve Since the angles are congruent by the definition of angle bisector, each angle is 60°.

Answer: Both angles are 60°.

CheckUse the Angle Addition Postulate to check your answer.

m1 + m2 = 120

120 = 120

Example 2
example 22

QUILTINGThe diagram shows one square for a particular quilt pattern. If mBAC = mDAE = 20, and BAE is a right angle, find mCAD.

A. 20

B. 30

C. 40

D. 50

Example 2
example 3

Given:

Prove:

Proofs Using Congruent Comp. or Suppl. Theorems

Example 3
example 31

Proof:

Statements Reasons

1. m3 + m1 = 180;1 and 4 form a linear pair.

1. Given

2. 1 and 4 are supplementary.

2. Linear pairs are supplementary.

3. 3 and 1 are supplementary.

3. Definition of supplementary angles

4. 3  4

4. s suppl. to same  are .

Proofs Using Congruent Comp. or Suppl. Theorems

Example 3
example 32

In the figure, NYR andRYA form a linear pair,AXY and AXZ form a linear pair, and RYA andAXZ are congruent. Prove that NYR and AXY are congruent.

Example 3
example 33

Statements Reasons

1. NYR and RYA, AXY and AXZ form linear pairs.

1. Given

2. NYR and RYA are supplementary. AXY and AXZ are supplementary.

2. If two s form a linear pair, then theyaresuppl.s.

3. Given

3. RYA  AXZ

?

4. ____________

4.NYR  AXY

Which choice correctly completes the proof?

Proof:

Example 3
example 34

A. Substitution

B. Definition of linear pair

C. s supp. to the same  or to  s are .

D. Definition of supplementary s

Example 3
example 4

Proof:

Statements Reasons

Use Vertical Angles

If 1 and 2 are vertical angles and m1 = d – 32 and m2 = 175 – 2d, find m1 and m2. Justify each step.

1. 1 and 2are vertical s.

1. Given

2. 1  2

2. Vertical Angles Theorem

3. Definition ofcongruent angles

3. m1 = m2

4. d – 32 = 175 – 2d

4. Substitution

Example 4
example 41

Use Vertical Angles

Statements Reasons

5. 3d – 32 = 175

5. Addition Property

6. 3d = 207

6. Addition Property

7. Division Property

7. d = 69

m1 = d – 32 m2 = 175 – 2d

= 69 – 32 or 37 = 175 – 2(69) or 37

Answer:m1 = 37 and m2 = 37

Example 4