INTRO TO ANGLE MEASUREMENT

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Measuring Angles• Angles are measured using a protractor,

which looks like a half-circle with markings around its edges.

• Angles are measured in units called degrees• 45 degrees, for example, is symbolized like

this:• Every angle on a protractor measures more

than 0 degrees and less than or equal to 180 degrees.

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A Protractor

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• The smaller the opening between the two sides of an angle, the smaller the angle measurement.

• The largest angle measurement (180 degrees) occurs when the two sides of the angle are pointing in opposite directions.

• To denote the measure of an angle we write an “m” in front of the symbol for the angle.

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• Here are some common angles and their measurements.

1 2

3

4

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Types of Angles• An acute angle is an angle that measures

less than 90 degrees.• A right angle is an angle that measures

exactly 90 degrees.• An obtuse angle is an angle that

measures more than 90 degrees.

acute right obtuse

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• A straight 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.

straight angleright angle

Adjacent Angles• Adjacent Angles

share a RAY and a VERTEX but no INTERIOR POINTS.

• Angles x and y do not share a ray.

• <DOC is adjacent to <COB, but it is not adjacent to <DOB. Can you tell why? (think about Point C)

Angles and Their Parts• An angle consists of two

different rays that have the same initial point. The rays are the sides of the angle. The initial point is the vertex of the angle.

• Point M is on the INTERIOR of <CAB.

• Point Q is on the EXTERIOR of <CAB.

• Point A is the VERTEX.• Points A, C, and B sit ON

the angle.

• M

• Q

Note:• The measure of ∠A is denoted by

m∠A. The measure of an angle can be approximated using a protractor, using units called degrees(°). For instance, ∠BAC has a measure of 50°, which can be written asm∠BAC = 50°.

B

AC

Reading a protractor.

A

O B

What is the measure of angle BOA?

113°

Angle Addition Postulate• If C is in the interior

of <ABD, then m<ABC + m<CBD = m<ABD.

• In other words, little angle + little angle = big angle.

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Angle Addition Post., Continued

• m<CAB + m<DAC = m<DAB, so…

• m<CAB + 53° = 64°

• m<CAB = 11°

• Find the m<DAB.• m<DAC + m<CAB =

m<DAB.• 35° + 30° = 65°

• If m<ABM = (3x + 2)° and m<MBC = (5x-4)° and m<ABC = 102°, find x and the angle measures.

• Angle Addition Postulate: • m<ABM + m<MBC = m<ABC (use subst.)• 3x + 2 + 5x – 4 = 102• 8x – 2 = 102• 8x = 104• x = 13; m<ABM = 41°; m<MBC = 61°