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.

4

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.

6

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:

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)