Name: ____________________________________________
Teacher: _____________
Date
11/10
Lesson
1
Topic
Proofs with Complementary and Supplementary Angles
11/11
11/12
-2
11/13
3
11/14
4
No School
More Proofs with Complementary and Supplementary
Angles
Parallel Line Theorems
Quiz 1
Proving Lines Parallel
11/17
5
More Parallel Line Proofs
11/18
6
Perpendicular Line Proofs
11/19
7
11/20
8
More Perpendicular Line Proofs
Quiz 2
Mixed Practice
11/21
9
Unit 5 Review
11/22
10
Unit 5 Test
11/23
Lesson 1: Complementary and Supplementary Angles.
Warm Up:
1. Two angles are complementary. One angle has a measure that is five times the measure of
the other angle. What is the measure, in degrees, of the larger angle?
Let’s examine some properties of complementary angles.

∠1 and ∠2 are complementary and so are ∠3 and ∠2. If m∠1 = 62, what
are the measures of the other angles?
Theorem:

If two angles are _________________________________________________________
then the angles are congruent
In the diagram, 𝑎 ⊥ 𝑏, ∠1 ≅ ∠4, and m∠1 =54, what are the measures of the other angles?
a
2 3
1
Theorem:
If two angles are congruent, then ____________________________________
_____________________________________________________________________
Here is the formal proof of one of the
theorems above:
Will these theorems work for Supplementary Angles too?
4
b
Complementary and Supplementary Angle Theorems
 If two angles are congruent, then their supplements are congruent
 If two angles are supplementary to the same angle, then the angles are congruent.
Before you use these theorems, you need to prove that the
angles are complementary or supplementary
So, how do you prove that angles are complementary or supplementary:
 If two angles have a sum of _________, then they are complementary
 If two angles have a sum of _________ , then they are supplementary
 If two angles form a ________________________________, they are supplementary.
 If two adjacent angles form a _________________________, then they are complementary.
Example 1:
Given: Lines l and m intersect
Prove: (1) 1 and 2 are supplementary
1 and 4 are supplementary
(2) m1 + m2 = m1 + m4
l
m
Example 2:
Given: DB  BF , AB  BC
Prove: ABD  CBF
Example 3:
Given: FBC  FDC
Prove: ABF  FDE
Lesson 2: More Proofs with Complementary and Supplementary Angles
Statement
Warm Up:
Given: mABD 
1
mABC ,
2
mABC  2mDBC
⃗⃗⃗⃗⃗⃗ bisects ABC
Prove: 𝐵𝐷
A
D
B
C
Example 1:
Given: 1 and 2 are complementary
3 and 4 are complementary
1  4
Prove: 2  5
C
G
F
2 3
1
4
E
A
5
B
D
Reason
Example 2:
Given: FDE and FBC are supplementary,
Prove: FDE  FBA
Example 3:
Given:
mYZW + mYWV =180
Prove: YZW  YWZ
Lesson 3: Parallel Line Theorems
Warm-up: List the angles involved
1. Alternate Interior Angles:
1
6
2. Corresponding Angles:
4
2
5
3
3. Interior Angles on the Same Side of the Transversal:
7
8
4. Vertical Angles:
5. Alternate Exterior Angles:
Parallel Lines in Proofs
We already know that
IF
2 lines are parallel
 ___________________________________________ are 
THEN
 ___________________________________________ are 
 ___________________________________________ are 
 Same-side interior angles are ______________________
We have used this in algebraic problems, now let’s look proofs.
Example 1:
Given: 𝑙 ∥ 𝑚
Prove: ∠3 ≅ ∠6
l
n
2
1
4
3
6
5 8
7
m
Example 2:
Given: 𝑙 ∥ 𝑚
Prove: 𝑚∠4 + 𝑚∠6 = 180
l
n
2
1
4
3
m
6
5 8
7
Example 3:
Given: 𝑙 ∥ 𝑚
Prove: ∠2 and ∠5 are supplementary
l
n
2
1
4
3
6
5 8
7
m
Lesson 4 – Proving Lines Parallel
Warm Up:
Given that lines m and n are
parallel, prove ∠𝟏 ≅ ∠𝟐
n
m
1
2
We have worked with proofs and algebraic problems when we know that the lines are parallel, now
we are going to go in the opposite direction:
Postulate: If two lines are cut by a transversal so that corresponding angles are congruent,
then the two lines are parallel.
1. Use the given information to prove that l ║ m.
Given: 4  8
Theorem: If two lines are cut by a transversal so that alternate interior angles are
congruent, then the two lines are parallel.
Theorem: If two lines are cut by a transversal so that alternate exterior angles are
congruent, then the two lines are parallel.
Theorem: If two lines are cut by a transversal so that same-side interior angles are
supplementary, then the two lines are parallel.
2. Given: p ║ r , 1  3
Prove: l ║ m
Theorem: If two lines are parallel to the same line, then they are parallel to each other.
3. Given: l ║ m, 3 and 4 are supplementary
Prove: l ║ n
Lesson 5 – More Parallel Line Proofs
Warm Up: Find all of the angles in the given diagram.
m 1 =
m 2 =
1
3
2 4
m 3 =
m 4 =
m 5 =
m 6 =
m 7 =
m 8 =
6
5
8
7
Example 1:
bisects PQS
Given:
1  4
Prove:
║
P
Q 1
2
3
R
T
4
S
B
2
Example 2: Write the theorem that applies to the given scenario.
If 𝑚∠1 = 72 and 𝑚∠2 = 108, then ̅̅̅̅
𝐴𝐷 ∥ ̅̅̅̅
𝐵𝐶
A
1
C
D
___________________________________________________________________________________________
__________________________________________________________________________________________
Example 3:
Given: 1  2
Prove: 3  4
3
m
2
1
4
n
Example 4:
Suppose the corresponding angles (1 and 2) on the far side of
the boat measure (4y – 2)° and (3y + 6)°, where y = 8. Show
that the oars are parallel.
Arithmetic:
Theorem that supports this:
________________________________________________________
_________________________________________________________
_________________________________________________________
Lesson 6: Perpendicular Line Proofs
Warm Up: For each of the statements below, write the theorem that proves that the two lines are
parallel.
1. 𝑚∠2 = 160 and 𝑚∠8 = 160
_______________________________________________________________________
_______________________________________________________________________
2. 𝑚∠2 = 140 and 𝑚∠6 = 140
__________________________________________________________________________________________________
__________________________________________________________________________________________________
3. 𝑚∠3 = 40 and 𝑚∠6 = 140
__________________________________________________________________________________________________
__________________________________________________________________________________________________
Perpendicular Lines
Theorem: If two intersecting lines form a linear pair of congruent angles, then the lines are
perpendicular.
Theorem: If a transversal is perpendicular to one of two parallel lines, then it is
perpendicular to both.
Theorem: If two lines are perpendicular to the same line, then the lines are parallel to each
other.
Example 1: Carpentry Application
A carpenter’s square forms a right angle. A
carpenter places the square so that one side is
parallel to an edge of a board, and then draws a line
along the other side of the square. Then he slides
the square to the right and draws a second line.
Why must the two lines be parallel?
__________________________________________________________________________________________________
__________________________________________________________________________________________________
Example 2:
Given: r || s, 1  2
Prove: r  t
Example 3:
Given: ⃡⃗⃗⃗⃗
𝐹𝐺 ⊥ ⃡⃗⃗⃗⃗
𝐺𝐻 , EHF  HFG
Prove: ⃡⃗⃗⃗⃗
𝐸𝐻 ⊥ ⃡⃗⃗⃗⃗
𝐺𝐻
Example 4:
Given: 1  2, p  q
Prove: p  r
Lesson 7: More Perpendicular Line Proofs
Warm Up:
If the 𝑚∠2 = 5𝑥 + 29 and 𝑚∠7 = 11𝑥 + 7, find 𝑚∠8.
1.
2.
Lesson 8: Mixed Practice
Put these cut-apart proofs back together!
Proof 1:
Proof 2:
Given: 𝑛 ∥ 𝑚, ∠2 is a right angle,
Given: ∠1 ≅ ∠2, 𝑚∠3 = 90
k
∠3 ≅ ∠4
Prove: 𝑘 ⊥ 𝑚
4
Prove: ℎ ∥ 𝑘
n
h
3
2
n
m
m
h
1
2
k
3
Challenge!
Directions: Find the mistakes in the proofs below:
(Note – all the statements are correct!)
⃡⃗⃗⃗ ⊥ ⃡⃗⃗⃗⃗
Given: ⃡⃗⃗⃗⃗
𝑃𝑄 ⊥ ⃡⃗⃗⃗⃗
𝐷𝐸, 𝑅𝑆
𝐷𝐸, ∠1 ≅ ∠2
⃡⃗⃗⃗⃗
⃡⃗⃗⃗⃗
Prove: 𝐴𝐵 ||𝑋𝑌
Proof 1:
Statement
⃡⃗⃗⃗⃗ ⊥ ⃡⃗⃗⃗⃗
⃡⃗⃗⃗ ⊥ 𝐷𝐸
⃡⃗⃗⃗⃗ , ∠1 ≅ ∠2
1. 𝑃𝑄
𝐷𝐸 , 𝑅𝑆
⃡⃗⃗⃗
2. ⃡⃗⃗⃗⃗
𝑃𝑄 ||𝑅𝑆
3. ∠1 ≅ ∠2
4. ∠2 ≅ ∠𝑆𝐸𝑌
5. ∠𝑃𝐷𝐸 ≅ ∠𝑆𝐸𝐷
6. ∠1 + ∠𝑃𝐷𝐸 ≅ ∠𝑆𝐸𝑌 + ∠𝑆𝐸𝐷
∠𝐴𝐷𝐸 ≅ ∠𝐷𝐸𝑌
⃡⃗⃗⃗⃗
⃡⃗⃗⃗⃗
7. 𝐴𝐵||𝑋𝑌
Reason
1. Given
2. If a transversal is perpendicular to one of two
parallel lines, then it is perpendicular to both.
3. Given
4. Vertical angles are congruent
5. If two lines are parallel, then alternate interior
angles are congruent.
6. Addition
7. If two lines are cut by a transversal so that
alternate exterior angles are congruent, then the
lines are parallel.
Proof 2:
Statement
⃡⃗⃗⃗⃗ ⊥ ⃡⃗⃗⃗⃗
⃡⃗⃗⃗ ⊥ 𝐷𝐸
⃡⃗⃗⃗⃗ , ∠1 ≅ ∠2
1. 𝑃𝑄
𝐷𝐸 , 𝑅𝑆
⃡⃗⃗⃗
2. ⃡⃗⃗⃗⃗
𝑃𝑄 ||𝑅𝑆
3. ∠1 ≅ ∠2
4. ∠2 ≅ ∠𝑆𝐸𝑌
5. ∠1 ≅ ∠𝑆𝐸𝑌
6. ∠𝑃𝐷𝐸 ≅ ∠𝑆𝐸𝐷
7. ∠1 + ∠𝑃𝐷𝐸 ≅ ∠𝑆𝐸𝑌 + ∠𝑆𝐸𝐷
∠𝐴𝐷𝐸 ≅ ∠𝐷𝐸𝑌
⃡⃗⃗⃗⃗ ||𝑋𝑌
⃡⃗⃗⃗⃗
8. 𝐴𝐵
Reason
1. Given
2. If two lines are cut by a transversal so that
corresponding angles are congruent, then the lines
are parallel.
3. Given
4. Vertical angles are congruent
5. Substitution
6. If two lines are parallel, then corresponding angles
are congruent.
7. Addition
8. If two lines are perpendicular to the same line,
then they are parallel to each other