UNIT 2: SEGMENTS AND ANGLES
2.1 Segment Bisectors
Example 1: Find Segment Lengths
M is the midpoint of
. Find AM and
MB.
P is the midpoint of RS. Find PS and RS.
You Try!
Find DE and EF
Find NP and MP
Example 2: Use Algebra with Segment Lengths
Line l is a segment bisector of MB. Find the
value of x.
You Try! Solve for x.
1) AB = 24, MB = 2x +4
2) TM = 3x + 5, MQ = x + 17
4) TM =
1
x  4 , TQ = 12
2
3) AM = 5x – 1, AB = 38
Example 3: A is the midpoint of BT. If BA = x2 and AT = -2x+8, find the length of BT.
You Try! E is the midpoint of PT. If PE = 2x2 + 5x and ET = 3, find the length of PE and
PT.
Example 4: Use the Midpoint Formula
a) Find the coordinates of the midpoint M of
with endpoints A(1, -3) and B(-4, 4).
b) Find the coordinates of the midpoint M of
with endpoints A(-2, -5) and B(7, 4).
c) Segment AB has a midpoint M. If A has coordinate (3,8) and M has coordinates (7,12),
find the coordinates of B.
You Try!
a) Find the coordinates of the midpoint M of
with endpoints T(4, 3) and S(8, 5).
b) Find the coordinates of the midpoint M of
with endpoints T(-5, 2) and S(7, -3).
c) Segment AB has a midpoint M. If A has coordinates (5,6) and M has coordinates (1,3), find
the coordinates of B.
2.2 Angle Bisectors
Example 1: Find Angle Measures
bisects ∠ABC, and m∠ABC = 110°.
Find m∠ABD and m∠DBC.
You Try! Find the missing values.
1) mAOB  20, mBOC  ?, mAOC  ?
2) mCOA  50, mAOB  ?, mCOB  ?
3) 2mAOC  m?
4)
1
mAOB  m ?
2
Example 2: Find Angle Measure and Classify an Angle.
bisects ∠LMN, and m∠LMP = 46°.
a. Find m∠PMN and m∠LMN.
b. Determine whether ∠LMN is acute, right, obtuse, or straight.
Explain
Example 3: Use Algebra with Angle Measures
bisects ∠PRS. Find the value of x.
Your Turn! BX is the bisector of ABC . Complete each equation.
a) ABC  4 x  12, mABX  24, x  ?
b) ABX  5 x, mXBC  3x  10, mABC  ?
c) ABC  5 x  18, mCBX  2 x  12, mABC  ?
Example 4: More Algebra with Angle Measures
AD bisects RAP . If the mRAD  x 2 and mDAP  8x  33 find the mRAP .
Your Turn! BX is the bisector of ABC . If mABX  x 2  4 x and mCBX  12 , find the
value of x and the mABC .
AD bisects RAP . If the mRAD  4x 2 and mDAP  23x  6 find the mRAP .
2.3 Complementary and Supplementary Angles
Example 1: Identify Complements and Supplements
a.
b.
Example 2: Identify Adjacent Angles
a.
b.
c.
c.
Example 3: Measures of Complements and Supplements
a. ∠A is a complement of ∠C, and m∠A = 47°. Find m∠C.
b. ∠P is a supplement of ∠R, and m∠R = 36°. Find m∠P.
c. m∠N = (2y + 20)˚, find the measure of its supplement.
d. An angle measures 3 degrees less than twice its complement. Find the measure of its
complement.
e. The sum of the measures of a complement and a supplement of an angle is 184˚. Find the
measure of the angle.
You Try!
a. ∠A is a complement of ∠C, and m∠A = 26°. Find m∠C.
b. ∠P is a supplement of ∠R, and m∠R = 123°. Find m∠P.
c. m∠G = (7x + 10)˚, find the measure of its supplement.
d. An angle is 10˚ more than 3 times the measure of its complement. Find the measure of its
complement.
e. The sum of the measures of a complement and a supplement of an angle is 176˚. Find the
measure of the angle.
Example 4: Use a Theorem.
∠7 and ∠8 are supplementary, and ∠8 and ∠9 are
supplementary. Name a pair of congruent angles. Explain
your reasoning.
2.4 Vertical Angles
Example 1: Identify Vertical Angles and Linear Pairs.
a.
b.
Example 2: Use the Linear Pair Postulate.
Find the measure of ∠RSU.
Example 3: Use the Vertical Angles Theorem.
Find m∠CED, m∠BEC, and m∠AED.
You Try!
The m∠4=122˚, find m∠1, m∠2, and m∠3.
c.
Example 5: Use Algebra with Vertical Angles
Find the value of y. The find
the measure of each angle.
You Try! Find the value of each variable.
a)
b)
Example 6: More Algebra with Vertical Angles
a)
b)
You Try! Find the value of each variable.
a)
b)
2.5 If-Then Statements and Deductive Reasoning
Deductive reasoning uses facts, definitions, accepted properties and the laws of logic to
make a logical argument. This form of reasoning differs from inductive reasoning, in
which previous examples and patterns are used to form a conjecture.
Example 1: Identify the hypothesis and the conclusion of each conditional.
hypothesis
conclusion
If two angles have the same measure, then the angles are congruent.
If two angles form a linear pair, then the angles are supplementary.
If the sum of the measures of two angles is 90˚, then the angles are complementary.
Example 2: Write if-then statements.
Every game on my computer is fun to play.
I will buy the CD if it costs less than $15.
I will purchase a school yearbook if it costs less than $20.
You cannot ride your bike if you have a flat tire.
School will be cancelled if it snows ten inches overnight.
Law of Detachment: If the hypothesis of a true if-then statement is true, then the
conclusion is also true.
Example 3: Use the Law of Detachment
If you study at least two hours for the test, then you will pass it. You study three
hours for the test.
If you wash the cotton T-shirt in hot water, then it will shrink. You wash the t-shirt in
hot water.
If x has a value of 4, then 3x +1 has a value of 13. The value of x is 4.
Law of Syllogism:
If ________________________________, then __________________________
If ________________________________, then __________________________
If ________________________________, then __________________________
Example 5: Using the Law of Syllogism.
Given: If the daily high temperature is 32˚F or less, then the water in the pipe is frozen. If the
water in the pipe is frozen, then the pipe will break.
Given: If the police catch Alfredo speeding, then Alfredo gets a ticket. If Alfredo drives a car,
then Alfredo drives too fast. If Alfredo drives too fast, then the police catch Tim speeding.
Given: If a number is divisible by 4, then the number is divisible by two. If a number is
divisible by 2, then the number is even.
Given: If the area of a square is 49 square inches, then the length of a side of the square is 7
inches. If the length of a side of a square is 7 inches, then the perimeter of the square is 28
inches.
2.6 Properties of Equality and Congruence
Properties of Equality and Congruence
Equality
Congruence
a=a
Reflexive Property
Symmetric Property
If a = b, then b = a
If
If
, then
, then
Transitive Property
If a = b and b = c,
then a = c
If
and
, then
and
, then
.
.
.
If
.
Properties of Equality and Real Numbers
If a = b , then a + c = b + c
Addition Property
Subtraction Property
Multiplication Property
If a = b, then a – c = b – c
If a = b, then
Division Property
Substitution Property
If a = b, then
If a = b, then b can replace a in any expression.
Distributive Property
a(b + c) = ab + ac
Example 1: Name the Properties of Equality and Congruence
a. If
then,
.
b. DE = DE
c. If ∠P is congruent to ∠Q and ∠Q is congruent to ∠R, then ∠P is
congruent to ∠R.
Example 2: Use Properties of Equality
In the diagram, N is the midpoint of segment MP, and P is the midpoint of segment
NQ. Show that MN = PQ.
Statement
MN = NP
Reason
NP = PQ
MN = PQ
Example 3: Identify the Properties of Equality that justify the indicated steps.
a)
2x – 3 = 17
Given
2x = 20
?
x = 10
?
1
x  5  10
2
1

2 x  5   10
2

x  10  20
x  30
b)
Example 4: Justify the Congruent Supplements Theorem.
Given: ∠1 and ∠2 are both supplementary to ∠3
Prove: ∠1 is congruent to ∠2.
Statement
Reason
m∠1+ m∠3 = 180˚
m∠2+ m∠3 = 180˚
m∠1+ m∠3 = m∠2+ m∠3
m∠1 = m∠2
∠1 ∠2
Your Turn!
Given: ∠1 and ∠2 are both complementary to ∠3
Prove: ∠1 is congruent to ∠2.
Statement
Reason
m∠1+ m∠3 = 90˚
m∠2+ m∠3 = 90˚
m∠1+ m∠3 = m∠2+ m∠3
m∠1 = m∠2
∠1 ∠2
Given
?
?
?