1-3 Measuring and Constructing Angles
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- Bertram Summers
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1 Bell-ringer 1. Draw AB and AC, where A, B, and C are noncollinear. Possible answer: 2. Draw opposite rays DE and DF. A B C F D E Solve each equation. 3. 2x x 4 + 3x 5 = x + 2 = 8x 10 4
2 Standard U1S3 Name and classify angles. Measure and construct angles and angle bisectors.
3 A transit is a tool for measuring angles. It consists of a telescope that swivels horizontally and vertically. Using a transit, a survey or can measure the angle formed by his or her location and two distant points. An angle is a figure formed by two rays, or sides, with a common endpoint called the vertex. You can name an angle several ways: by its vertex, by a point on each ray and the vertex, or by a number.
4 The set of all points between the sides of the angle is the interior of an angle. The exterior of an angle is the set of all points outside the angle. Angle Name R, SRT, TRS, or 1 You cannot name an angle just by its vertex if the point is the vertex of more than one angle. In this case, you must use all three points to name the angle, and the middle point is always the vertex.
5 Example 1: Naming Angles A surveyor recorded the angles formed by a transit (point A) and three distant points, B, C, and D. Name three of the angles. Possible answer: BAC CAD BAD
6 Check It Out! Example 1 Write the different ways you can name the angles in the diagram. RTQ, T, STR, 1, 2
7 The measure of an angle is usually given in degrees. Since there are 360 in a circle, one degree is of a circle. When you use a protractor to measure angles, you are applying the following postulate.
8 You can use the Protractor Postulate to help you classify angles by their measure. The measure of an angle is the absolute value of the difference of the real numbers that the rays correspond with on a protractor. If OC corresponds with c and OD corresponds with d, m DOC = d c or c d.
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10 Example 2: Measuring and Classifying Angles Find the measure of each angle. Then classify each as acute, right, or obtuse. A. WXV m WXV = 30 WXV is acute. B. ZXW m ZXW = = 100 ZXW = is obtuse.
11 Check It Out! Example 2 Use the diagram to find the measure of each angle. Then classify each as acute, right, or obtuse. a. BOA m BOA = 40 BOA is acute. b. DOB m DOB = 125 DOB is obtuse. c. EOC m EOC = 105 EOC is obtuse.
12 Congruent angles are angles that have the same measure. In the diagram, m ABC = m DEF, so you can write ABC DEF. This is read as angle ABC is congruent to angle DEF. Arc marks are used to show that the two angles are congruent. The Angle Addition Postulate is very similar to the Segment Addition Postulate that you learned in the previous lesson.
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14 Example 3: Using the Angle Addition Postulate m DEG = 115, and m DEF = 48. Find m FEG m DEG = m DEF + m FEG 115 = 48 + m FEG = m FEG Add. Post. Substitute the given values. Subtract 48 from both sides. Simplify.
15 Check It Out! Example 3 m XWZ = 121 and m XWY = 59. Find m YWZ. m YWZ = m XWZ m XWY m YWZ = m YWZ = 62 Add. Post. Substitute the given values. Subtract.
16 An angle bisector is a ray that divides an angle into two congruent angles. JK bisects LJM; thus LJK KJM.
17 Example 4: Finding the Measure of an Angle KM bisects JKL, m JKM = (4x + 6), and m MKL = (7x 12). Find m JKM.
18 Step 1 Find x. Example 4 Continued m JKM = m MKL Def. of bisector (4x + 6) = (7x 12) x + 18 = 7x 4x 4x 18 = 3x 6 = x Substitute the given values. Add 12 to both sides. Simplify. Subtract 4x from both sides. Divide both sides by 3. Simplify.
19 Example 4 Continued Step 2 Find m JKM. m JKM = 4x + 6 = 4(6) + 6 = 30 Substitute 6 for x. Simplify.
20 Check It Out! Example 4a Find the measure of each angle. QS bisects PQR, m PQS = (5y 1), and m PQR = (8y + 12). Find m PQS. Step 1 Find y. Def. of bisector Substitute the given values. 5y 1 = 4y + 6 y 1 = 6 y = 7 Simplify. Subtract 4y from both sides. Add 1 to both sides.
21 Check It Out! Example 4a Continued Step 2 Find m PQS. m PQS = 5y 1 = 5(7) 1 = 34 Substitute 7 for y. Simplify.
22 Exit Slip Classify each angle as acute, right, or obtuse. 1. XTS 2. WTU acute right 3. K is in the interior of LMN, m LMK =52, and m KMN = 12. Find m LMN. 64
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