Warm up 9/2/14

Warm up 9/2/14 Answer the question and draw a picture if you can:

1) What is a right angle?

2) What is an acute angle?

3) What is an obtuse angle?

4) What is a vertical angle?

Angle Pair Relationships

Angle Pair Relationship Essential Questions

How do I prove geometric theorems involving lines, angles?

Standard: MCC9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Straight AnglesStraight Angles

___________ are two rays that are part of a the same line and have only their endpoints in common.

Opposite rays

XY Z

XY and XZ are ____________.opposite rays

The figure formed by opposite rays is also referred to as a ____________. A straight angle measures 180 degrees.straight angle

Naming AnglesNaming Angles

R

S

T

vertex

side

side

There are several ways to name this angle.

1) Use the vertex and a point from each side.

SRT or TRS

The vertex letter is always in the middle.

2) Use the vertex only.

R

If there is only one angle at a vertex, then theangle can be named with that vertex.

3) Use a number.

1

1

AnglesAngles

Definitionof Angle

An angle is a figure formed by two noncollinear rays that have a common endpoint.

E

D

F

2

Symbols: DEF

2

E

FED

Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle.

Types of Angles

A

right angle m A = 90

acute angle 0 < m A < 90

A

obtuse angle 90 < m A < 180

A

Angle MeasureAngle Measure

Classify each angle as acute, obtuse, or right.

110°

90°40°

50°

130° 75°

Obtuse

Obtuse

Acute

Acute Acute

Right

Angle MeasureAngle Measure

Adjacent Angles Adjacent Angles

When you “split” an angle, you create two angles.

D

A

C

B1

2

The two angles are called _____________adjacent angles

1 and 2 are examples of adjacent angles. They share a common ray.

Name the ray that 1 and 2 have in common. ____BD

adjacent = next to, joining.

Adjacent Angles Adjacent Angles

Definition of

Adjacent

Angles

Adjacent angles are angles that:

M

J

N

R1

21 and 2 are adjacent

with the same vertex R and

common side RM

A) share a common side

B) have the same vertex, and

C) have no interior points in common

Adjacent AnglesAdjacent Angles

Determine whether 1 and 2 are adjacent angles.

No. They have a common vertex B, but _____________no common side

1 2

B

12

G

Yes. They have the same vertex G and a common side with no interior points in common.

N

1

2J

L

No. They do not have a common vertex or ____________a common side

The side of 1 is ____LN

JNThe side of 2 is ____

Linear Pairs of AnglesLinear Pairs of Angles

Definition of

Linear Pairs

Two angles form a linear pair if and only if (iff):

1 and 2 are a linear pair.

A) they are adjacent and

B) their noncommon sides are opposite rays

C

A DB

1 2

AD form and BDBA

180 2 1

Linear Pairs of AnglesLinear Pairs of AnglesIn the figure, and are opposite rays.CM CE

1

2

M

43 E

H

T

A

C

1) Name the angle that forms a linear pair with 1.

ACE

ACE and 1 have a common side the same vertex C, and opposite rays

and

CA

CM CE

2) Do 3 and TCM form a linear pair? Justify your answer.

No. Their noncommon sides are not opposite rays.

When two lines intersect, ____ angles are formed.four

12

34

There are two pair of nonadjacent angles.

These pairs are called _____________.vertical angles

Vertical AnglesVertical Angles

Definition of

VerticalAngles

Two angles are vertical iff they are two nonadjacent angles formed by a pair of intersecting lines.

12

34

Vertical angles:

1 and 3

2 and 4


Theorem 3-1

Vertical AngleTheorem

Vertical angles are congruent.

1

4

3

2mn

1 3

2 4


Find the value of x in the figure:

The angles are vertical angles.

So, the value of x is 130°.130°

x°


Find the value of x in the figure:

The angles are vertical angles.

(x – 10) = 125.(x – 10)°

125°

x – 10 = 125.

x = 135.


Definition of

Complementary

Angles

30°

A

BC

60°D

E

F

Two angles are complementary if and only if (iff) The sum of their degree measure is 90.

mABC + mDEF = 30 + 60 = 90

Complementary and Supplementary AnglesComplementary and Supplementary Angles

30°

A

BC

60°D

E

F

If two angles are complementary, each angle is a complement of the other.

ABC is the complement of DEF and DEF is the complement of ABC.

Complementary angles DO NOT need to have a common side or even the same vertex.


15°H

75° I

Some examples of complementary angles are shown below.

mH + mI = 90

mPHQ + mQHS = 9050°

H

40°Q

P

S

30°60°T

UV

WZ

mTZU + mVZW = 90


Definition of

Supplementary

Angles

If the sum of the measure of two angles is 180, they form a special pair of angles called supplementary angles.

Two angles are supplementary if and only if (iff) the sum of their degree measure is 180.

50°

AB

C

130°

D

E F

mABC + mDEF = 50 + 130 = 180


105°H

75° I

Some examples of supplementary angles are shown below.

mH + mI = 180

mPHQ + mQHS = 18050°

H

130°

Q

P S

mTZU + mUZV = 180

60°120°

T

UV

W

Z

60° and

mTZU + mVZW = 180


Suppose A B and mA = 52.

Find the measure of an angle that is supplementary to B.

A52°

B52° 1

B + 1 = 1801 = 180 – B1 = 180 – 521 = 128°

Congruent AnglesCongruent Angles

1) If m1 = 2x + 3 and the m3 = 3x + 2, then find the m3

2) If mABD = 4x + 5 and the mDBC = 2x + 1, then find the mEBC

3) If m1 = 4x - 13 and the m3 = 2x + 19, then find the m4

4) If mEBG = 7x + 11 and the mEBH = 2x + 7, then find the m1

x = 17; 3 = 37°

x = 29; EBC = 121°

x = 16; 4 = 39°

x = 18; 1 = 43°

A B C

D

E

G

H

12

34

Congruent AnglesCongruent Angles

Warm up 9/2/14

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