1.4: Measuring Segments and Angles

The numerical location of a point on a number line. On a number line length AB = AB = |B - A| 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) Vocabulary 1.4 Coordinate Length Congruent Segments Midpoint B A C D E 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. The Segment Addition Postulate: If three points A,B,and C are collinear and B is between A and C, then AB + BC = AC

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 Find which two of the segments XY, ZY, and ZW are congruent. Because XY = ZW, XY ZW. GEOMETRY LESSON 1-4 Measuring Segments and Angles 1-4

Use the Segment Addition Postulate to write an equation. AN + NB = AB Segment Addition Postulate (2x – 6) + (x + 7) = 25 Substitute. 3x + 1 = 25Simplify the left side. 3x = 24Subtract 1 from each side. x = 8Divide each side by 3. AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25. If AB = 25, find the value of x. Then find AN and NB. AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15 Substitute 8 for x. GEOMETRY LESSON 1-4 Measuring Segments and Angles 1-4

Use the definition of midpoint to write an equation. RM = MTDefinition of midpoint 5x + 9 = 8x – 36Substitute. 5x + 45 = 8x Add 36 to each side. 45 = 3x Subtract 5x from each side. 15 = x Divide each side by 3. RM and MT are each 84, which is half of 168, the length of RT. M is the midpoint of RT. Find RM, MT, and RT. RM = 5x + 9 = 5(15) + 9 = 84 MT = 8x – 36 = 8(15) – 36 = 84 Substitute 15 for x. RT = RM + MT = 168 GEOMETRY LESSON 1-4 Measuring Segments and Angles 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 m  COD = |x-y|º O A B C D y  77º x  51º 0º 180º m  COD = |x-y| = | | = | -26 | = 26º

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: Measures exactly 90º Measure is GREATER than 90º Measure is LESS than 90º Measure is exactly 180º ---this is a line Angles with the same measure. 1 2  FAD,  FBC,  1  FAD  ADE  FAB

Measure AnglesUse a Protractor 1 m  1 = 40º

The Angle Addition Postulate A B C O m  AOB = 60º m  BOC = | |º = 60 º and m  AOC = 120 º The Angle Addition Postulate says that as long as  AOB and  BOC do not overlap, then m  AOC = m  AOB +m  BOC = 120 D Find m  AOB, m  BOC and m  AOC

Name the angle below in four ways. The name can be the vertex of the angle: 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. The name can be the number between the sides of the angle: 3. GEOMETRY LESSON 1-4 Measuring Segments and Angles 1-4

Because 0 < 80 < 90, 2 is acute. m 2 = 80 Use a protractor to measure each angle. m 1 = 110 Because 90 < 110 < 180, 1 is obtuse. Find the measure of each angle. Classify each as acute, right, obtuse, or straight. GEOMETRY LESSON 1-4 Measuring Segments and Angles 1-4

Use the Angle Addition Postulate to solve. m 1 + m 2 = m ABCAngle Addition Postulate m 2 = 88Substitute 42 for m 1 and 88 for m ABC. m 2 = 46Subtract 42 from each side. Suppose that m 1 = 42 and m ABC = 88. Find m 2. GEOMETRY LESSON 1-4 Measuring Segments and Angles 1-4

The numerical location of a point on a number line. On a number line length AB = AB = |B - A| 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) Re Cap: Coordinate Length Congruent Segments Midpoint B A C D E 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. 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: Measures exactly 90º Measure is GREATER than 90º Measure is LESS than 90º Measure is exactly 180º ---this is a line Angles with the same measure. 1 2  FAD,  FBC,  1  FAD  ADE  FAB

Re Cap 3 A B C O m  AOB = 60º m  BOC = | |º = 60 º and m  AOC = 120 º The Angle Addition Postulate says that as long as  AOB and  BOC do not overlap, then m  AOC = m  AOB +m  BOC = 120 D Find m  AOB, m  BOC and m  AOC

Use the figure below for Exercises 4–6. 4. Name 2 two different ways. 5. Measure and classify 1, 2, and BAC. 6. Which postulate relates the measures of 1, 2, and BAC? 14 Angle Addition Postulate DAB, BAD Use the figure below for Exercises If XT = 12 and XZ = 21, then TZ = If XZ = 3x, XT = x + 3, and TZ = 13, find XZ. 3. Suppose that T is the midpoint of XZ. If XT = 2x + 11 and XZ = 5x + 8, find the value of x °, right; 30°, acute; 120°, obtuse GEOMETRY LESSON 1-4 Extra Practice 1-4