1.2. Angle Relationships and Similar. Mrs. Poland January 23, Geometric Properties Triangles
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1 1.1 Angles Basic Terminology Degree Measure Standard Position Coterminal Angles 1.2 Angle Relationships and Similar Triangles Geometric Properties Triangles Mrs. Poland January 23, 2013
2 Objectives Objective #1: Students will be able to use vocabulary to describe situations and answer questions. Success criterion #1: I can use vocabulary to describe situations and answers. Objective #2: Students will be able to express angles in decimal degrees and degrees, minutes and seconds. Success criterion #2: I can express angle measures in appropriate units. Objective #3: Students will be able to use geometric relationships to solve problems. Success criterion #3: I can use geometric properties of lines and angles to solve problems. Success criterion #4: I can use geometric properties of triangles to solve problems.
3 Basic Terminology An angle s measure is generated by a rotation about the vertex. The ray in its initial position is called the initial side of the angle. The ray in its location after the rotation is the terminal side of the angle.
4 Basic Terminology Positive angle: The rotation of the terminal side of an angle is counterclockwise. Negative angle: The rotation of the terminal side is clockwise.
5 Measuring Angles The most common unit for measuring angles is the degree. A complete rotation of a ray gives an angle whose measure is 360. of complete rotation gives an angle whose measure is 1.
6 Example 1a FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES Find the measure of each marked angle. Since the two angles form a right angle, they are complementary. Combine terms. Divide by 9. Determine the measure of each angle by substituting 10 for x:
7 Example 1b FINDING MEASURES OF COMPLEMENTARY AND SUPPLEMENTARY ANGLES (continued) Find the measure of each marked angle. Since the two angles form a straight angle, they are supplementary. The angle measures are and.
8 Degrees, Minutes, Seconds One minute is 1/60 of a degree. One second is 1/60 of a minute.
9 Example 2 Converting between decimal and degrees, minutes and seconds o Convert to decimal degrees 74 o o 14_ = o = = o Convert to degrees, minutes and seconds o = o o = (60) o = o = (60) o =
10 Example 3 Finding the sum and difference Find the sum or difference. (a) (b) Add degrees and minutes separately. Write 90 as 89 60ʹ.
11 Standard Position An angle is in standard position if its vertex is at the origin and its initial side is along the positive x-axis.
12 Quadrantal Angles Angles in standard position with their terminal sides along the x-axis or y-axis, such as angles with measures 90, 180, 270, and so on.
13 Coterminal Angles Coterminal angles are angles that share the same initial and terminal sides and differ by a multiple of 360 degrees.
14 Example 4 FINDING MEASURES OF COTERMINAL ANGLES Find the angle of least possible positive measure coterminal with an angle of 908. Add or subtract 360 as many times as needed to obtain an angle with measure greater than 0 but less than 360. An angle of 908 is coterminal with an angle of 188.
15 Example 4 FINDING MEASURES OF COTERMINAL ANGLES (continued) Find the angle of least possible positive measure coterminal with an angle of 75. Add or subtract 360 as many times as needed to obtain an angle with measure greater than 0 but less than 360. An angle of 75 is coterminal with an angle of 285.
16 Example 4 FINDING MEASURES OF COTERMINAL ANGLES (continued) Find the angle of least possible positive measure coterminal with an angle of 800. The least integer multiple of 360 greater than 800 is An angle of 800 is coterminal with an angle of 280.
17 Coterminal Angles To find an expression that will generate all angles coterminal with a given angle, add integer multiples of 360 to the given angle. For example, the expression for all angles coterminal with 60 is
18 Example 5 FINDING ANGLE MEASURES Find the measure of angles 1, 2, 3, and 4, given that lines m and n are parallel. Angles 1 and 4 are alternate exterior angles, so they are equal. Subtract 3x. Add 40. Divide by 2. Angle 1 has measure Substitute 21 for x.
19 Example 5 FINDING ANGLE MEASURES (continued) Angle 4 has measure Angle 2 is the supplement of a 65 angle, so it has measure. Angle 3 is a vertical angle to angle 1, so its measure is 65. Substitute 21 for x.
20 Example 6 APPLYING THE ANGLE SUM OF A TRIANGLE PROPERTY The measures of two of the angles of a triangle are 48 and 61. Find the measure of the third angle, x. The sum of the angles is 180. Add. Subtract 109. The third angle of the triangle measures 71.
21 Example 7 FINDING ANGLE MEASURES IN SIMILAR TRIANGLES In the figure, triangles ABC and NMP are similar. Find the measures of angles B and C. Since the triangles are similar, corresponding angles have the same measure. B corresponds to M, so angle B measures 31. C corresponds to P, so angle C measures 104.
22 Example 8 FINDING SIDE LENGTHS IN SIMILAR TRIANGLES In the figure, triangles ABC and DFE are similar. Find the measures of sides DF and EF. Since the triangles are similar, corresponding sides are proportional. DF corresponds to AB, and DE corresponds to AC, so Side DF has length 12. EF corresponds to CB, so Side EF has length 16.
23 Example 9 FINDING THE HEIGHT OF A FLAGPOLE Firefighters at a station need to measure the height of the station flagpole. They find that at the instant when the shadow of the station is 18 m long, the shadow of the flagpole is 99 ft long. The station is 10 m high. Find the height of the flagpole. Since the two triangles are similar, corresponding sides are proportional. Lowest terms The flagpole is 55 feet high.
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