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1.4: Measuring Segments and Angles Vocabulary 1.4 •The numerical location of a point on a number line. •Coordinate •On a number line length AB = AB = |B - A| •Length •Congruent Segments •Sets of points that are of the same length. Symbol is: •The location of the middle of a segment. The midpoint divides a segment into two equal halves. •On a number line, midpoint of AB = 1/2 (B+A) •Midpoint The length of a segment on a number line is determined using The Ruler Postulate: The points on a line can be put in one-to-one correspondence with the real number line so that the distance between any two points is the absolute value of the difference of the corresponding numbers. A -8 B -6 -4 -2 -1 0 C 2 E D 4 6 8 The Segment Addition Postulate: If three points A,B,and C are collinear and B is between A and C, then AB + BC = AC Measuring Segments and Angles GEOMETRY LESSON 1-4 Find which two of the segments XY, ZY, and ZW are congruent. Use the Ruler Postulate to find the length of each segment. XY = | –5 – (–1)| = | –4| = 4 ZY = | 2 – (–1)| = |3| = 3 ZW = | 2 – 6| = |–4| = 4 Because XY = ZW, XY ZW. 1-4 Measuring Segments and Angles GEOMETRY LESSON 1-4 If AB = 25, find the value of x. Then find AN and NB. Use the Segment Addition Postulate to write an equation. AN + NB = AB Segment Addition Postulate (2x – 6) + (x + 7) = 25 Substitute. 3x + 1 = 25 3x = 24 x=8 AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15 Simplify the left side. Subtract 1 from each side. Divide each side by 3. Substitute 8 for x. AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25. 1-4 Measuring Segments and Angles GEOMETRY LESSON 1-4 M is the midpoint of RT. Find RM, MT, and RT. Use the definition of midpoint to write an equation. RM = MT Definition of midpoint 5x + 9 = 8x – 36 Substitute. 5x + 45 = 8x Add 36 to each side. 45 = 3x Subtract 5x from each side. 15 = x Divide each side by 3. RM = 5x + 9 = 5(15) + 9 = 84 MT = 8x – 36 = 8(15) – 36 = 84 Substitute 15 for x. RT = RM + M T = 168 RM and MT are each 84, which is half of 168, the length of RT. 1-4 The Protractor Postulate Let OA and OB be opposite rays in a plane, OA, OB, and all the rays with endpoint O that can be drawn on one side of AB can be paired with the real numbers from 0º to 180º so that: a. OA is paired with 0º and OB is paired with 180º. b. If OC is paired with x and OD is paired with y, then mCOD = |x-y|º D C y 77º mCOD = |x-y| = | 51 - 77 | = | -26 | = 26º x 51º 180º B O A 0º Vocabulary 1.4, cont. •Angle •Right Angle •Obtuse Angle •Acute Angle •Straight Angle •Congruent Angles •Formed by two rays with the same endpoint. •The rays: sides •Common endpoint: the vertex •Name: FAD , FBC, 1 •Measures exactly 90º •Measure is GREATER than 90º FAD •Measure is LESS than 90º ADE •Measure is exactly 180º ---this is a line FAB •Angles with the same measure. 1 2 Measure Angles Use a Protractor 1 m 1 = 40º The Angle Addition Postulate Find m AOB, m BOC and m AOC B A C O D m AOB = 60º m BOC = | 60 - 120 |º = 60 º and m AOC = 120 º The Angle Addition Postulate says that as long as AOB and BOC do not overlap, then mAOC = m AOB +m BOC = 120 Measuring Segments and Angles GEOMETRY LESSON 1-4 Name the angle below in four ways. The name can be the number between the sides of the angle: The name can be the vertex of the angle: 3. G. Finally, the name can be a point on one side, the vertex, and a point on the other side of the angle: AGC, CGA. 1-4 Measuring Segments and Angles GEOMETRY LESSON 1-4 Find the measure of each angle. Classify each as acute, right, obtuse, or straight. Use a protractor to measure each angle. m 1 = 110 Because 90 < 110 < 180, m 2 = 80 Because 0 < 80 < 90, 1 is obtuse. 2 is acute. 1-4 Measuring Segments and Angles GEOMETRY LESSON 1-4 Suppose that m 1 = 42 and m ABC = 88. Find m 2. Use the Angle Addition Postulate to solve. m 1+m 2=m ABC Angle Addition Postulate. 42 + m 2 = 88 Substitute 42 for m m 2 = 46 Subtract 42 from each side. 1-4 1 and 88 for m ABC. Re Cap: •The numerical location of a point on a number line. •Coordinate •On a number line length AB = AB = |B - A| •Length •Congruent Segments •Sets of points that are of the same length. Symbol is: •The location of the middle of a segment. The midpoint divides a segment into two equal halves. •On a number line, midpoint of AB = 1/2 (B+A) •Midpoint The length of a segment on a number line is determined using The Ruler Postulate: The points on a line can be put in one-to-one correspondence with the real number line so that the distance between any two points is the absolute value of the difference of the corresponding numbers. A -8 B -6 -4 -2 -1 0 C 2 E D 4 6 8 The Segment Addition Postulate: If three points A,B,and C are collinear and B is between A and C, then AB + BC = AC Recap 2. •Angle •Right Angle •Obtuse Angle •Acute Angle •Straight Angle •Congruent Angles •Formed by two rays with the same endpoint. •The rays: sides •Common endpoint: the vertex •Name: FAD , FBC, 1 •Measures exactly 90º •Measure is GREATER than 90º FAD •Measure is LESS than 90º ADE •Measure is exactly 180º ---this is a line FAB •Angles with the same measure. 1 2 Re Cap 3 Find m AOB, m BOC and m AOC B A C O D m AOB = 60º m BOC = | 60 - 120 |º = 60 º and m AOC = 120 º The Angle Addition Postulate says that as long as AOB and BOC do not overlap, then mAOC = m AOB +m BOC = 120 Extra Practice GEOMETRY LESSON 1-4 Use the figure below for Exercises 1-3. Use the figure below for Exercises 4–6. 1. If XT = 12 and XZ = 21, then TZ = 7. 9 2. If XZ = 3x, XT = x + 3, and TZ = 13, find XZ. 24 3. Suppose that T is the midpoint of XZ. If XT = 2x + 11 and XZ = 5x + 8, find the value of x. 14 4. Name 2 two different ways. DAB, BAD 5. Measure and classify 1, 2, and BAC. 90°, right; 30°, acute; 120°, obtuse 6. Which postulate relates the measures of 1, 2, and BAC? Angle Addition Postulate 1-4