Download Packet 1 for Unit 2 M2 Geo

M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
ASSIGNMENTS FOR PART 1 OF UNIT 2 LINES AND ANGLES
Part 1 of Unit 2 includes sections 1-4, 1-5, and 2-8 from our textbook.
Due
Number
2A
2B
2C
2D
Description
Topics
Section 1-4:
Vocabulary: degrees, acute angle, obtuse
angle, right angle, angle bisector
Name and label angles using a number, single
p. 41-42 # 9, 10, 12, 18, 20 – 23 all, 43 letter, or three letters
p. 44 # 56
Identify sides and vertex of an angle
Use a protractor to measure angles in degrees
Classify an angle as right, acute, or obtuse
Use algebra to solve problems involving an
angle bisector
Section 1-5:
Vocabulary: adjacent angles, linear pair,
vertical angles, complementary,
p. 51-52 # 8 – 17 all, 36 – 41 all
supplementary, perpendicular
Identify pairs of angles that have specific
relationships
Section 1-5:
p. 51 # 19 – 26 all
Use algebra to solve problems involving angle
pairs
Section 2-8:
p. 156 – 157 # 1 – 4 all, 6, 8 – 10 all, 13 Use two-column proofs to prove statements
about angle pairs.
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
1-4
1-4
Angle Measure
Part 1: Naming Angles
Figure 1
If two rays have a common endpoint, they form an angle. The rays
are the sides of the angle. The common endpoint is the vertex.
1. Name the sides of the angle shown in Figure 1.
2. Name the vertex of the angle shown in Figure 1.
3. The angle in Figure 1 can be named as ∠ A, ∠ BAC, ∠ CAB, or ∠ 1. Use this information to
write three different rules for naming angles:
a. __________________________________________________________________________
b. __________________________________________________________________________
c. __________________________________________________________________________
4. Use Figure 2 to answer the questions below:
a. Name all the different angles that have R as their vertex.
Avoid naming the same angle more than once.
Figure 2
b. Name the sides of ∠ 1.
A right angle is an angle whose measure is 90 ° . An acute angle has measure less than 90 ° .
An obtuse angle has measure greater than 90 ° but less than 180 ° .
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
5. Classify each angle in Figure 3 as right, acute, or obtuse. Then use a protractor to
measure the angle to the nearest degree.
a. ∠ ABD
b. ∠ DBC
c. ∠ EBC
Figure 3
Part 2: Congruent Angles
Figure 4
Angles that have the same measure are congruent angles.
A ray that divides an angle into two congruent angles is
is the angle bisector
called an angle bisector. In Figure 4,
of ∠ MPR, and ∠ MPN ≅ ∠ NPR. Point N is in the interior of
∠ MPR.
In the diagram, arcs are drawn inside these angles to show
that they are congruent.
6. Follow the steps below to solve the problem: In Figure 4, if m ∠ MPN = 2x + 14 and
m ∠ NPR = x + 34, find x, and find m ∠ NPR.
a. Since
is an angle bisector of ∠ MPR, then ∠ MPN ≅ ∠ NPR. Write an equation about
x showing that the measures of these angles are equal.
b. Solve your equation for x.
c. Use the value of x from part (b) to find m ∠ NPR.
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
7. In Figure 5,
and
are opposite rays. (This means
bisects ∠ PQT.
that they lie on the same line.) Also,
a. Write an equation, and solve: If m ∠ PQT = 60 and
m ∠ PQS = 4x + 14, find the value of x.
Figure 5
b. Write an equation, and solve: If m ∠ PQS = 3x + 13 and m ∠ SQT = 6x – 2, find m ∠ PQT.
c. Name a point in the interior of ∠ PQT.
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
1-5
Angle Relationships
1-5
Part 1: Pairs of Angles
Adjacent angles are two angles that lie in the same
plane and have a common vertex and a common side,
but no common interior points.
1. Name a pair of adjacent angles in the diagram at
right.
2. Name a different pair of adjacent angles in the same diagram.
3. The chart below shows pairs of angles that are vertical angles on the left and pairs of
angles that are not vertical angles on the right.
Pairs of Vertical Angles
NOT Pairs of Vertical Angles
∠1 and ∠2
∠3 and ∠4
∠AED and ∠BEC
∠AEC and ∠DEB
∠1 and ∠2
∠3 and ∠4
∠5 and ∠6
∠7 and ∠8
∠9 and ∠10
Use the examples above to write a definition for vertical angles. Use the words “sides”
and “vertex” in your definition.
Vertical angles are two angles that…
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
4. The chart below shows pairs of angles that form a linear pair on the left and pairs of
angles that do not form a linear pair on the right.
Linear Pairs
NOT Linear Pairs
∠1 and ∠2
∠3 and ∠4
∠AED and ∠AEC
∠BED and ∠DEA
∠1 and ∠2
∠3 and ∠4
∠5 and ∠6
∠A and ∠B
Use the examples above to write a definition for a linear pair. Use the words “sides” and
“vertex” in your definition.
A linear pair is a pair of angles that…
5. The chart below shows pairs of angles that are complementary angles on the left and
pairs of angles that are not complementary angles on the right.
NOT Complementary Angles
Complementary Angles
m∠1 + m∠2 = 90°
∠G and ∠H
∠1 and ∠2
∠3 and ∠4
∠1 and ∠2
∠3 and ∠4
Use the examples above to write a definition for complementary angles.
Complementary angles are a two angles that…
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
6. The chart below shows pairs of angles that are supplementary angles on the left and pairs
of angles that are not complementary angles on the right.
Supplementary Angles
m∠3 + m∠4 = 180°
NOT Supplementary Angles
m∠4 + m∠5 ≠ 180°
∠1 and ∠2
∠3 and ∠4
∠1, ∠2, and ∠3
∠4 and ∠5
Use the examples above to write a definition for supplementary angles.
Supplementary angles are a two angles that…
Figure 6
7. Use the Figure 6 to answer the following questions:
a. Name a pair of vertical angles.
b. Name a pair of adjacent angles.
c. Name a pair of supplementary angles.
d. Name a pair of complementary angles.
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
Part 2: Algebra and Perpendicular Lines
Lines, rays, and segments that form four right angles are perpendicular.
The right angle symbol □ indicates that the lines are perpendicular. In the
is perpendicular to
, or
⊥
.
figure at the right,
8. Follow the steps below to solve the problem: Find x so that
are perpendicular.
and
a. If
⊥
, then m ∠ DZP = 90 ° . Write an equation showing
that the measures of ∠ DZQ and ∠ QZP add up to 90° .
b. Solve your equation for x.
c. Check your answer by finding the measures of ∠DZQ and ∠QZP and adding them.
⊥
9. In Figure 7, find the value of x and y so that
8
.
Figure 7
M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
Part 3: Properties of Angle Pairs
10. In Figure 8, C lies between points A and B on
.
a. Name the linear pair of angles in Figure 8.
b. Find the sum of the measures of the linear pair in
Figure 8.
Figure 8
c. Use a straightedge to draw another linear pair of angles.
d. Use a protractor to measure the two angles you drew, and find their sum.
e. Complete this statement: If two angles form a linear pair, then their measures…
11. Use Figure 9 to answer the following:
Figure 9
a. Find each measure:
m∠DEA = ______
m∠AEC = ______
m∠CEB = ______
b. Name two pairs of vertical angles in Figure 9. What do you notice about their
measures?
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
12. Use Figure 10 to answer the following:
a. Find each measure:
m∠HJG = ______
m∠IJF = ______
m∠GJI = ______
b. Name two pairs of vertical angles in Figure 10. What do you notice about their
measures?
c. Complete the statement: Vertical angles have measures that…
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
2-8
2-8
Proving Angle Relationships
Angle Addition
Two adjacent angles can be added to form a single angle with measure
equal to the sum of their measures.
Example: In the diagram at right, m ∠ PQR + m ∠ RQS = m ∠ PQS.
Linear pairs add
up to 180.
If two angles form a linear pair, then they are supplementary.
Example: If ∠ 1 and ∠ 2 form a linear pair, then m ∠ 1 + m ∠ 2 = 180.
Two angles that
form a right
angle add up to
90.
If the noncommon sides of two adjacent angles form a right angle,
then the angles are complementary.
Example: If
⊥
, then m ∠ 3 + m ∠ 4 = 90.
These examples show how your new rules can be used.
Example 1: If ∠ 1 and ∠ 2 form a linear pair and m ∠ 2 = 115, find m ∠ 1.
m ∠ 1 + m ∠ 2 = 180
m ∠ 1 + 115 = 180
m ∠ 1 = 65
Linear pairs add up to 180.
Substitution
Subtraction Property
Example 2: If ∠1 and ∠2 form a right angle and m∠2 = 20, find m∠1.
m ∠ 1 + m ∠ 2 = 90
m ∠ 1 + 20 = 90
m ∠ 1 = 70
Two angles that forma right angle add up to 90.
Substitution
Subtraction Property
Write an equation and solve to find the measure of each numbered angle. Name the reason
that justifies your work.
1.
2.
m ∠ 7 = 5x + 5,
m∠8 = x – 5
3.
m ∠ 5 = 5x, m ∠ 6 = 4x + 6,
m ∠ 7 = 10x,
m ∠ 8 = 12x – 12
11
m ∠ 11 = 11x,
m ∠ 13 = 10x + 12
M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
The Reflexive, Symmetric, and Transitive Properties all hold true for angles. The following
theorems also hold true for angles.
Congruent Supplements Theorem
Angles supplementary to the same angle or congruent angles are
congruent.
Congruent Complements
Theorem
Angles complimentary to the same angle or to congruent angles are
congruent.
Vertical Angles Theorem
If two angles are vertical angles, then they are congruent.
Theorem 2.9
Perpendicular lines intersect to form four right angles.
Theorem 2.10
All right angles are congruent.
Theorem 2.11
Perpendicular lines form congruent adjacent angles.
Theorem 2.12
If two angles are congruent and supplementary, then each angle is a
right angle.
Theorem 2.13
If two congruent angles form a linear pair, then they are right angles.
Example: Write a two-column proof.
Given: ∠ ABC and ∠ CBD are complementary.
∠ DBE and ∠ CBD form a right angle.
Prove: ∠ ABC ≅ ∠ DBE
Statements
Reasons
1. ∠ABC and ∠CBD are complementary.
∠DBE and ∠CBD form a right angle.
1. Given
2. ∠DBE and ∠CBD are complementary.
2. Complement Theorem
3. ∠ABC ≅ ∠DBE
3.
s
complementary to the same ∠ or ≅
Complete each proof.
4. Given:
⊥ ; ∠ 1 and ∠ 3 are complementary.
Prove: ∠ 2 ≅ ∠ 3
Proof:
Statements
a.
Reasons
a. ______________
⊥
b. _________________
b. Def. of ⊥
c. m ∠ ABC = 90
c. Def. of right angle
d. m ∠ ABC =m ∠ 1 + m ∠ 2
d. ______________
e. 90 = m ∠ 1 + m ∠ 2
e. Substitution
f. ∠ 1 and ∠ 2 are complementary
g. _________________
f. _______________
g. Given
h. ∠ 2 ≅ ∠ 3
h. ______________
12
s
are ≅.
M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
5. Given: ∠ 1 and ∠ 2 form a linear pair.
m ∠ 1 + m ∠ 3 = 180
Prove: ∠ 2 ≅ ∠ 3
Proof:
Statements
a. ∠ 1 and ∠ 2 form a linear pair.
m ∠ 1 + m ∠ 3 = 180
b. _________________
c. ∠ 1 is suppl. to ∠ 3.
d. _________________
Reasons
a. Given
b. Suppl. Theorem
c. ________________
d. ∠ s suppl. to the same
∠ or ≅ ∠ s are ≅.
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
PRACTICE FOR SECTIONS 1-4, 1-5, 2-8
For # 1-12, use the figure at the right.
Name the vertex of each angle.
1. ∠ 4
2. ∠ 1
3. ∠ 2
4. ∠ 5
7. ∠ STV
8. ∠ 1
11. ∠ WTS
12. ∠ 2
Name the sides of each angle.
5. ∠ 4
6. ∠ 5
Write another name for each angle.
9. ∠ 3
10. ∠ 4
Classify each angle as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest
degree.
13. ∠ NMP
14. ∠ OMN
15. ∠ QMN
16. ∠ QMO
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
In the figure,
and
are opposite rays,
bisects ∠ EBC.
Write an equation, and solve to find the indicated measures.
17. If m ∠ EBD = 4x + 16 and m ∠ DBC = 6x + 4, find m ∠ EBD.
18. If m ∠ EBD = 4x − 8 and m ∠ EBC = 5x + 20, find the value of x and m ∠ EBC.
For # 19- 24, use the figure at the right. Name an angle or angle pair that satisfies each condition.
19. Name two acute vertical angles.
20. Name two obtuse vertical angles.
21. Name a linear pair.
22. Name two acute adjacent angles.
23. Name an angle complementary to ∠ EKH.
24. Name an angle supplementary to ∠ FKG.
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
25. Write an equation, and solve: Find the measures of an angle and its complement if one angle measures 24
degrees more than the other.
26. Write an equation, and solve: The measure of the supplement of an angle is 36 less than the measure of
the angle. Find the measures of the angles..
For # 27-28, use the figure at the right.
27. If m ∠ RTS = 8x + 18, find the value of x so that
⊥
.
28. If m ∠ PTQ = 3y –10 and m ∠ QTR = y, find the value of y so that ∠ PTR is a right angle.
Determine whether each statement can be assumed from the figure. Explain.
29. ∠ WZU is a right angle.
30. ∠ YZU and ∠ UZV are supplementary.
31. ∠ VZU is adjacent to ∠ YZX.
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M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
Find the measure of each numbered angle, and give the reason(s) that justify your work.
32. m ∠ 2 = 57
33. m ∠ 5 = 22
34. m ∠ 1 = 38
35. m ∠ 13 = 4x + 11,
m ∠ 14 = 3x + 1
36. ∠ 9 and ∠ 10 are
complementary.
∠ 7 ≅ ∠ 9, m ∠ 8 = 41
37. m ∠ 2 = 4x – 26,
m ∠ 3 = 3x + 4
38. Complete the following proof.
Given: ∠ QPS ≅ ∠ TPR
Prove: ∠ QPR ≅ ∠ TPS
Proof:
Statements
Reasons
a.______________________________________
b. m ∠ QPS = m ∠ TPR
c. m ∠ QPS = m ∠ QPR + m ∠ RPS
m ∠ TPR = m ∠ TPS + m ∠ RPS
d. ______________________________________
e. ______________________________________
f. ______________________________________
a. ________________________________________
b. ________________________________________
c. ________________________________________
17
d. Substitution
e. _______________________________________
f. _______________________________________
M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
REVIEW FOR SECTIONS 1-4, 1-5, 2-8
For # 1-10, use the figure at the right.
Name the vertex of each angle.
1. ∠ 5
2. ∠ 3
3. ∠ 8
4. ∠ NMP
7. ∠ MOP
8. ∠ OMN
Name the sides of each angle.
5. ∠ 6
6. ∠ 2
Write another name for each angle.
9. ∠ QPR
10. ∠ 1
Classify each angle as right, acute, or obtuse. Then use a protractor to measure the angle to the nearest
degree.
11. ∠ UZW
12. ∠ YZW
13. ∠ TZW
14. ∠ UZT
In the figure,
and
are opposite rays,
bisects ∠ DCF, and
15. If m ∠ DCE = 4x + 15 and m ∠ ECF = 6x – 5, find m ∠ DCE.
16. If m ∠ FCG = 9x + 3 and m ∠ GCB = 13x – 9, find m ∠ GCB.
18
bisects ∠ FCB.
M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
Name an angle or angle pair that satisfies each condition.
17. Name two obtuse vertical angles.
18. Name a linear pair with vertex B.
19. Name an angle not adjacent to, but complementary to ∠ FGC.
20. Name an angle adjacent and supplementary to ∠ DCB.
21. Write an equation, and solve: Two angles are complementary. The measure of one angle is 21 more than
twice the measure of the other angle. Find the measures of the angles.
22. Write an equation, and solve: If a supplement of an angle has a measure 78 less than the measure of the
angle, what are the measures of the angles?
For # 23-24, use the figure at the right.
23. If m ∠ FGE = 5x + 10, find the value of x so that
⊥
.
24. If m ∠ BGC = 16x - 4 and m ∠ CGD = 2x + 13,find the value of x so that ∠ BGD is a right angle
.
19
M2 GEOMETRY PACKET 1 FOR UNIT 2 LINES AND ANGLES
Determine whether each statement can be assumed from the figure. Explain.
25. ∠ NQO and ∠ OQP are complementary.
26. ∠ SRQ and ∠ QRP is a linear pair.
27. ∠ MQN and ∠ MQR are vertical angles.
Find the measure of each numbered angle, and give the reasons that justify your work.
28. m ∠ 1 = x + 10
m ∠ 2 = 3x + 18
29. m ∠ 4 = 2x – 5
m ∠ 5 = 4x – 13
20
30. m ∠ 6 = 7x – 24
m ∠ 7 = 5x + 14