7-1 7-2 Angles 7-1 7-2 Angles PAPA
Measurement of an AngleMeasurement of an Angle
2
To denote the measure of an angle we write an
“m” in front of the symbol for the angle.Here are some common angles and their measurements.
3
1 2
3
4
1 45m 2 90m
3 135m
4 180m
Congruent AnglesCongruent Angles
• So, two angles are congruent if and only if they have the same measure.
• So, The angles below are congruent.
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if and only if .A B m A m B
Means Means CongruentCongruent Means EqualMeans Equal
Types of AnglesTypes of Angles
• An acuteacute angle is an angle that measures less than 90 degrees.
• A rightright angle is an angle that measures exactly 90 degrees.
• An obtuseobtuse angle is an angle that measures more than 90 degrees.
5
acute right obtuse
• A straightstraight angle is an angle that measures 180 degrees. (It is the same as a line.)
• When drawing a right angle we often mark its opening as in the picture below.
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straight angle
right angle
Types of AnglesTypes of Angles
Perpendicular LinesPerpendicular Lines• Two lines are perpendicular if
they intersect to form a right intersect to form a right angle.angle. See the diagram.
• Suppose angle 2 is the right angle. Then since angles 1 and 2 are supplementary, angle 1 is a right angle too. Similarly, angles 3 and 4 are right angles.
• So, perpendicular lines intersect to form fourfour right angles right angles.
7
12
3 4
Perpendicular LinesPerpendicular Lines
• The symbol for perpendicularity is• So, if lines m and n are perpendicular, then we write
.
.m n
m
nm n
Adjacent AnglesAdjacent AnglesAdjacent angles share a common vertex and
one common side.Adjacent angles are “side by side”
and share a common ray.
45º15º
Adjacent AnglesAdjacent AnglesThese are examples of adjacent angles.
55º
35º
50º130º
80º 45º
85º20º
Adjacent AnglesAdjacent AnglesThese angles are NOT adjacent.
45º55º
50º100º
35º
35º
Vertical AnglesVertical Angles• Two angles formed by intersecting lines and have no
sides in common but share a common vertex. • Are congruent.
When 2 lines When 2 lines intersect, they intersect, they make vertical make vertical
angles.angles.
75º
75º
105º105º
Common Common VertexVertex
Vertical Vertical angles are angles are opposite opposite
one one another.another.
75º
75º
105º105º
Vertical AnglesVertical Angles
Vertical Vertical angles are angles are opposite opposite
one one another.another.
Vertical AnglesVertical Angles
75º
75º
105º105º
Vertical angles are congruent Vertical angles are congruent (equal).(equal).
30º150º
150º30º
Vertical AnglesVertical Angles
Vertical AnglesVertical Angles
1 4
Two angles that are opposite angles. Vertical angles are congruent.
1 2
33 44
5 6
7 8
2 3
5 8,
6 7
Name the Vertical AnglesName the Vertical Angles
Supplementary AnglesSupplementary AnglesAdd up to 180Add up to 180º.º.
60º120º
40º
140º
Adjacent and Supplementary Angles
Supplementary Angles
but not Adjacent
Supplementary AnglesSupplementary Angles
• Two angles are supplementarysupplementary if their measures add up to
• If two angles are supplementary each angle is the supplementsupplement of the other.
• If two adjacent angles together form a straight angle as below, then they are supplementary.
18
180 .
1 2
1 and 2 are
supplementary
Complementary AnglesComplementary AnglesAdd up to 90Add up to 90º.º.
70º
20º20º
70º
Adjacent and Complementary Angles
Complem
entary Angles
but not Adjacent
Complementary AnglesComplementary Angles
• Two angles are if their measures add upcomplementarycomplementary to
• If two angles are complementary, then each angle is called the complementcomplement of the other.
• If two adjacent angles together form a right angle as below, then they are complementary.
20
90 .
12
A
BC
1 and 2 are
complementary
if is a
right angle
ABC
Supplementary vs. ComplementarySupplementary vs. ComplementaryHow do I rememberHow do I remember??
The way I remember is this:
• C comes before S in the alphabet.
• 90 comes before 180 when I count.
• Complementary is 90, Supplementary is 180.
Guess Who?
• I am an angle.
Guess Who?
• I am an angle.• I have 180°
Guess Who?
• I am an angle.• I have 180°• I look like this:
Supplementary Complementary
Guess Who?
• I am two adjacent angles.
Guess Who?
• I am two adjacent angles.• I look like an “L” with a line in the middle.
Guess Who?
• I am two adjacent angles.• I look like an “L” with a line in the middle.• I add up to 90°• I look like this:
Guess Who?
• I am two adjacent angles.• I look like an “L” with a line in the middle.• I add up to 90°• I look like this:
Complementary Supplementary
Guess Who?
Complementary Supplementary
Guess Who?
Complementary Supplementary
Review
• Complementary angles are…….
Review
• Complementary angles are…….
Review
• Supplementary Angles are…..
Practice Time!
Find the missing angle
55
x
I know that these angles are complementary.
They must add up to 90°
So……
90 – 55 = 35
The missing angle is 35
You try.
20x
Are they supplementary or complementary?
Find the missing side.
You try.
20x
Are they supplementary or complementary?
complementary
Find the missing side.
90 – 20 = 70
The missing angle
Is 70
One More
50
x
Find the missing angle
120x
I know these are supplementary angles.
Supplementary angles add up to 180.
The given angle is 120. So…..
180 – 120 = 60
The missing angle is 60
Find the missing angle
130x
What kind of angles?
What’s the missing angle?
Find the missing angle
130 x
What kind of angles?
Supplementary
What’s the missing angle?
Adds up to 180, so…..
180 – 130 = 50
Find the missing angle
x30
Do this one on your own.
Directions: Identify each pair of angles as
adjacent, vertical, supplementary, complementary,
or none of the above.
#1
60º120º
#1
60º120º
Supplementary Angles
Adjacent Angles
#2
60º30º
#2
60º30º
Complementary Angles
#3
75º75º
#3
75º75º
Vertical Angles
#4
60º40º
#4
60º40º
None of the above
#5
60º
60º
#5
60º
60º
Vertical Angles
#6
45º135º
#6
45º135º
Supplementary Angles
Adjacent Angles
#7
65º
25º
#7
65º
25º
Complementary Angles
Adjacent Angles
#8
50º90º
#8
50º90º
None of the above
Directions:Determine the missing angle.
#1
45º?º
#1
45º135º
#2
65º
?º
#2
65º
25º
#3
35º
?º
#3
35º
35º
#4
50º
?º
#4
50º
130º
#5
140º
?º
#5
140º
140º
#6
40º
?º
#6
40º
50º
Transversal
• Definition: A line that intersects two or more lines in a plane at different points is called a transversal.
• When a transversal t intersects line n and m, eight angles of the following types are formed:
Exterior anglesInterior anglesConsecutive interior anglesAlternative exterior anglesAlternative interior anglesCorresponding angles
tm
n
Corresponding AnglesCorresponding Angles: Two angles that occupy
corresponding positions.
75
2 6
1 2
3 4
5 6
7 8
1 5
3 7 4 8
The corresponding angles are the ones at the same location at each intersection
Angles and Parallel Lines
• If two parallel lines are cut by a transversal, then the following pairs of angles are congruent.
1. Corresponding angles
2. Alternate interior angles
3. Alternate exterior angles
Proving Lines Parallel
• If two lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
DC
BA
Alternate Angles
• Alternate Interior Angles: Two angles that lie between parallel lines on opposite sides of the transversal (but not a linear pair).
• Alternate Exterior Angles: Two angles that lie outside parallel lines on opposite sides of the transversal.
Lesson 2-4: Angles and Parallel Lines 78
3 6, 4 5
2 7, 1 8
1 2
3 4
5 6
7 8
Example: If line AB is parallel to line CD and s is parallel to t, find the measure of all the angles when m< 1 = 100°. Justify your answers.
Lesson 2-4: Angles and Parallel Lines 79
m<2=80° m<3=100° m<4=80°
m<5=100° m<6=80° m<7=100° m<8=80°
m<9=100° m<10=80° m<11=100° m<12=80°
m<13=100° m<14=80° m<15=100° m<16=80°
t
16 15
1413
12 11
109
8 7
65
34
21
s
DC
BA
Proving Lines Parallel
• If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
DC
BA
Ways to Prove Two Lines Parallel
• Show that corresponding angles are equal.• Show that alternative interior angles are equal.• In a plane, show that the lines are perpendicular to the
same line.
Homework
Pg 305 #6-14e, 18-32e (just answers)Pg 309 #6-24e (just answers)