Angle Addition Postulate (Geometry 3_3)

Students learn the Angle Addition Postulate and use it to solve problems.

1. The Angle Addition PostulateThe Angle Addition Postulate
You will learn to find the measure of an angle and the bisector
of an angle.
NOTHING NEW!

2. 1) Draw an acute,
an obtuse, or
a right angle.
Label the
angle RST.
R
T
S
The Angle Addition PostulateThe Angle Addition Postulate

3. 1) Draw an acute,
an obtuse, or
a right angle.
Label the
angle RST.
R
T
S
2) Draw and label
a point X in the
interior of the
angle. Then
draw SX.
X
The Angle Addition PostulateThe Angle Addition Postulate

4. 1) Draw an acute,
an obtuse, or
a right angle.
Label the
angle RST.
R
T
S
2) Draw and label
a point X in the
interior of the
angle. Then
draw SX.
X
3) For each angle, find mRSX, mXST, and RST.
The Angle Addition PostulateThe Angle Addition Postulate

5. 1) Draw an acute,
an obtuse, or
a right angle.
Label the
angle RST.
R
T
S
2) Draw and label
a point X in the
interior of the
angle. Then
draw SX.
X
3) For each angle, find mRSX, mXST, and RST.
30°
45°
The Angle Addition PostulateThe Angle Addition Postulate

6. 1) Draw an acute,
an obtuse, or
a right angle.
Label the
angle RST.
R
T
S
2) Draw and label
a point X in the
interior of the
angle. Then
draw SX.
X
3) For each angle, find mRSX, mXST, and RST.
30°
45°
75°
The Angle Addition PostulateThe Angle Addition Postulate

7. R
T
S
X
30°
45°
75°
1) How does the sum of mRSX and mXST compare to mRST ?
The Angle Addition PostulateThe Angle Addition Postulate

8. R
T
S
X
30°
45°
75°
= mRST = 75
mXST = 30
+ mRSX = 45
1) How does the sum of mRSX and mXST compare to mRST ?
Their sum is equal to the measure of RST .
The Angle Addition PostulateThe Angle Addition Postulate

9. R
T
S
X
30°
45°
75°
= mRST = 75
mXST = 30
+ mRSX = 45
1) How does the sum of mRSX and mXST compare to mRST ?
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

10. R
T
S
X
30°
45°
75°
= mRST = 75
mXST = 30
+ mRSX = 45
1) How does the sum of mRSX and mXST compare to mRST ?
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 two
smaller angles is equal to the measure
of the larger angle.
The Angle Addition PostulateThe Angle Addition Postulate

11. For any angle PQR, if A is in the interior of PQR, then
mPQA + mAQR = mPQR.
Postulate 3-3
Angle
Addition
Postulate
2
1
A
R
P
Q
The Angle Addition PostulateThe Angle Addition Postulate

12. For any angle PQR, if A is in the interior of PQR, then
mPQA + mAQR = mPQR.
Postulate 3-3
Angle
Addition
Postulate
2
1
A
R
P
Q m1 + m2 = mPQR.
The Angle Addition PostulateThe Angle Addition Postulate

13. For any angle PQR, if A is in the interior of PQR, then
mPQA + mAQR = mPQR.
Postulate 3-3
Angle
Addition
Postulate
2
1
A
R
P
Q m1 + m2 = mPQR.
There are two equations that can be derived using Postulate 3 – 3.
The Angle Addition PostulateThe Angle Addition Postulate

14. For any angle PQR, if A is in the interior of PQR, then
mPQA + mAQR = mPQR.
Postulate 3-3
Angle
Addition
Postulate
2
1
A
R
P
Q m1 + m2 = mPQR.
There are two equations that can be derived using Postulate 3 – 3.
m1 = mPQR – m2
m2 = mPQR – m1
The Angle Addition PostulateThe Angle Addition Postulate

15. For any angle PQR, if A is in the interior of PQR, then
mPQA + mAQR = mPQR.
Postulate 3-3
Angle
Addition
Postulate
2
1
A
R
P
Q m1 + m2 = mPQR.
There are two equations that can be derived using Postulate 3 – 3.
m1 = mPQR – m2
m2 = mPQR – m1
These equations are true no matter where A is located
in the interior of PQR.
The Angle Addition PostulateThe Angle Addition Postulate

16. 2
1
Y
Z
X
W
Find m2 if mXYZ = 86 and m1 = 22.
The Angle Addition PostulateThe Angle Addition Postulate

17. 2
1
Y
Z
X
W
Find m2 if mXYZ = 86 and m1 = 22.
m2 + m1 = mXYZ Postulate 3 – 3.
The Angle Addition PostulateThe Angle Addition Postulate

18. 2
1
Y
Z
X
W
Find m2 if mXYZ = 86 and m1 = 22.
m2 = mXYZ – m1
m2 + m1 = mXYZ Postulate 3 – 3.
The Angle Addition PostulateThe Angle Addition Postulate

19. 2
1
Y
Z
X
W
Find m2 if mXYZ = 86 and m1 = 22.
m2 = mXYZ – m1
m2 = 86 – 22
m2 + m1 = mXYZ Postulate 3 – 3.
The Angle Addition PostulateThe Angle Addition Postulate

20. 2
1
Y
Z
X
W
Find m2 if mXYZ = 86 and m1 = 22.
m2 = mXYZ – m1
m2 = 86 – 22
m2 = 64
m2 + m1 = mXYZ Postulate 3 – 3.
The Angle Addition PostulateThe Angle Addition Postulate

21. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
The Angle Addition PostulateThe Angle Addition Postulate

22. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD Postulate 3 – 3.
The Angle Addition PostulateThe Angle Addition Postulate

23. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD Postulate 3 – 3.
2x + (5x – 6) = 120 Substitution
The Angle Addition PostulateThe Angle Addition Postulate

24. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD Postulate 3 – 3.
2x + (5x – 6) = 120 Substitution
7x – 6 = 120 Combine like terms
The Angle Addition PostulateThe Angle Addition Postulate

25. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD 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

26. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD 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

27. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD 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
mABC = 2x
mABC = 2(18)
mABC = 36
The Angle Addition PostulateThe Angle Addition Postulate

28. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD 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
mABC = 2x
mABC = 2(18)
mABC = 36
mCBD = 5x – 6
mCBD = 5(18) – 6
mCBD = 90 – 6
mCBD = 84
The Angle Addition PostulateThe Angle Addition Postulate

29. 2x°
(5x – 6)°
B
D
C
A
Find mABC and mCBD if mABD = 120.
mABC + mCBD = mABD 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
mABC = 2x
mABC = 2(18)
mABC = 36
mCBD = 5x – 6
mCBD = 5(18) – 6
mCBD = 90 – 6
mCBD = 84
36 + 84 = 120
The Angle Addition PostulateThe Angle Addition Postulate

30. Just as every segment has a midpoint that bisects the segment, every angle
has a ___ that bisects the angle.
The Angle Addition PostulateThe Angle Addition Postulate

31. Just as every segment has a midpoint that bisects the segment, every angle
has a ___ that bisects the angle.ray
The Angle Addition PostulateThe Angle Addition Postulate

32. Just as every segment has a midpoint that bisects the segment, every angle
has a ___ that bisects the angle.ray
This ray is called an ____________ .
The Angle Addition PostulateThe Angle Addition Postulate

33. Just as every segment has a midpoint that bisects the segment, every angle
has a ___ that bisects the angle.ray
This ray is called an ____________ .angle bisector
The Angle Addition PostulateThe Angle Addition Postulate

34. Just as every segment has a midpoint that bisects the segment, every angle
has a ___ that bisects the angle.ray
This ray is called an ____________ .angle bisector
The Angle Addition PostulateThe Angle Addition Postulate

35. The bisector of an angle is the ray with its endpoint at the
vertex of the angle, extending into the interior of the
angle.
The bisector separates the angle into two angles of equal
measure.
Definition of
an Angle
Bisector
2
1
A
R
P
Q
The Angle Addition PostulateThe Angle Addition Postulate

36. The bisector of an angle is the ray with its endpoint at the
vertex of the angle, extending into the interior of the
angle.
The bisector separates the angle into two angles of equal
measure.
Definition of
an Angle
Bisector
2
1
A
R
P
Q
QAis the bisector of PQR.
The Angle Addition PostulateThe Angle Addition Postulate

37. The bisector of an angle is the ray with its endpoint at the
vertex of the angle, extending into the interior of the
angle.
The bisector separates the angle into two angles of equal
measure.
Definition of
an Angle
Bisector
2
1
A
R
P
Q
m1 = m2
QAis the bisector of PQR.
The Angle Addition PostulateThe Angle Addition Postulate

38. If bisects CAN and mCAN = 130, find 1 and 2.AT
1
2
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

39. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1
2
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

40. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1 + 2 = CANPostulate 3 - 3
1
2
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

41. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1 + 2 = CANPostulate 3 - 3
1 + 2 = 130 Replace CAN with 130
1
2
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

42. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1 + 2 = CANPostulate 3 - 3
1 + 2 = 130 Replace CAN with 130
1 + 1 = 130 Replace 2 with 1
1
2
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

43. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1 + 2 = CANPostulate 3 - 3
1 + 2 = 130 Replace CAN with 130
1 + 1 = 130 Replace 2 with 1
2(1) = 130 Combine like terms
1
2
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

44. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1 + 2 = CANPostulate 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
1
2
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

45. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1 + 2 = CANPostulate 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
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

46. If bisects CAN and mCAN = 130, find 1 and 2.AT
ATSince bisects CAN, 1 = 2.
1 + 2 = CANPostulate 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
A
C
N
T
The Angle Addition PostulateThe Angle Addition Postulate

47. The Angle Addition PostulateThe Angle Addition Postulate