Download My CCSD

Segment and Angle Addition
Postulates and Proofs
Unit 3 Lesson 14 Notes
Line Segment
A line segment consists of two points
called endpoints of the segment and
all the points between them.
A
D
A piece of spaghetti is a physical
model of a line segment.
H
In order for you to say that a point B is
between two points A and C, all three points
must lie on the same line, and AB + BC = AC.
Given: Plane Figure
R
Conclusion:
D
C
Using Segment Addition Postulate,
answer the following questions:
Example 1: If DT = 60, find the value of x. Then find DS and
ST.
Segment Addition Continued:
Example 2: If EG = 100, find the value of x. Then find EF and
FG.
Example 3: Using the Segment Addition Postulate
M is between N and O.
Find NO.
Example 4: Draw a picture and solve for the
missing segment.
B is between A and C, AC = 14 and BC = 11.4. Find. AB
Example 5: Draw a picture and solve for the
missing segment.
Find RT if S is between R and T.
RS = 2x + 7, ST = 28 and RT = 4x.
Midpoint of a segment:
The midpoint of a segment is the point that bisects, or
divides, the segment into two congruent segments.
Example: C is the
midpoint of BD and
AF
Segment bisector:
Example 6: BC = 3x + 2 and CD = 5x – 10. Solve for x. What is
the length of BD?
Segment bisector:
Example 7: AC = 5x - 16 and CF = 2x – 4. Solve for x. What is
the length of AF?
Angle Addition Postulate
First, let’s recall some previous information from last week….
We discussed the Segment Addition Postulate, which
stated that we could add the lengths of adjacent segments
together to get the length of an entire segment.
For example:
J
K
L
JK + KL = JL
If you know that JK = 7 and KL = 4,
then you can conclude that JL = 11.
The Angle Addition Postulate is very similar, yet applies to
angles. It allows us to add the measures of adjacent angles
together to find the measure of a bigger angle…
Angle Addition Postulate
If B lies on the interior of AOC,
then mAOB + mBOC = mAOC.
B
A
mAOC = 115
50
O
65
C
Example 1:
D
Example 2:
G
114
K
95
19
H
Given: mGHK = 95
mGHJ = 114.
Find: mKHJ.
J
134°
A
46°
B
C
This is a special example, because the
two adjacent angles together create a
straight angle.
Predict what mABD + mDBC equals.
ABC is a straight angle, therefore
mABC = 180.
The Angle Addition Postulate
tells us:
mABD + mDBC = mABC
mGHK + mKHJ = mGHJ
95 + mKHJ = 114
mKHJ = 19.
Plug in what
you know.
Solve.
mABD + mDBC = 180
So, if mABD = 134,
46
then mDBC = ______
R
Given:
mRSV = x + 5
mVST = 3x - 9
mRST = 68
V
Find x.
S
T
Set up an equation using the Angle
Addition Postulate.
mRSV + mVST = mRST
x + 5 + 3x – 9 = 68
Solve.
4x- 4 = 68
4x = 72
x = 18
Plug in
what you
know.
Extension: Now that you
know x = 18, find mRSV
and mVST.
mRSV = x + 5
mRSV = 18 + 5 = 23
mVST = 3x - 9
mVST = 3(18) – 9 = 45
Check:
mRSV + mVST = mRST
23 + 45 = 68
B
C
mBQC = x – 7 mCQD = 2x – 1 mBQD = 2x + 34
Find x, mBQC, mCQD, mBQD.
mBQC = x – 7
mBQC = 42 – 7 = 35
Q
D
mBQC + mCQD = mBQD
x – 7 + 2x – 1 = 2x + 34
3x – 8 = 2x + 34
x – 8 = 34
x = 42
mCQD = 2x – 1
mCQD = 2(42) – 1 = 83
mBQD = 2x + 34
mBQD = 2(42) + 34 = 118
Check:
mBQC + mCQD = mBQD
35 + 83 = 118
x = 42
mCQD = 83
mBQC = 35 mBQD = 118
Theorem 2.2 Properties of Angle Congruence
 Angle congruence is reflexive, symmetric, and transitive.
 Examples:
 Reflexive: For any angle A, A ≅ A.
 Symmetric: If A ≅ B, then B ≅ A
 Transitive: If A ≅ B and B ≅ C, then A ≅ C.
Ex. 1: Transitive Property of Angle Congruence
 Prove the Transitive Property of Congruence for angles
C
Given: A ≅ B, B ≅ C
Prove: A ≅ C
B
A
Ex. 1: Transitive Property of Angle Congruence
Statement:
1. A ≅ B, B ≅ C
2. mA = mB
3. mB = mC
4. mA = mC
5. A ≅ C
Reason:
1. Given
2. Def. Cong. Angles
3. Def. Cong. Angles
4. Transitive property
5. Def. Cong. Angles
Ex. 2: Using the Transitive Property
Given: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3
Prove: m1 = 40
1
4
2
3
Ex. 2:
Statement:
1. m3 ≅ 40, 1 ≅ 2,
2 ≅ 3
2.
3.
4.
1 ≅ 3
m1 = m 3
m1 = 40
Reason:
1. Given
Trans. Prop of Cong.
3. Def. Cong. Angles
4. Substitution
2.
Theorem 2.3
All right angles are congruent.
Example 3: Proving Theorem 2.3
Given: 1 and 2 are right angles
Prove: 1 ≅ 2
Ex. 3:
Statement:
1. 1 and 2 are right
angles
2. m1 = 90, m2 = 90
3. m1 = m2
4. 1 ≅ 2
Reason:
1. Given
Def. Right angle
3. Transitive property
4. Def. Cong. Angles
2.
Properties of Special Pairs of Angles

Theorem 2.4: Congruent Supplements. If two angles
are supplementary to the same angle (or to congruent
angles), then they are congruent.
If m1 + m2 = 180 AND m2 + m3 = 180, then 1 ≅
3.
Congruent Complements Theorem

Theorem 2.5: If two angles are complementary to the
same angle (or congruent angles), then the two angles
are congruent.
If m4 + m5 = 90 AND m5 + m6 = 90, then 4 ≅
6.
Proving Theorem 2.4
Given: 1 and 2 are supplements, 3 and 4 are
supplements, 1 ≅ 4
Prove: 2 ≅ 3
1
2
3
4
Ex. 4:
Statement:
1.
2.
3.
4.
5.
6.
7.
1 and 2 are supplements, 3 and 4
are supplements, 1 ≅ 4
m 1 + m 2 = 180; m 3 + m 4
= 180
m 1 + m 2 =m 3 + m 4
m 1 = m 4
m 1 + m 2 =m 3 + m 1
m 2 =m 3
2 ≅ 3
Reason:
1.
Given
2.
Def. Supplementary angles
3.
Transitive property of equality
4.
Def. Congruent Angles
5.
Substitution property
6.
Subtraction property
7.
Def. Congruent Angles
Postulate 12: Linear Pair Postulate
If two angles form a linear pair, then they are supplementary.
1
2
m 1 + m 2 = 180
Example 5: Using Linear Pairs
In the diagram, m8 = m5 and m5 = 125.
Explain how to show m7 = 55
5
6
7
8
Solution:
 Using the transitive property of equality m8 = 125. The
diagram shows that m 7 + m 8 = 180. Substitute 125
for m 8 to show m 7 = 55.
Vertical Angles Theorem
 Vertical angles are congruent.
2
1
3
4
1 ≅ 3; 2 ≅ 4
Proving Theorem 2.6
Given: 5 and 6 are a linear pair, 6 and 7 are a linear
pair
Prove: 5 7
5
6
7
Ex. 6: Proving Theorem 2.6
Statement:
1.
5 and 6 are a linear pair, 6 and 7
Reason:
1.
Given
2.
Linear Pair postulate
3.
Congruent Supplements Theorem
are a linear pair
2.
3.
5 and 6 are supplementary, 6
and 7 are supplementary
5 ≅ 7