Angle Addition Postulate (Geometry 3_3)

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



R

TS

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

X



R

TS


X

3) For each angle, find mRSX, mXST, and RST.



R

TS


X


30°

45°



R

TS


X


30°

45°

75°


R

TS

X

30°

45°

75°

1) How does the sum of mRSX and mXST compare to mRST ?


R

TS

X

30°

45°

75°

= mRST = 75

mXST = 30

+ mRSX = 45


Their sum is equal to the measure of RST .


R

TS

X

30°

45°

75°

= mRST = 75

mXST = 30

+ mRSX = 45


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



R

TS

X

30°

45°

75°

= mRST = 75

mXST = 30

+ mRSX = 45


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


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


Postulate 3-3

Angle Addition Postulate

For any angle PQR, if A is in the interior of PQR, thenmPQA + mAQR = mPQR.

2

1

A

R

P

Q


Postulate 3-3



2

1

A

R

P

Q m1 + m2 = mPQR.


Postulate 3-3



2

1

A

R

P

Q m1 + m2 = mPQR.

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


Postulate 3-3



2

1

A

R

P

Q m1 + m2 = mPQR.


m1 = mPQR – m2

m2 = mPQR – m1


Postulate 3-3



2

1

A

R

P

Q m1 + m2 = mPQR.


m1 = mPQR – m2

m2 = mPQR – m1

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


2

1Y

Z

X

W

Find m2 if mXYZ = 86 and m1 = 22.


2

1Y

Z

X

W


m2 + m1 = mXYZ Postulate 3 – 3.


2

1Y

Z

X

W


m2 = mXYZ – m1



2

1Y

Z

X

W


m2 = mXYZ – m1

m2 = 86 – 22



2

1Y

Z

X

W


m2 = mXYZ – m1

m2 = 86 – 22

m2 = 64



2x°

(5x – 6)°

B

DC

A

Find mABC and mCBD if mABD = 120.


2x°

(5x – 6)°

B

DC

A


mABC + mCBD = mABD Postulate 3 – 3.


2x°

(5x – 6)°

B

DC

A



2x + (5x – 6) = 120 Substitution


2x°

(5x – 6)°

B

DC

A




7x – 6 = 120 Combine like terms


2x°

(5x – 6)°

B

DC

A





7x = 126 Add 6 to both sides


2x°

(5x – 6)°

B

DC

A





7x = 126

x = 18

Add 6 to both sides

Divide each side by 7


2x°

(5x – 6)°

B

DC

A





7x = 126

x = 18

Add 6 to both sides


mABC = 2x

mABC = 2(18)

mABC = 36


2x°

(5x – 6)°

B

DC

A





7x = 126

x = 18

Add 6 to both sides


mABC = 2x

mABC = 2(18)

mABC = 36

mCBD = 5x – 6

mCBD = 5(18) – 6

mCBD = 90 – 6

mCBD = 84


2x°

(5x – 6)°

B

DC

A





7x = 126

x = 18

Add 6 to both sides


mABC = 2x

mABC = 2(18)

mABC = 36

mCBD = 5x – 6

mCBD = 5(18) – 6

mCBD = 90 – 6

mCBD = 84

36 + 84 = 120


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


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



This ray is called an ____________ .



This ray is called an ____________ .angle bisector



This ray is called an ____________ .angle bisector


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


Definition of

an Angle Bisector



2

1

A

R

P

Q

QA is the bisector of PQR.


Definition of

an Angle Bisector



2

1

A

R

P

Q

m1 = m2

QA is the bisector of PQR.


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

1

2

AC

N

T



ATSince bisects CAN, 1 = 2.

1

2

AC

N

T




1 + 2 = CAN Postulate 3 - 3

1

2

AC

N

T





1 + 2 = 130 Replace CAN with 130

1

2

AC

N

T






1 + 1 = 130 Replace 2 with 1

1

2

AC

N

T






1 + 1 = 130 Replace 2 with 1

2(1) = 130 Combine like terms

1

2

AC

N

T






1 + 1 = 130 Replace 2 with 1


(1) = 65 Divide each side by 21

2

AC

N

T






1 + 1 = 130 Replace 2 with 1



Since 1 = 2

1

2

AC

N

T






1 + 1 = 130 Replace 2 with 1



Since 1 = 2, 2 = 65

1

2

AC

N

T



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Angle Addition Postulate (Geometry 3_3)