Measuring AnglesMeasuring Angles

Geometry vs AlgebraGeometry vs Algebra

Segments are CongruentSegments are Congruent– Symbol Symbol [ [ ] ] – AB AB CD CD 1 1 22

Lengths of segments are equal.Lengths of segments are equal.– Symbol Symbol [ = ][ = ]– AB = CDAB = CD– mm1 = m1 = m22

AnglesAngles

Formed by 2 rays with the same endpointFormed by 2 rays with the same endpoint– Vertex of the AngleVertex of the Angle

Symbol: Symbol: [ [ ] ]

Name it by:Name it by:– Its Vertex Its Vertex AA– A number A number 11– Or by 3 Points Or by 3 Points BACBAC

- Vertex has to be in the middle- Vertex has to be in the middle

A

1

B

C

How many How many s can you find? Name them.s can you find? Name them.

3 3 ss ADB or ADB or BDABDA BDC or BDC or CDBCDB ADC or ADC or CDACDA

Notice D (the vertex) is always in the middle.Notice D (the vertex) is always in the middle.

Can’t use Can’t use DD

But But 1 or 1 or 2 could be added.2 could be added.

A B

C

D

1 2

Classifying Angles by their MeasuresClassifying Angles by their Measures

Acute Right Obtuse

x°

x < 90°

x°

x = 90°

x°

x > 90°

Straight x°

x = 180°

Postulate 1-7Postulate 1-7Protractor PostulateProtractor Postulate

Let OA & OB be opposite rays in a plane, & Let OA & OB be opposite rays in a plane, & all the rays with endpoint O that can be all the rays with endpoint O that can be drawn on one side of AB can be paired with drawn on one side of AB can be paired with the real number from 0 to 180.the real number from 0 to 180.

A BO

DC

Postulate 1-8Postulate 1-8Angle Addition PostulateAngle Addition Postulate

If point B is in the interior of If point B is in the interior of MAD, then MAD, then

mmMAB + mMAB + mBAD = mBAD = mMADMAD

M B

D

A

If If MAD is a straight MAD is a straight , then , then

mmMAB + mMAB + mBAD = mBAD = mMAD = 180MAD = 180°°

M

B

D

A

Finding Finding measures (m measures (m ))

Find mFind mTSW ifTSW if– mmRSW = 130RSW = 130°°– mmRST = 100°RST = 100°

R S

TW

mmRST + mRST + mTSW = mTSW = mRSWRSW

100 + m100 + mTSW = 130TSW = 130

mmTSW = 30TSW = 30°°

AdditionAdditionmmXYZ = 150XYZ = 150

1 = 3x - 15

2 = 2x - 10

x

Y

Z

m1 + m2 = mXYZ

(3x - 15) + (2x – 10) = 150

5x – 25 = 150

5x = 175

x = 35

Adjacent AnglesAdjacent Angles

Adjacent angles – Adjacent angles – two coplanar angles two coplanar angles with a common side, a common vertex, with a common side, a common vertex, and no common interior points.and no common interior points.

1 2

3 4

and

and

Vertical AnglesVertical Angles

Vertical angles – Vertical angles – two angles whose sides two angles whose sides are opposite rays.are opposite rays.

1 2

3 4

and

and

Complementary AnglesComplementary AnglesComplementary angles – Complementary angles – two angles two angles whose measures have a sum of 90°.whose measures have a sum of 90°.– Each angle is called the Each angle is called the complementcomplement of the of the

other.other.

1 2and

Aand B

Supplementary AnglesSupplementary Angles

Supplementary angles – Supplementary angles – two angles two angles whose measures have a sum of 180°.whose measures have a sum of 180°.– Each angle is called the Each angle is called the supplementsupplement of the of the

other.other.

3 4and

Band C

Identifying Angle PairsIdentifying Angle PairsIs the statement true or false? Is the statement true or false?

a. a. are adjacent angles. are adjacent angles.

b.b. are vertical angles. are vertical angles.

c.c. are complementary. are complementary.

BFDand CFD AFBand EFD AFEand BFC

Perpendicular LinesPerpendicular Lines

Perpendicular lines – Perpendicular lines – intersecting lines intersecting lines that form right anglesthat form right angles

Linear PairsLinear PairsA A linear pairlinear pair is a pair of adjacent angles whose is a pair of adjacent angles whose noncommon sides are opposite rays.noncommon sides are opposite rays.– The angles of a linear pair form a straight angle.The angles of a linear pair form a straight angle.

Finding Missing Angle MeasuresFinding Missing Angle Measuresare a linear pair.are a linear pair.

What are the measures ofWhat are the measures of ? ?

KPLand JPL 2 24, 4 36.m KPL x andm JPL x

KPLand JPL

Finding Missing Angle MeasuresFinding Missing Angle Measures

180m KPL m JPL (2 24) (4 36) 180x x

6 60 180x 6 120x

20x 2 24m KPL x 2(20) 24

4 36m JPL x 4(20) 36 80 36 116

40 24 64

Angle BisectorAngle BisectorAn An angle bisectorangle bisector is a ray that divides an is a ray that divides an angle into angle into two congruent anglestwo congruent angles..– Its endpoint is at the angle vertex.Its endpoint is at the angle vertex.– Within the ray, a segment with the same endpoint is Within the ray, a segment with the same endpoint is

also an angle bisector.also an angle bisector.The ray or segment bisects the angle.The ray or segment bisects the angle.

Using an Angle Bisector to Find Angle Using an Angle Bisector to Find Angle MeasuresMeasures

bisects . If ,bisects . If ,

what iswhat is

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DAB 58m DAB ?m DAC

m CAB m DAC 58

m DAB m CAB m DAC 58 58 116