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. m3 m5
2. Given: line t || line s and neither is perpendicular to line g. Which of the following statements is false? s t A. m2 m5 180 g 5 6 3 1 7 8 B. m1 m7 4 2 C. m3 m5 180
1
2
r
3
B. m5 m4
5
4
C. m2 m3 180
s
6
D. m2 m4
D. m2 m3
3. In the diagram ⃡⃗⃗⃗ ∥ ⃡⃗⃗⃗⃗⃗ and mYRH 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 mRIP mSMY ? A. Alternate Exterior Angles Theorem B. Alternate Interior Angles Theorem C. Consecutive Interior Angles Theorem D. Corresponding ∠ Postulate
mMHF 100 mRHM 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
