Geometry unit 1.3
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This document contains several examples and problems involving concepts related to number lines, segments, midpoints, angles, and coordinate planes. It defines key terms like length, midpoint, acute/obtuse/right angles, and provides examples of using formulas and properties like the segment addition postulate to solve problems involving finding lengths, midpoints, missing values, and classifying angles. Several word problems are included for practice applying these geometric concepts.
Geometry unit 1.3
- 2. The numerical location of a point on a number line. Length : On a number line length AB = AB = |B - A| Midpoint : On a number line, midpoint of AB = 1/2 (B+A) A B C D E -8 -6 -4 -2 -1 0 2 4 6 8
- 3. Find which two of the segments XY, ZY, and ZW are congruent. 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.
- 4. If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC. A B C
- 5. 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 Simplify the left side. 3x = 24 Subtract 1 from each side. x = 8 Divide each side by 3. AN = 2x – 6 = 2(8) – 6 = 10 NB = x + 7 = (8) + 7 = 15 Substitute 8 for x. AN = 10 and NB = 15, which checks because the sum of the segment lengths equals 25.
- 6. 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 + MT = 168 RM and MT are each 84, which is half of 168, the length of RT.
- 7. 1. T is in between of XZ. If XT = 12 and XZ = 21, then TZ = ? 2. T is the midpoint of XZ. If XT = 2x +11 and XZ = 5x + 8, find the value of x. Quiz
- 8. Answers: 1. T is in between of XZ. If XT = 12 and XZ = 21, then TZ = ? 21 – 12 = 9, TZ = 9 2. T is the midpoint of XZ. If XT = 2x +11 and XZ = 5x + 8, find the value of x. Since T is a midpoint of XZ, 2*XT = XZ 2(2x + 11) = 5x + 8 4x+22=5x+8 X=14
- 9. On a number line formula: a b 2 On a coordinate plane x x y y x , y 1 2 1 2 m m formula: 2 , 2
- 10. QS has endpoints Q(3, 5) and S(7, -9). Find the coordinates of its midpoint M. The midpoint of AB is M(3, 4). One endpoint is A(-3, -2). Find the coordinates of the other endpoint B.
- 11. Answers 1) QS has endpoints Q(3, 5) and S(7, -9). Find the coordinates of its midpoint M. ((3+7)/2, (5+-9)/2) The midpoint is (5,-2). 2) The midpoint of AB is M(3, 4). One endpoint is A(-3, -2). Find the coordinates of the other endpoint B (-3+x)/2 = 3, -3 + x = 6, x = 9 (-2+y)/2 = 4, -2 + y = 8, y = 10 The other endpoint is (9,10).
- 12. FAD , FBC, 1 •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 ADE FAB •Angles
- 13. Name the angle below in four ways. The name can be the number between the sides of the angle: 3 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.
- 14. 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 1 and 88 for m ABC. m 2 = 46 Subtract 42 from each side.
- 15. Use the figure below for Exercises 4–6. 4. Name 2 two different ways. DAB and BAD 5. Measure and classify 1, 2, and BAC. 6. Which postulate relates the measures of 1, 2, and BAC? 14 Angle Addition Postulate Use the figure below for Exercises 1-3. 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. 90°, right; 30°, acute; 120°, obtuse
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