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1.4 INTRO TO ANGLE MEASUREMENT 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: 45 • Every angle on a protractor measures more than 0 degrees and less than or equal to 180 degrees. 2 A Protractor 3 • 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. 4 • Here are some common angles and their measurements. m1 45 1 m2 90 2 m3 135 3 4 m4 180 5 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 6 • 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. right angle straight angle 7 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. vertex • Points A, C, and B sit ON the angle. • Rays: AC and AB make up the angle •Q C •M sides B A Note: • The measure of A is denoted by mA. 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 as B mBAC = 50°. A C Reading a protractor. A O What m<BOA? 103° B Naming Angles • We could name this angle using the points or the number at the vertex. 1 • There are 4 names we could this angle— • <E—since point E is the vertex and there is ONLY ONE angle using that vertex. • <FED or <DEF— notice that the vertex is the middle point mentioned. • <1—for the number given at the vertex. 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. 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 • 3x + 2 + 5x – 4 = 102 (use subst.) • 8x – 2 = 102 (CLT) • 8x = 104 (Add. POE) • x = 13 (Div. POE) • m<ABM = 41°; m<MBC = 61°