Download Unit 5: Angles

Unit 5: Angles
Name: ________________
Teacher: ____________
Period: ______
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Aim: SWBAT Identify Complementary, Supplementary, and Vertical angles.
DO NOW: Read and answer the following question
Angle: Formed by 2 rays with a common endpoint.
-The common endpoint is called the VERTEX of the angle.
-You can name an angle by using the vertex point, or by using one point on
each ray with the vertex point in the middle.
Ex: ∠ ∠
A
B
C
Adjacent Angles – Two angles are adjacent angles (side-by-side) if they have a common side,
the same vertex and DO NOT overlap.
1
∠ 1 and ∠ 2 are adjacent angles
2
Classifying Angles:
-To measure an angle, we use a protractor. The unit of measure is degrees.
-There are 360 degrees in a full rotation.
DO NOW:
Name the following angle 4 ways:
D
1
F
E
2
Angle Relationships
Complementary Angles – Two angles are complementary if the SUM of their angle measures is
90°. Complementary angles form corners (right angles).
30°
60°
Right angle
Adjacent complementary angles
30° and 60° angles are complementary because 30° + 60° = 90°
Supplementary Angles - Two angles are supplementary if the SUM of their angle measures is
180˚. Supplementary angles form straight lines.
140°
Line 180°
40°
Adjacent supplementary angles
40° and 140° angles are supplementary because 40° + 140° = 180°.
Vertical Angles – Vertical angles are congruent ( ≅ ) angles formed by the intersection of two
lines. They are opposite each other and have congruent ( ≅ ) measurements.
1
2
3
4
∠ 1 ≅ ∠ 4 and ∠ 2 ≅ ∠ 3
Why? They are vertical angles.
If vertical angles formed by two intersecting lines are right angles the lines are said to be
Perpendicular lines.
The symbol for
perpendicular is ⊥.
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CLASSWORK:
Find the measures of the COMPLEMENTS of the following angles.
1) 30°
2) 70° _____
60°
3) 25° _____
4) 18° _____
Find the measures of the SUPPLEMENTS of the following angles.
5) 60° 120°
6) 150° _____
7) 90° _____
8) 45° _____
In each figure, find the value of x:
9)
10)
x
65
70
11)
x
12)
155
x
x
123
4
In the following diagram,
.
⊥
B
A
C
F
G
E
D
1) Name a right angle:___________________
2) Name an acute angle:__________________
3) Name an obtuse angle:_________________
4) Name a straight angle:_________________
5) Name a pair of adjacent angles:_______________
6) Name a pair of complementary angles: _____________________
7) Name a pair of supplementary angles:______________________
8) If the m∡BGC = 55˚, find the m∡BGA._____________________
9) If the m∡BGA = 35˚, find the m∡EGD______________________
10) If the m∡BGC = 55˚, find the m∡BGF______________________
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HW: Angle Relationships
State the angle relationship. Find the value of x in each figure.
1)
2)
3)
Angle Relationship:
______________
Angle Relationship:
____________
Angle Relationship:
______________
x = _____
x = _____
x = _____
Find the complement of each of the following angles.
4) 45° ________
5) 82° ________
6) 23° _________
Find the supplement of each of the following angles.
7) 173° ________
8) 75° ________
9) 92° _________
10) Name 2 pairs of vertical angles in the figure below.
11) Use the figure from #10 to name 2 pairs of supplementary angles.
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12) Use the diagram and the information given to answer the following questions:
a) Name a pair of complementary angles:________________
b) Name a pair of supplementary angles:_________________
c) Name a pair of vertical angles:______________________
d) Find the measure of ∡3. __________
e) Find the measure of ∡8. __________
f) Find the measure of ∡4. __________
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AIM: SWBAT identify the relationships of angles formed by two parallel lines and a
transversal.
Parallel Lines – lines in the same plane that DO NOT intersect
Transversal – a line that intersects two lines to form eight angles
← transversal
1
2
3 4
5 6
7 8
Interior Angles: ∠3, ∠4, ∠5, ∠6
(inside parallel lines)
Exterior Angles: ∠ 1, ∠2, ∠7, ∠8
(outside parallel lines)
Alternate Interior Angles – Interior angles found on opposite sides of the transversal.
When two parallel lines are cut by a transversal the alternate interior angles are congruent.
Examples: ∠ 3 & ∠6, ∠4 & ∠5
Alternate Exterior Angles - Exterior angles found on opposite sides of the transversal.
When two parallel lines are cut by a transversal the alternate exterior angles are congruent.
Examples: ∠1 & ∠8, ∠2 & ∠7
Corresponding Angles – angles that hold the same position on two different lines cut by the
transversal. When two parallel lines are cut by a transversal the corresponding angles are
congruent. Examples: ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8
Vertical Angles – angles formed by the intersection of two lines. They are opposite each other
and have congruent angle measurements.
Examples: ∠1 & ∠4, ∠2 & ∠3, ∠5 & ∠8, ∠6 & ∠7
Supplementary Angles – two angles whose sum is 180°. Supplementary angles form straight
lines.
Examples: ∠ 1 & ∠2, ∠3 & ∠4, ∠5 & ∠6, ∠7 & ∠8, ∠3 & ∠1, ∠ 4 & ∠2, ∠7 & ∠5, ∠8 & ∠6
Consecutive Interior Angles - The pairs of angles on one side of the transversal but
inside the two lines are called Consecutive Interior Angles. Consecutive Interior Angles are
supplementary.
Examples: ∠ 4 & ∠6, ∠3 & ∠5
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Classwork – Angle Relationships
Use the information given and shown in the diagram to answer the following questions.
ALWAYS START WITH THE GIVEN INFO!
1) ∠ 2 and ∠ ____ are corresponding angles.
2) ∠ 3 and ∠ ____ are alternate interior angles.
3) ∠ 5 and ∠ ____ are alternate exterior angles.
4) ∠ 4 and ∠ ____ or ____ are supplementary angles.
5) ∠ 7 and ∠ ____ are consecutive interior angles.
6) ∠7 and ∠ ____ are vertical angles.
7) ∠ 6 is supplementary to ∠ ____ and ∠ ____.
8) Find the m ∠ 1. ____ Why? ______________________________________________
9) Find the m∠ 6. ____ Why? ______________________________________________
10) Find the m ∠ 4. ____ Why?______________________________________________
_______________________________________________
11) Find the m ∠ 2. ____ Why? _____________________________________________
_____________________________________________
_____________________________________________
12) Are ∠'s 1 and 7 congruent? Why or why not?_________________________________
13) I know ∠6 and ∠ 5 are ________________ angles, therefore, if I know the measure of
∠ 5, I can find the measure of angle 6.
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HW: Parallel Lines cut by a Transversal
Use the information given and shown in the diagram to answer the following questions.
1) ≮8 and ≮ ____ are alternate exterior angles.
2) ≮ 4 and ≮ ____ are alternate interior angles.
3) ≮ 2 and ≮ ____ are vertical angles.
4) ≮ 5 and ≮ ____ are alternate interior angles.
5) ≮ 3 and ≮ ____ are alternate exterior angles.
6) ≮ 2 and ≮ ____ are corresponding angles.
7) ≮ 2 and ≮ ____ are consecutive interior angles.
8) ≮ 4 and ≮ 1 are adjacent ______________ angles.
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State the given relationship between the angles: alternate interior, alternate exterior,
corresponding, vertical or supplementary AND Find the measure of the missing angle.
9) ≮ 1 and ≮ 2 are _____________________________ angles.
Find the m ≮ 2. _________
10) ≮ 1 and ≮ 8 are _____________________________ angles.
Find the m ≮ 8. _________
11) ≮ 1 and ≮ 4 are _____________________________ angles.
Find the m ≮ 4. _________
12) ≮ 4 and ≮ 6 are _____________________________ angles.
Find the m ≮ 6. _________
13) ≮ 4 and ≮ 7 are ______________________________ angles.
Find the m ≮ 7. _______
14) ≮ 5 and ≮ 4 are _____________________________ angles.
Find the m ≮ 5. _________
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AIM: SWBAT identify the relationships of angles formed by 2 parallel lines & a
transversal AND find the missing angle measure.
d
6 3
5 4
8 1
7 2
m
Given:
line m ll line n
d is a transversal
m ∠ 6 is 128˚
n
Use the information given and shown in the diagram to answer the following questions.
1) List the 2 pairs of alternate interior angles. ________________________________
2) List the 4 pairs of corresponding angles. ___________________________________
3) List the 2 pairs of alternate exterior angles. ________________________________
4) List the 4 pairs of vertical angles. _________________________________________
5) List 2 pairs of consecutive interior angles. ___________________________________
6) Why are ∠ 1 and ∠ 7 congruent? ___________________________________________
7) ∠ 6 and ∠ 5 are ____________________ angles and _______________________ angles.
8) Why are ∠ 4 and ∠ 8 congruent? ___________________________________________
9) Are ∠’s 3 and 7 congruent? Why? __________________________________________
10) Find the m ∠ 3. ______ Justify ____________________________________________
11) Find the m ∠ 2. ______ Justify ____________________________________________
12) Find the m ∠ 5. ______ Justify ____________________________________________
13) Are all alternate interior and exterior angles congruent? Why or why not?
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Use the diagram to answer questions 14-20.
14) Find the m ∠ 1. ______ Justify ____________________________________________
15) Find the m ∠ 5. ______ Justify ____________________________________________
16) Find the m ∠ 3. ______ Justify ____________________________________________
17) Find the m ∠ 7. ______ Justify ____________________________________________
18) Find the m ∠ 2. ______ Justify ____________________________________________
19) Find the m ∠ 4. ______ Justify ____________________________________________
20) Find the m ∠ 6. ______ Justify ____________________________________________
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For problems 21 – 29, the figure at the right shows p ||q, m ∠ 1 = 78° and m ∠ 2 = 47°. Find
the measures of the following angles.
21) m ∠ 1 = 78o
p
22) m ∠ 2 = 47 o
q
7
9 8
23) m ∠3 = _____
24) m ∠ 4 = _____
2
3
6
1
4
5
25) m ∠ 5 = _____
26) m ∠ 6 = _____
27) m ∠ 7 = _____
28) m ∠ 8 = _____
29) m ∠ 9 = _____
30) If
∥
, what is the angle relationship shown by ∡AGH and
∡GHD?________________
Use that relationship to set up an equation and find the measure
of∡GHD ALGEBRAICALLY.
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HW: Finding the measure of angles formed by transversals
Use the information given and shown in the diagram to answer the following questions.
ALWAYS START WITH THE GIVEN INFO!
1) ∠ 2 and ∠ ______ are corresponding angles.
2) ∠ 3 and ∠ ______ are alternate exterior angles.
3) ∠ 1 and ∠ ______ are alternate interior angles.
4) ∠7 and ∠ ______ are vertical angles.
5) ∠ 4 and ∠ ______ are consecutive interior angles.
6) ∠ 6 is supplementary to ∠ ______ and ∠ ______.
7) Find the m ∠ 1. ______ Why? ______________________________________________
8) Find the m∠ 6. ______ Why? ______________________________________________
9) Find the m ∠ 8. ______ Why? ______________________________________________
10) Find the m ∠ 2. ______Why? ___________________________________________
11) Find the m ∠ 7. ______ Why? _____________________________________________
12) Are ∠'s 5 and 3 congruent? Why or why not?_________________________________
13) Are ∠'s 4 and 2 congruent? Why or why not?_________________________________
15
Refer to the figure at the right. Line a is parallel to line b and m∠2 is
145°. Find each given angle measure. Justify your answer.
m∠9 _____ Why?_____________________________
14)
55°; Sample answer: ∠2 and angles 9 and 10 are vertical angles. So, m∠9 + m∠10 = 145°. So, m∠9
145 − 90 or 55°.
m∠7_____ Why?_____________________________
15)
145°; Sample answer: ∠2 and ∠7 are alternate interior
angles. So, m∠7 = 145°.
16) m∠3
_____ Why?_____________________________
35°; Sample answer: ∠1 and ∠2 are supplementary. So, m∠1
= 35°. ∠1 and ∠3 are corresponding angles, so m∠3 = 35°.
17) m∠4______ Why?_____________________________
State the angle relationship and solve for the missing angles ALGEBRAICALLY.
18)
19)
Angle Relationship - ___________
Angle Relationship - __________________
20)
21)
Angle Relationship - __________________
Angle Relationship-__________________
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