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3.2 angles and parallel lines ink.notebook
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September 29, 2016
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3.2 Angles and
Parallel Lines
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3.2 angles and parallel lines ink.notebook
Lesson Objectives
Standards
Lesson Notes
3.2 Angles and Parallel Lines
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September 29, 2016
Lesson Objectives
Standards
Lesson Notes
After this lesson, you should be able to successfully use angle and line relationships to find missing angles.
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Corresponding Angles Postulate
Lesson Objectives
Standards
Lesson Notes
G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
G.CO.9 Prove theorems about lines and angles.
If two parallel lines are cut by a transversal,
then the pairs of corresponding angles (angles
that have corresponding positions) are
____________________.
Â
1 2
3 4
5 6
7 8
n
B)
A)
116¡ Press the tabs to view details.
m
(x + 21)¡ 75¡ 15x¡ 2
3.2 angles and parallel lines ink.notebook
September 29, 2016
Consecutive Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of consecutive interior angles
(angles that lie between the two lines and are
on the same side of the transversal) are
________________________.
Â
1 2
3 4
m
5 6
7 8
C)
(3x ­ 27)¡ n
D)
102¡ 116¡ (9x + 19)¡ Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of alternate interior angles
(angles that lie between the two lines and are
on opposite sides of the transversal) are
Â
___________________.
1 2
3 4
m
5 6
7 8
n
F)
E)
6x¡ 72¡ 7x¡ 98¡ 3
3.2 angles and parallel lines ink.notebook
September 29, 2016
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal,
then the pairs of alternate exterior angles
(angles that lie outside the two lines and are
on opposite sides of the transversal) are
__________________.
Â
1 2
3 4
m
5 6
7 8
G)
21x¡ n
H)
(x ­ 18)¡ 110¡ 147¡ Ç
If , then Corr.
Ú's are
Ç
≅
If , then Alt. Int.
Ú's are
≅
If Ç, then Consec Int.
Ú's are SUPP
Ç
If , then Alt. EXT.
Ú's are
≅
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3.2 angles and parallel lines ink.notebook
September 29, 2016
In the figure, m∠9 = 80 and m∠5 = 68. Find the measure of each angle. Tell which postulate(s) or theorem(s) you used.
1. ∠12
1 2
4 3
2. ∠1
9 10
12 11
5 6
8 7
3. ∠4 13 14
16 15
w
v
4. ∠3
p
q
5. ∠7
6. ∠16
Angles that are ¤
alternate interior angles
vertical angles
alternate exterior angles
corresponding angles
Angles that are supplementary
adjacent angles that form a linear pair
consecutive interior angles
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3.2 angles and parallel lines ink.notebook
September 29, 2016
p
1 2
4 3
5 6
8 7
7. In the figure, mÚ2 = 75. Find the
m measures of the remaining angles.
n
mÚ1 = Ú1 and Ú2 form ____________________
mÚ3 = Ú3 and Ú2 form ____________________ mÚ4 = Ú4 and Ú2 are _____________________
mÚ5 = Ú5 and Ú3 are _____________________
mÚ6 = Ú6 and Ú2 are _____________________
mÚ7 = Ú7 and Ú3 are _____________________
mÚ8 =
Ú8 and Ú6 are _____________________
corresponding Ús
vertical Ús
linear pair
alt int Ús
9. Find the value of x.
8. If m∠1 = 3x + 15, m∠2 = 4x − 5, and m∠3 = 5y, find the value of x and y
q
p
1
2
4
(3x ­ 7)¡ r
3
(x + 5)¡ s
6
3.2 angles and parallel lines ink.notebook
September 29, 2016
10. Find the value of the variable(s) in each figure. Explain your reasoning.
3x¡ 2y¡ 4y¡ (5x ­ 20)¡ IFF = If and only if
A theorem or postulate that is true
for both the conditional statement
and the inverse statement.
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3.2 angles and parallel lines ink.notebook
September 29, 2016
Example 2: Prove that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
Given: p Ç q
On the
Worksheet
Prove: ∠1 ≅ ∠2
GIVEN
CORRESPONDING ANGLES ≅
VERTICAL ANGLES ≅
SUBSTITUTION
In the figure, m∠7 = 100. Find the measure of each angle.
1. m∠9
3. m∠8
Practice
5. m∠5
2. m∠6
4. m∠2
6. m∠11
6
5
87
1 2
4 3
m
10
9
11
12
u
t
s
8
3.2 angles and parallel lines ink.notebook
September 29, 2016
In the figure, m∠3 = 75 and m∠10 = 105. Find the measure of each angle. Tell which postulate(s) or theorem(s) you used
7. m∠2
8. m∠5
3
1
w
9. m∠7
2
56
10. m∠15
x
Find the value of the variable(s) in each figure. Explain your reasoning.
4
78
13.
12
11 16
15
10
9
14
13
z
y
11. m∠14
12. m∠9
Find the value of the variable(s) in each figure. Explain your reasoning.
14.
Find the value of the variable(s) in each figure. Explain your reasoning.
15.
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3.2 angles and parallel lines ink.notebook
September 29, 2016
Find the value of the variable(s) in each figure. Explain your reasoning.
Find the value of the variable(s) in each figure. Explain your reasoning.
16.
17.
Find the value of the variable(s) in each figure. Explain your reasoning.
19. Find m∠1. (Hint: Draw an auxiliary line.)
18.
(9x + 12)¡ 3x¡ (4y ­ 10)¡ 50¡ 1
100¡ 10
3.2 angles and parallel lines ink.notebook
20. Find m∠1. (Hint: Draw an auxiliary line.)
September 29, 2016
21. RAMPS A parking garage ramp rises to connect two horizontal levels of a parking lot. The ramp makes a 10° angle with the horizontal. What is the measure of angle 1 in the figure?
62¡ 1
144¡ 22. CITY ENGINEERING Seventh Avenue runs perpendicular to both 1st and 2nd Streets, which are parallel. However, Maple Avenue makes a 115° angle with 2nd Street. What is the measure of angle 1?
Answers: 1. 100 3. 80 5. 100 7. 105, Corresponding Û Post or Consec Int Û Thm 9. 105, Supp Thm or Alt Int Û Thm 11. 75, Supp Thm or Consec Int Û Thm 13. 5x + 40 = 180, x = 28, Supp Thm 3y – 1 + 40 = 180, y = 47, Correspond Û Post or Consec Int Û Thm 15. 9x + 21 = 11x – 1, x = 11, Correspond Û Post 5y – 5 + 120 = 180, y = 13, Supp Thm 17. 3x + 90 = 180, x = 30, Supp Thm 30 = 2y, y = 15, Alt Int Û Thm 180 = 30 + z, z = 150, Supp Thm 19. 150¡ 21. 170¡
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3.2 angles and parallel lines ink.notebook
September 29, 2016
PG 174: #’s 13 – 30, 38 – 43, 51 – 53, 55 – 57, 59, 61 – 63 12
3.2 angles and parallel lines ink.notebook
September 29, 2016
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