Download ANGLE PAIRS in two lines cut by a transversal

Name ________________________________ Per_____
2.1 Angle Relationships in Parallel Lines
Vocabulary
Parallel lines
Skew lines
Perpendicular lines
Transversal
Example 1:
1.
2.
Fill in the blank with parallel, perpendicular, or skew
Fill in the blank with parallel, perpendicular, or skew.
(b) ̅ is ________ to ̅̅̅̅̅. (c) ̅̅̅̅ is __________to ̅̅̅̅.
(b) ̅̅̅̅ is _________ to ̅̅̅̅. (c) ̅̅̅̅ is __________to ̅̅̅̅ .
ANGLE PAIRS in two lines cut by a transversal
Corresponding angles
Consecutive (same side) interior angles
• corresponding
positions.
• same side
• between the two lines
Alternate interior angles
Alternate exterior angles
• alternate sides
• between the two lines
• alternate sides
• outside the two lines
Other angle relationships that you will need to remember…
Vertical angles
Linear Pair
• opposite ∠s with the
same vertex
• adjacent ∠s that make
a straight line
1
Example 2: Classify the pair of numbered angles.
1.
2.
3.
4.
7
5
6
8
6. Identify all pairs of the following angles.
5. Identify the relationship between each pair of angles, if
any.
b. Alternate interior angles
2
1
8
3 4
6 5
c. Consecutive interior angles
7
d. Alternate exterior angles
1) ∠1 and ∠7
4) ∠3 and ∠8
e. Vertical Angles
2) ∠4 and ∠6
5) ∠3 and ∠5
3) ∠8 and ∠7
6) ∠2 and ∠4
a. Corresponding angles
f. Linear Pairs
WHEN LINES ARE PARALLEL! (magic happens…HARRY POTTER!)
Corresponding Angles Postulate
If two parallel lines are cut by a
transversal, then pairs of
corresponding angles
a____________.
a
1
b
2
Statements
1. 𝑎 ∥ 𝑏
Reasons
1.
2. ∠____ ≅ ∠____
2.
Statements
1. 𝑎 ∥ 𝑏
1.
2. ∠____ ≅ ∠____
2.
Alternate Interior Angles Theorem
If two parallel lines are cut by a
transversal, then pairs of
alternate interior angles are
_______________.
a
3
b
4
Reasons
Alternate Exterior Angles Theorem
If two parallel lines are cut by a
transversal, then pairs of alternate
exterior angles are __________.
a
5
b
6
Statements
1. 𝑎 ∥ 𝑏
1.
Reasons
2. ∠____ ≅ ∠____
2.
Consecutive Interior Angles Theorem
If two parallel lines are cut by a
transversal, then pairs of consecutive
interior angles are
____________________.
a
7
b
8
2
Statements
1. 𝑎 ∥ 𝑏
1.
Reasons
2. ∠___ & ∠___ are supp.
3.
2.
3.
Example 3: Use the diagram below to find the angle measures. Explain your reasoning.
1. If the ∠
the ∠
4. If the
∠
∠
2. If the ∠
the ∠
what is
what is the
5. If the
∠
what is
∠
what is the
Example 4: Finding all the angle measures.
If ∥ and ∠
, find the measures of all the angles formed by
the parallel lines cut by the transversal.
p
q
𝑚∠
𝑚∠
𝑚∠
𝑚∠
𝑚∠
𝑚∠
𝑚∠
𝑚∠
3. If the
∠
∠
what is the
6. If the ∠
the ∠
what is
DO YOU NOTICE A
PATTERN???? Describe it!
E
THE HARRY POTTER SCAR!
1.
2.
3.
4.
Mark any angle with a dot
Find its vertical ∠ and mark it with a dot
Copy the same dot pattern on the other parallel
Connect the dots
B
D
A
• If they both have a dot or are both blank (SAME) → _____________
F
• If one has a dot and the other it blank (DIFFERENT) → _____________
Example 5: If ̅̅̅̅ ∥ ̅̅̅̅, are the angles congruent or supplementary?
1. ∠
and ∠
2. ∠
and ∠
D
B
F
H
C
G
E
3. ∠
and ∠
4. ∠
A
Example 6: Solve for x and explain your reasoning.
1.
2.
3
and ∠
2. ∠
and ∠
5. ∠
and ∠
2.2 Converses
Vocabulary
Conditional Statement
Ex: “If you have visited the statue of Liberty, then you have been to New York.”
Converse
Ex:
Example 1: Write the converse of the given statement.
1. If an animal has wings, then it can fly.
2. If you are student, then you have a student I.D. card.
3. All sharks have a boneless skeleton.
4. All police officers eat donuts.
Example 2: (a) Write the converse of the true statement. (b)Then decide whether the converse is true
or false. If false, provide a counterexample.
1. If an animal is an owl, then it is also a bird.
2. If two lines form right angles, then they are
perpendicular.
3. If an angle measures 130°, then it is obtuse.
4. If two angles are adjacent, then they are congruent.
Checkpoint
1. Find
below.
E a counterexample to the statement
E
E
E
If two angles are supplementary, then they are formed by two parallel lines cut by a transversal.
a.
b. B
c. B
d.
B
A
A
1
1
2
F
A
2
A
1
D
D
D
2
F
B
1
2
F
F
2. Write the converse of the statement below. Then determine whether each statement is true or false. If false, give a
counterexample.
Conditional Statement:
If two angles are right angles, then they are congruent.
Converse:____________________________________________________
4
T or F
T or F
2.3 Parallel & Perpendicular Lines
Example 1: Solve for x and explain your reasoning.
1.
2.
3.
4.
(4x – 8)
(3x + 10)
(11x + 2y)
Baby Proofs
1. Given: ∥ ;
Prove: ∠
M
∠
1
I
2. Given: ∥ ;
J Prove: ∠
a
a
3
b
L
J
K
2
K
M
∠
4
b
L
N
3. Given: ∥ ;
Prove: ∠
N
∠
4. Given:
Prove:
5
∥ ;
∠
∠
MORE PROOFS
1. Given: ∥ ; ∠
∠
Prove:
∥
∠
∠
∠
;
3. Given: ̅̅̅̅ ∥ ̅̅̅̅
∠
Statements
; ∠
Reasons
1.
∠
Statements
1. ̅̅̅̅ ∥ ̅̅̅̅
∠
Reasons
∠
1.
∠
∠
Prove:
∥
∠
∠
∥ ;
Statements
1.
∥
∥ , ∠
3
m
2
Reasons
1.
1
r
1.
n
m
Prove:
∥
∠
4
3
4. Given:
1.
Reasons
∠
n
2
Prove:
∥ ;
m
1
2. Given:
Statements
1.
n
s
6
PERPENDICULAR LINES
• So if two adjacent angles are
Two lines that form four
formed by
m
________ ________.
lines, then they
are_______________.
n
Example 2: Perpendicular Lines
⃗⃗⃗⃗⃗ , complete the sentence using your new vocabulary.
1. a. Given ⃗⃗⃗⃗⃗
∠
b. If
V
is a _________angle, because the definition of______________________.
∠
and
∠
T
the find the value of x. Explain.
R
2. If ̅̅̅̅
S
̅̅̅̅ , solve for x and y. Explain.
P
F
4x-2 °
5x+11 °
E
y°
N
Q
LET”S KEEP PRACTICING THOSE ANGLE NAMES!
Name the angle pair. Then state if they are congruent or supplementary.
̅̅̅̅ ∥ ̅̅̅̅
a. ∠
∠
e. ∠
∠
b. ∠
∠
f. ∠
∠
c. ∠
∠
g. ∠
∠
d. ∠
∠
h. ∠
∠
7
G
O
2.4 Perpendicular lines + Proofs
PERPENDICULAR LINES in proof
Given: 𝑚
Statements
1. 𝑚
𝑛
1
_________→ _________ →_________
m
n
𝑛
Reasons
1.
2.
2.
3.
3.
RIGHT ANGLES CONGRUENCE THEOREM
Statements
a
1. 𝑎
All right angles are ___________.
4x-2 °
1 2
5x+11 °
b
𝑏
Reasons
1.
2.
2.
3.
3.
y°
_________→ _________ →_________
Example 2: Using Perpendicular lines in a proof.
⃡⃗⃗⃗⃗ ,
1. Given: ⃡⃗⃗⃗⃗
Statements
∠
1. ⃡⃗⃗⃗⃗ ⃡⃗⃗⃗⃗ , ∠
Prove: ∠
2. ∠
K
is a right angle
T
3. Given: ̅̅̅̅ ̅̅̅̅ ;
̅̅̅̅ ̅̅̅̅
Prove: ∠ ≅ ∠
D
B
C
2.
3.
4.
4. Angle Addition Postulate
5.
5.
6.
6.
1. ̅̅̅̅
A
1.
3.
P
A
Reasons
Statements
̅̅̅̅ ; ̅̅̅̅
Reasons
̅̅̅̅
1.
PROVING LINES PARALLEL
**REMEMBER: Magic happens only if the lines are parallel, so…
You can use angle measures to PROVE lines are parallel!
When to use the CONVERSE!!!
a ||b
⇨
∠ ≌∠
______________________________
∠ ≌∠
⇨
a ||b
______________________________
8
Example#1: Determine whether each set of lines are parallel or not. Explain!
a)
b)
c)
d)
111°
a
O
PROOFS
1. Given: ∠ ≅ ∠
Prove: ∥
AI
M
R
1
2
m
b
1
3
P
4
b
a
Q
2. Given:
∥ ; ∠
12
Prove: ∠
n
N
2
P
N
2. Given: ∠ ≅ ∠ ;
∠R
AI
Prove: ∠
AJ
Statements
Reasons
1. ∠ ≅ ∠ ; ∠
1.
c
1
3
d
P
4
2
Q
N
3. Given : ∠10 ≌ ∠9 ;
AJ
∠
∠
Prove:
9
10
Statements
,
1.
∠10 ≌ ∠9 ;
∠
,
Statements
1. ∠ and ∠ are supplementary, m∠
Prove: m∠
b
5 6
8 7
1.
b
m∠
1 9
10 3
∠
a
1 2
4 3
5 6
8 7
2. Given: ∠ and ∠ are
supplementary,
a
Reasons
2
4
9
Reasons
1.
Remember
this???
Transitive Property
3. Given: ∠ ≅ ∠ , ∠ ≅ ∠ ,
∠
Prove: ∠
If a = b and b =c, then _______________
Statements
1. ∠ ≅ ∠ , ∠ ≅ ∠ ,
Reasons
∠
1.
3
5
6
2
1
PERPENDICULAR TRANSVERSAL THEOREM
If
and
,
then __________
1. 𝑚
Statements
𝑡; 𝑛 𝑡
Reasons
,
Prove:
Statement
1.
2.
2 1
3 4
s
4.
k
∠
,
1. Given
∥
2.
3.
∠
4.
g
5. Given:
Prove:
,
Reason
3.
6 5
7 8
h
2.
,
∠
∠
,
Statement
,
∠
∠
1.
,
,
Reason
∠
1.
g
f
7
6 8
3 1
2 4
5
m
n
6. Given: m || n;
Prove: ∠
∠
Statement
1.
3
4
∠
Reason
1.
m
5
n
10
n
t
1. given
2.
Proofs
4. Given:
m
2.5 Review + Multiple Choice
1. In the diagram line r is parallel to line s. Which of the
following statements must be true?
A. m3  m5
2. Given: line t || line s and neither is perpendicular to line
g. Which of the following statements is false?
s
t
A. m2  m5  180
g
5 6
3
1
7 8
B. m1  m7
4
2
C. m3  m5  180
1
2
r
3
B. m5  m4
5
4
C. m2  m3  180
s
6
D. m2  m4
D. m2  m3
3. In the diagram ⃡⃗⃗⃗ ∥ ⃡⃗⃗⃗⃗⃗ and mYRH  100 . Which of
the following conclusions does not have to be true?
A.
B.
C.
D.
4. Based on the diagram, which theorem or postulate would
support the statement mRIP  mSMY ?
A. Alternate Exterior
Angles Theorem
B. Alternate Interior
Angles Theorem
C. Consecutive Interior
Angles Theorem
D. Corresponding ∠ Postulate
mMHF  100
mRHM  80
SRT and MHF are alternate exterior angles
SRY and RHV are alternate interior angles
5. In the diagram below, ∠ ≅ ∠ . Which of the following
must be true?
r
s
1
A.
2 5 6
7
3
B. ∠
∠
4
8
t
C. ∠
∠
D.
∠
6. Which type of angles are a counterexample to the
conjecture below?
“If two lines are parallel, then each pair of angles are
A
A
supplementary”.
A. ∠ ∠
B. ∠ ∠
C. ∠ ∠
D. ∠ ∠
∠
7. In the diagram to the right, ⃡⃗⃗
which angles are congruent?
⃡⃗⃗⃗⃗ and ⃡⃗⃗
⃡⃗⃗⃗ then
C
D
3
4
8. In the diagram below,
∠
∠
B
B
the following does not have to be true?
A. ∠
,∠
A.
∠
∠
B. ∠
,∠
B.
∠
∠
C. ∠
∠
C.
∥
D. ∠
,∠
D.
∠
. Which of
r
1
2 5 6
s
4
7
3
8
t
∠
10. In the diagram below, which pair of angles are alternate
interior angles?
H
9. Allison wanted to solve for , so she set up the equation
. What would her
reasoning be?
J
𝑥
“If two parallel lines are intersected by
a transversal, then…
1 2
𝑥
A.∠
and ∠
B. ∠
and ∠
S
V
R
M
T
A. linear pairs are supplementary.”
B. corresponding angles are supplementary.”
C. alternate interior angles are congruent.”
D. consecutive (same-side) interior angles are supplementary.”
11
C. ∠
and ∠
D. ∠
and ∠
K
G
X
L
Y
11. Use the diagram to determine which of the pair of
angles is alternate exterior angles.
A. ∠1 and ∠15
B. ∠9 and ∠15
a
b
1 2
3 4
8 7
6 5
12. To solve for x in the diagram below, Betty used the
equation
.
Betty can justify her
equation by the
following statement:
m
C. ∠4 and ∠11
11 12
9 10
14 13
D. ∠2 and ∠8
16
“If two parallel lines are intersected by a transversal, then ...
n
15
A.
B.
C.
D.
13. Use the diagram to determine which of the pair of
angles is corresponding angles.
a
b
A. ∠2 and ∠10
B. ∠8 and ∠11
C. ∠4 and ∠10
D. ∠10 and ∠12
EXTRA PRACTICE
1. Given: ∠
∠
,
Prove:
m
1
l
1
4 3
1 2
1 2
6 5
3 4
8 7
9 10
14 13
16
alternate interior angles are congruent.
alternate exterior angles are congruent.
corresponding angles are congruent.
consecutive interior angles are supplementary.
14. Use the diagram to determine which of the pair of
angles is consecutive interior angles.
A. ∠3 and ∠11
B. ∠13 and ∠16
C. ∠9 and ∠13
D. ∠10 and ∠13
m
n
11 12
15
,
Statements
1.
∠
,
∠
n
1
p
1
2. Solve for x and y. Explain your reasoning for each equation you set up!
(14x – 22)°
(2x + 5y)°
(11x + 14)°
12
b
1 2
6 5
3 4
8 7
9 10
14 13
16
Reasons
,
6 7
5 8
a
1.
m
11 12
15
n
2.5 Perimeter & Area
Formulas for Perimeter (P), Area (A), and Circumference (C)
Rectangle or Square
Triangle
P = _________
P = ______________
A = _____
A = _______
b = ________, h = ________
b = ________, h = ________
Circle
**NOTE**
• Height is always _______________ to the base
C = _______
• Perimeter, Circumference:
A = _______
r = __________
_______ units (Ex:
= __________
)
• Area:
Example 1: Find perimeter, circumference, and area
1. Find the perimeter and area of the rectangle.
2. Find the circumference and area of the circle. Leave
your answers in terms of π.
3. Find the perimeter and area of the figure.
4. Find the perimeter and area of the figure.
10 yd
5. Find the area and circumference of the circle inscribed in
the square.
6. Find the perimeter and area of the figure.
15 yd
12 yd
13
AREA ON A COORDINATE PLANE (2 methods)
• Subtract values
• Count the units
(4, 10)
(4, 2)
Example 2: Find the area of the figure shown.
1.
(2, 10)
2.
(10, 10)
(13, 12)
(2, 4)
(2, 3)
(13, 2)
(13, 4)
(10, 3)
3.
4.
(4, 7)
(9, 7)
(4, -3)
(9, -3)
(7,7)
(1, 2)
(5, 2)
Example 3: Find unknown length
1. The base of a triangle is 12 feet. It’s area is 36
square feet. Find the height of the triangle.
3. The perimeter of a square is 128 inches.
a. Find the length of one side of the square.
2. The area of a rectangle is 243 square meters. The rectangle is three
times its width. Find the length and width of the rectangle.
4. The circumference of a circle is 14 centimeters. Find
the area of the circle.
b. Then find the area of the square.
14
SPIRAL REVIEW
I. Points, Lines, Planes…
• Collinear:
Use the diagram below.
a. Name a point that is collinear with C, S, and P.
b. Name a point that is coplanar with A, C, and
D.
c. Circle the correct set of 3 collinear points.
B, K, L
K, M, L
P, I, L
G, S, C
• Coplanar:
d. Circle the correct set of 4 coplanar points.
G, O, J, P
K, I, J, F
L, C, I, O
A, C, S, C
II. Addition Postulates
1. A is between H and T. If HA =
,
AT =
, and HT=35, solve for x and explain.
2. If ∠
∠
find
∠
∠
EXTRA PROOF PRACTICE
1. Given: ∠ ≅ ∠ ;
∠
∠
Prove:
∠
∠
and ∠
are complementary
Prove: ∠
2. Given:
15
, and
and explain.
2.7 Composite Area
PARTIALLY SHADED…
FULLY SHADED…
–
=
Arearegion =
=
Arearegion =
Areashaded – Areaunshaded
+
Areashaded + Areashaded
Example 1: Fins the area of the shaded region.
1. Find the area of the shaded region.
2. Find the area of the shaded region. (Round to the tenths)
18 in
8m
6 in
8m
4 in
3. Find the area of the shaded region
4. Find the area of the shaded region
●
2 cm
6 yd
11 cm
6 yd
5. Find the area of the figure below comprised of a
rectangle and a semicircle.
●
6. Find the area of the figure below comprised of a square
and a right triangle.
16
Example 2: Composite figures
1. Find the area and perimeter of the figure below if all line
segments meet at right angles. (Figure not drawn to scale)
2. Find the area and perimeter of the figure below if all line
segments meet at right angles. (Figure not drawn to scale)
SPIRAL REVIEW
For every 90° → 1 quadrant over
Rotations
Switch #’s
1. If
is rotated 90
2. If
is rotated 180
counterclockwise about the origin,
clockwise about the origin, then
then what would be the coordinates
what are the coordinates of its
of the new point?
image?
Translations
1. If
translates to
translated to what point?
then
is
PRACTICE MAKES PERFECT!
1. In the figure, ̅̅̅̅ and ̅ are intersected by ̅̅̅̅.
∠
and which of the following angles are
known as corresponding angles?
K
A. ∠𝐽𝑀𝑁
1 2 B. ∠𝐽𝑀𝐿
3
4
N
D. ∠𝐼𝑀𝐿
2. If
translates to
translated to what point?
M
then
is
2. You are planting grass on a rectangular plot of land. You are also
building a fence around the edge of the plot. The plot is 45 yards
long and 30 yards wide. How much area do you need to cover
with grass s eed? How many yards of fencing do you need?
H
G
C. ∠𝑁𝑀𝐼
3. If
then what are the
coordinates of its image after a
rotation 90 clockwise about the
origin?
J
I
L
3. Solve for x and explain your reasoning.
17
K
4.
a. Write an equation that can be used to find the value of x and justify your
equation.
H
6x+8 °
7x-12 °
b. Find the value of x.
3x-8 °
H
c. Find the measure of one of the acute angles.
L
5. Find the area of the triangle formed by the
coordinates
and
.
6. a) Write the converse of the statement.
If two angles formed by parallel line cut by a
transversal are corresponding angles, then they are
congruent.
b) Is the converse true or false? If false, give a
counterexample.
∠
∠
and ∠
are complementary
Prove: ∠
2. Given:
3. Given: ∠ ≅ ∠ ,
Prove: ∠
∠
a
1
2
b
3 6
4 5
r
7 10
8 9
s
18
19