The Angle Addition PostulateThe Angle Addition Postulate

You will learn to find the measure of an angle and the bisectorof an angle.

NOTHING NEW!

1) Draw an acute, an obtuse, or a right angle. Label the angle RST.

R

TS

The Angle Addition PostulateThe Angle Addition Postulate

1) Draw an acute, an obtuse, or a right angle. Label the angle RST.

R

TS

2) Draw and label a point X in the interior of the angle. Then draw SX.

X

The Angle Addition PostulateThe Angle Addition Postulate

1) Draw an acute, an obtuse, or a right angle. Label the angle RST.

R

TS

2) Draw and label a point X in the interior of the angle. Then draw SX.

X

3) For each angle, find m<RSX, m<XST, and <RST.

The Angle Addition PostulateThe Angle Addition Postulate

1) Draw an acute, an obtuse, or a right angle. Label the angle RST.

R

TS

2) Draw and label a point X in the interior of the angle. Then draw SX.

X

3) For each angle, find m<RSX, m<XST, and <RST.

30°

45°

The Angle Addition PostulateThe Angle Addition Postulate

1) Draw an acute, an obtuse, or a right angle. Label the angle RST.

R

TS

2) Draw and label a point X in the interior of the angle. Then draw SX.

X

3) For each angle, find m<RSX, m<XST, and <RST.

30°

45°

75°

The Angle Addition PostulateThe Angle Addition Postulate

R

TS

X

30°

45°

75°

1) How does the sum of m<RSX and m<XST compare to m<RST ?

The Angle Addition PostulateThe Angle Addition Postulate

R

TS

X

30°

45°

75°

= m<RST = 75

m<XST = 30

+ m<RSX = 45

1) How does the sum of m<RSX and m<XST compare to m<RST ?

Their sum is equal to the measure of <RST .

The Angle Addition PostulateThe Angle Addition Postulate

R

TS

X

30°

45°

75°

= m<RST = 75

m<XST = 30

+ m<RSX = 45

1) How does the sum of m<RSX and m<XST compare to m<RST ?

2) Make a conjecture about the relationship between the two smaller angles and the larger angle.

Their sum is equal to the measure of <RST .

The Angle Addition PostulateThe Angle Addition Postulate

R

TS

X

30°

45°

75°

= m<RST = 75

m<XST = 30

+ m<RSX = 45

1) How does the sum of m<RSX and m<XST compare to m<RST ?

2) Make a conjecture about the relationship between the two smaller angles and the larger angle.

Their sum is equal to the measure of <RST .

The sum of the measures of the twosmaller angles is equal to the measureof the larger angle.

The Angle Addition PostulateThe Angle Addition Postulate

Postulate 3-3

Angle Addition Postulate

For any angle PQR, if A is in the interior of <PQR, thenm<PQA + m<AQR = m<PQR.

2

1

A

R

P

Q

The Angle Addition PostulateThe Angle Addition Postulate

Postulate 3-3

Angle Addition Postulate

For any angle PQR, if A is in the interior of <PQR, thenm<PQA + m<AQR = m<PQR.

2

1

A

R

P

Q m<1 + m<2 = m<PQR.

The Angle Addition PostulateThe Angle Addition Postulate

Postulate 3-3

Angle Addition Postulate

For any angle PQR, if A is in the interior of <PQR, thenm<PQA + m<AQR = m<PQR.

2

1

A

R

P

Q m<1 + m<2 = m<PQR.

There are two equations that can be derived using Postulate 3 – 3.

The Angle Addition PostulateThe Angle Addition Postulate

Postulate 3-3

Angle Addition Postulate

For any angle PQR, if A is in the interior of <PQR, thenm<PQA + m<AQR = m<PQR.

2

1

A

R

P

Q m<1 + m<2 = m<PQR.

There are two equations that can be derived using Postulate 3 – 3.

m<1 = m<PQR – m<2

m<2 = m<PQR – m<1

The Angle Addition PostulateThe Angle Addition Postulate

Postulate 3-3

Angle Addition Postulate

For any angle PQR, if A is in the interior of <PQR, thenm<PQA + m<AQR = m<PQR.

2

1

A

R

P

Q m<1 + m<2 = m<PQR.

There are two equations that can be derived using Postulate 3 – 3.

m<1 = m<PQR – m<2

m<2 = m<PQR – m<1

These equations are true no matter where A is locatedin the interior of <PQR.

The Angle Addition PostulateThe Angle Addition Postulate

2

1Y

Z

X

W

Find m<2 if m<XYZ = 86 and m<1 = 22.

The Angle Addition PostulateThe Angle Addition Postulate

2

1Y

Z

X

W

Find m<2 if m<XYZ = 86 and m<1 = 22.

m<2 + m<1 = m<XYZ Postulate 3 – 3.

The Angle Addition PostulateThe Angle Addition Postulate

2

1Y

Z

X

W

Find m<2 if m<XYZ = 86 and m<1 = 22.

m<2 = m<XYZ – m<1

m<2 + m<1 = m<XYZ Postulate 3 – 3.

The Angle Addition PostulateThe Angle Addition Postulate

2

1Y

Z

X

W

Find m<2 if m<XYZ = 86 and m<1 = 22.

m<2 = m<XYZ – m<1

m<2 = 86 – 22

m<2 + m<1 = m<XYZ Postulate 3 – 3.

The Angle Addition PostulateThe Angle Addition Postulate

2

1Y

Z

X

W

Find m<2 if m<XYZ = 86 and m<1 = 22.

m<2 = m<XYZ – m<1

m<2 = 86 – 22

m<2 = 64

m<2 + m<1 = m<XYZ Postulate 3 – 3.

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

2x + (5x – 6) = 120 Substitution

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

2x + (5x – 6) = 120 Substitution

7x – 6 = 120 Combine like terms

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

2x + (5x – 6) = 120 Substitution

7x – 6 = 120 Combine like terms

7x = 126 Add 6 to both sides

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

2x + (5x – 6) = 120 Substitution

7x – 6 = 120 Combine like terms

7x = 126

x = 18

Add 6 to both sides

Divide each side by 7

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

2x + (5x – 6) = 120 Substitution

7x – 6 = 120 Combine like terms

7x = 126

x = 18

Add 6 to both sides

Divide each side by 7

m<ABC = 2x

m<ABC = 2(18)

m<ABC = 36

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

2x + (5x – 6) = 120 Substitution

7x – 6 = 120 Combine like terms

7x = 126

x = 18

Add 6 to both sides

Divide each side by 7

m<ABC = 2x

m<ABC = 2(18)

m<ABC = 36

m<CBD = 5x – 6

m<CBD = 5(18) – 6

m<CBD = 90 – 6

m<CBD = 84

The Angle Addition PostulateThe Angle Addition Postulate

2x°

(5x – 6)°

B

DC

A

Find m<ABC and m<CBD if m<ABD = 120.

m<ABC + m<CBD = m<ABD Postulate 3 – 3.

2x + (5x – 6) = 120 Substitution

7x – 6 = 120 Combine like terms

7x = 126

x = 18

Add 6 to both sides

Divide each side by 7

m<ABC = 2x

m<ABC = 2(18)

m<ABC = 36

m<CBD = 5x – 6

m<CBD = 5(18) – 6

m<CBD = 90 – 6

m<CBD = 84

36 + 84 = 120

The Angle Addition PostulateThe Angle Addition Postulate

Just as every segment has a midpoint that bisects the segment, every anglehas a ___ that bisects the angle.

The Angle Addition PostulateThe Angle Addition Postulate

Just as every segment has a midpoint that bisects the segment, every anglehas a ___ that bisects the angle.ray

The Angle Addition PostulateThe Angle Addition Postulate

Just as every segment has a midpoint that bisects the segment, every anglehas a ___ that bisects the angle.ray

This ray is called an ____________ .

The Angle Addition PostulateThe Angle Addition Postulate

Just as every segment has a midpoint that bisects the segment, every anglehas a ___ that bisects the angle.ray

This ray is called an ____________ .angle bisector

The Angle Addition PostulateThe Angle Addition Postulate

Just as every segment has a midpoint that bisects the segment, every anglehas a ___ that bisects the angle.ray

This ray is called an ____________ .angle bisector

The Angle Addition PostulateThe Angle Addition Postulate

Definition of

an Angle Bisector

The bisector of an angle is the ray with its endpoint at thevertex of the angle, extending into the interior of the angle.

The bisector separates the angle into two angles of equalmeasure.

2

1

A

R

P

Q

The Angle Addition PostulateThe Angle Addition Postulate

Definition of

an Angle Bisector

The bisector of an angle is the ray with its endpoint at thevertex of the angle, extending into the interior of the angle.

The bisector separates the angle into two angles of equalmeasure.

2

1

A

R

P

Q

QA is the bisector of <PQR.

The Angle Addition PostulateThe Angle Addition Postulate

Definition of

an Angle Bisector

The bisector of an angle is the ray with its endpoint at thevertex of the angle, extending into the interior of the angle.

The bisector separates the angle into two angles of equalmeasure.

2

1

A

R

P

Q

m<1 = m<2

QA is the bisector of <PQR.

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

<1 + <2 = <CAN Postulate 3 - 3

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

<1 + <2 = <CAN Postulate 3 - 3

<1 + <2 = 130 Replace <CAN with 130

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

<1 + <2 = <CAN Postulate 3 - 3

<1 + <2 = 130 Replace <CAN with 130

<1 + <1 = 130 Replace <2 with <1

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

<1 + <2 = <CAN Postulate 3 - 3

<1 + <2 = 130 Replace <CAN with 130

<1 + <1 = 130 Replace <2 with <1

2(<1) = 130 Combine like terms

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

<1 + <2 = <CAN Postulate 3 - 3

<1 + <2 = 130 Replace <CAN with 130

<1 + <1 = 130 Replace <2 with <1

2(<1) = 130 Combine like terms

(<1) = 65 Divide each side by 21

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

<1 + <2 = <CAN Postulate 3 - 3

<1 + <2 = 130 Replace <CAN with 130

<1 + <1 = 130 Replace <2 with <1

2(<1) = 130 Combine like terms

(<1) = 65 Divide each side by 2

Since <1 = <2

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

If bisects <CAN and m<CAN = 130, find <1 and <2.AT

ATSince bisects <CAN, <1 = <2.

<1 + <2 = <CAN Postulate 3 - 3

<1 + <2 = 130 Replace <CAN with 130

<1 + <1 = 130 Replace <2 with <1

2(<1) = 130 Combine like terms

(<1) = 65 Divide each side by 2

Since <1 = <2, <2 = 65

1

2

AC

N

T

The Angle Addition PostulateThe Angle Addition Postulate

The Angle Addition PostulateThe Angle Addition Postulate