Geometry Short Cycle 1 Exam Review

Transcription

1 Name: lass: ate: I: Geometry Short ycle 1 Exam Review Multiple hoice Identify the choice that best completes the statement or answers the question. 1. Name a plane that contains. a. plane R c. plane WRT b. plane WT d. plane R 2. Find the length of. a. = 7 c. = 7 b. = 9 d. = 8 3. The tip of a pendulum at rest sits at point. uring an experiment, a physics student sets the pendulum in motion. The tip of the pendulum swings back and forth along part of a circular path from point to point. uring each swing the tip passes through point. Name all the angles in the diagram. a. O, O c. O, O, O, O b. O, O, O d. O, O, O 1

2 Name: I: 4. Name all pairs of vertical angles. a. MLN and JLM ; JLK and KLN b. JLK and MLN ; JLM and KLN c. JKL and MNL; JML and KNL d. JLK and JLM ; KLN and MLN 5. There are four fruit trees in the corners of a square backyard with 30-ft sides. What is the distance between the apple tree and the plum tree P to the nearest tenth? a ft c ft b ft d ft 6. Identify the transformation. Then use arrow notation to describe the transformation. a. The transformation is a 90 rotation. ' ' ' b. The transformation is a 45 rotation. ' ' ' c. The transformation is a reflection. ' ' ' d. The transformation is a translation. ' ' ' 2

3 Name: I: 7. Name three collinear points. a. P, G, and N c. R, P, and G b. R, P, and N d. R, G, and N 8. etermine if the biconditional is true. If false, give a counterexample. figure is a square if and only if it is a rectangle. a. The biconditional is true. b. The biconditional is false. rectangle does not necessarily have four congruent sides. c. The biconditional is false. ll squares are parallelograms with four 90 angles. d. The biconditional is false. rectangle does not necessarily have four 90 angles. 9. Solve the equation 4x 6 = 34. Write a justification for each step. 4x 6 = 34 Given equation [1] 4x = 40 Simplify. 4x = [2] x = 10 Simplify. a. [1] Substitution Property of Equality; [2] ivision Property of Equality b. [1] ddition Property of Equality; [2] ivision Property of Equality c. [1] ivision Property of Equality; [2] Subtraction Property of Equality d. [1] ddition Property of Equality; [2] Reflexive Property of Equality 1

4 Name: I: 10. Write a justification for each step. m JKL = 100 m JKL = m JKM + m MKL [1] 100 = (6x + 8) + (2x 4) Substitution Property of Equality 100 = 8x + 4 Simplify. 96 = 8x Subtraction Property of Equality 12 = x [2] x = 12 Symmetric Property of Equality a. [1] Transitive Property of Equality [2] ivision Property of Equality b. [1] ngle ddition Postulate [2] ivision Property of Equality c. [1] ngle ddition Postulate [2] Simplify. d. [1] Segment ddition Postulate [2] Multiplication Property of Equality 11. Identify the property that justifies the statement. and EF. So EF. a. Reflexive Property of ongruence c. Symmetric Property of ongruence b. Substitution Property of Equality d. Transitive Property of ongruence 12. Two angles with measures (2x 2 + 3x 5) and (x x 7) are supplementary. Find the value of x and the measure of each angle. a. x = 5; 60 ; 30 c. x = 5; 60 ; 120 b. x = 6; 85 ; 95 d. x = 4; 40 ; 90 4

5 Name: I: 13. Use the given paragraph proof to write a two-column proof. Given: is a right angle. 1 3 Prove: 2 and 3 are complementary. Paragraph proof: Since is a right angle, m = 90 by the definition of a right angle. y the ngle ddition Postulate, m = m 1 + m 2. y substitution, m 1 + m 2 = 90. Since 1 3, m 1 = m 3 by the definition of congruent angles. Using substitution, m 3 + m 2 = 90. Thus, by the definition of complementary angles, 2 and 3 are complementary. omplete the proof. Two-column proof: Statements Reasons 1. is a right angle Given 2. m = efinition of a right angle 3. m = m 1 + m 2 3. [1] 4. m 1 + m 2 = Substitution 5. m 1 = m 3 5. [2] 6. m 3 + m 2 = Substitution 7. 2 and 3 are complementary. 7. efinition of complementary angles a. [1] Substitution [2] efinition of congruent angles b. [1] ngle ddition Postulate [2] efinition of congruent angles c. [1] ngle ddition Postulate [2] efinition of equality d. [1] Substitution [2] efinition of equality 5

6 Name: I: 14. Identify the transversal and classify the angle pair 11 and 7. a. The transversal is line l. The angles are corresponding angles. b. The transversal is line l. The angles are alternate interior angles. c. The transversal is line n. The angles are alternate exterior angles. d. The transversal is line m. The angles are corresponding angles. 15. Find m. a. m = 40 c. m = 35 b. m = 45 d. m = 50 6

7 Name: I: 16. In a swimming pool, two lanes are represented by lines l and m. If a string of flags strung across the lanes is represented by transversal t, and x = 10, show that the lanes are parallel. a. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are alternate interior angles and they are congruent, so the lanes are parallel by the lternate Interior ngles Theorem. b. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are alternate interior angles, and they are congruent, so the lanes are parallel by the onverse of the lternate Interior ngles Theorem. c. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are corresponding angles and they are congruent, so the lanes are parallel by the onverse of the orresponding ngles Postulate. d. 3x + 4 = 3(10) + 4 = 34 ; 4x 6 = 4(10) 6 = 34 The angles are same-side interior angles and they are supplementary, so the lanes are parallel by the onverse of the Same-Side Interior ngles Theorem. 17. Use slopes to determine whether the lines are parallel, perpendicular, or neither. and for (3,5), ( 2,7), (10,5), and (6,15) a. neither c. parallel b. perpendicular 18. etermine whether the lines 12x + 3y = 3 and y = 4x + 1 are parallel, intersect, or coincide. a. intersect c. parallel b. coincide 19. is between and E. E = 6x, = 4x + 8, and E = 27. Find E. a. E = 17.5 c. E = 105 b. E = 78 d. E = 57 7

8 Name: I: 20. Find the measure of O. Then, classify the angle as acute, right, or obtuse. a. m O = 125 ; obtuse c. m O = 90 ; right b. m O = 35 ; acute d. m O = 160 ; obtuse 21. bisects, m = (7x 1), and m = (4x + 8). Find m. a. m = 22 c. m = 40 b. m = 3 d. m = Find the measure of the complement of M, where m M = 31.1 a c b d Find the measure of the supplement of R, where m R = (8z + 10) a. (170 8z) c b. (190 8z) d. (80 8z) 24. M is the midpoint of N, has coordinates ( 6, 6), and M has coordinates (1, 2). Find the coordinates of N. a. (8, 10) c. ( 2 1 2, 2) b. ( 5, 4) d. (8 1 2, 91 2 ) 8

9 Name: I: 25. Write the converse, inverse, and contrapositive of the conditional statement, If an animal is a bird, then it has two eyes. Find the truth value of each. a. onverse: If an animal is not a bird, then it does not have two eyes. The converse is false. Inverse: If an animal does not have two eyes, then it is not a bird. The inverse is true. ontrapositive: If an animal is a bird, then it has two eyes. The contrapositive is true. b. onverse: If an animal has two eyes, then it is a bird. The converse is false. Inverse: If an animal is not a bird, then it does not have two eyes. The inverse is false. ontrapositive: If an animal does not have two eyes, then it is not a bird. The contrapositive is true. c. onverse: If an animal does not have two eyes, then it is not a bird. The converse is true. Inverse: If an animal is not a bird, then it does not have two eyes. The inverse is true. ontrapositive: If an animal has two eyes, then it is a bird. The contrapositive is false. d. onverse: ll birds have two eyes. The converse is true. Inverse: ll animals have two eyes. The inverse is true. ontrapositive: ll birds are animals, and animals have two eyes. The contrapositive is true. 26. For the conditional statement, write the converse and a biconditional statement. If a figure is a right triangle with sides a, b, and c, then a 2 + b 2 = c 2. a. onverse: If a figure is not a right triangle with sides a, b, and c, then a 2 + b 2 c 2. iconditional: figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. b. onverse: If a 2 + b 2 = c 2, then the figure is a right triangle with sides a, b, and c. iconditional: figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. c. onverse: If a 2 + b 2 c 2, then the figure is not a right triangle with sides a, b, and c. iconditional: figure is not a right triangle with sides a, b, and c if and only if a 2 + b 2 c 2 d. onverse: If a 2 + b 2 c 2, then the figure is not a right triangle with sides a, b, and c. iconditional: figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. 9

10 Name: I: 27. gardener has 26 feet of fencing for a garden. To find the width of the rectangular garden, the gardener uses the formula P = 2l + 2w, where P is the perimeter, l is the length, and w is the width of the rectangle. The gardener wants to fence a garden that is 8 feet long. How wide is the garden? Solve the equation for w, and justify each step. P = 2l + 2w Given equation 26 = 2(8) + 2w [1] 26 = w Simplify. 16 = = 2w Subtraction Property of Equality Simplify = 2w 2 [2] 5 = w Simplify. w = 5 Symmetric Property of Equality a. [1] Substitution Property of Equality [2] ivision Property of Equality The garden is 5 ft wide. b. [1] Simplify [2] ivision Property of Equality The garden is 5 ft wide. c. [1] Substitution Property of Equality [2] Subtraction Property of Equality The garden is 5 ft wide. d. [1] Subtraction Property of Equality [2] Simplify The garden is 5 ft wide. 28. Give an example of corresponding angles. a. 8 and 4 c. 3 and 6 b. 4 and 1 d. 5 and 7 10

11 Name: I: 29. Use the information m 1 = (3x + 30), m 2 = (5x 10), and x = 20, and the theorems you have learned to show that l Ä m. a. y substitution, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. y the Substitution Property of Equality, m 1 = m 2 = 90. y the onverse of the lternate Interior ngles Theorem, l Ä m. b. y substitution, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. Since 1 and 2 are alternate interior angles, m 1 = m 2 = 180. y the onverse of the Same-Side Interior ngles Theorem, l Ä m. c. y substitution, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. Since 1 and 2 are same-side interior angles, m 1 = m 2 = 180. y the onverse of the Same-Side Interior ngles Theorem, l Ä m. d. Since 1 and 2 are same-side interior angles, m 1 = 3(20) + 30 = 90 and m 2 = 5(20) 10 = 90. y substitution, m 1 = m 2 = 90. y the onverse of the lternate Interior ngles Theorem, l Ä m. 30. Ä for (4, 5), ( 2, 3), (x, 2), and (6, y). Find a set of possible values for x and y. Ï a. Ê Ë Áx, yˆ y = 1 x 4 Ô Ì, x 6 Ô Ï 3 c. ÔÊ Ì x, y ÓÔ Ô Ë Á ˆ y = 3x 20, y 2 Ô ÓÔ Ô Ï b. Ê Ë Á x, y ˆ y = 1 x 4 Ô Ô Ï Ì d. ÔÊ Ì x, y ÓÔ 3 Ô Ë Á ˆ y = 3x 20, x 2 Ô ÓÔ Ô 11

12 I: Geometry Short ycle 1 Exam Review nswer Section MULTIPLE HOIE 1. NS: plane can be described by any three noncollinear points. Of the choices given, only points W, R, and T are noncollinear. Thus, lies in plane WRT. Points,, and R are collinear. plane can be described by any three noncollinear points. Points W,, and T are collinear. plane can be described by any three noncollinear points. plane can be described by any three noncollinear points. PTS: 1 IF: asic REF: 192eb53a df-9c7d f0d2ea OJ: Identifying Points and Lines in a Plane LO: MTH MTH TOP: 1-1 Understanding Points, Lines, and Planes KEY: point line plane MS: OK 1 2. NS: = 8 ( 1) = = 7 = 7 The length of a segment is always positive. Find the absolute value of the difference of the coordinates. Find the absolute value of the difference of the coordinates. PTS: 1 IF: asic REF: 19313ea df-9c7d f0d2ea OJ: Finding the Length of a Segment LO: MTH TOP: 1-2 Measuring and onstructing Segments KEY: segment length MS: OK 1 1

13 I: 3. NS: O is another name for O, O is another name for O, and O is another name for O. Thus the diagram contains three angles. What is the name for the angle that describes the change in position from point to point? ngle O is another name for angle O, and angle O is another name for angle O. What is the name for the angle that describes the change in position from point to point? Point O is the vertex of all the angles in the diagram. PTS: 1 IF: verage REF: 193aa df-9c7d f0d2ea OJ: Naming ngles LO: MTH TOP: 1-3 Measuring and onstructing ngles KEY: angle MS: OK 1 4. NS: The vertical angle pairs are JLK and MLN, and JLM and KLN. These angles appear to have the same measure. These angles are adjacent, not vertical. Vertical angles share a common vertex, the point of intersection of the two lines. The vertex is the middle letter in the angle's name. These angles are adjacent, not vertical. PTS: 1 IF: asic REF: 194b518a df-9c7d f0d2ea OJ: Identifying Vertical ngles LO: MTH TOP: 1-4 Pairs of ngles KEY: vertical angles MS: OK 1 1

14 I: 5. NS: Set up the yard on a coordinate plane so that the apple tree is at the origin, the fig tree F has coordinates (30, 0), the plum tree P has coordinates (30, 30), and the nectarine tree N has coordinates (0, 30). The distance between the apple tree and the plum tree is P. P = Ê Ë Áx 2 x 1 ˆ 2 + Ê Ë Á y 2 y 1 ˆ 2 = ( 30 0) 2 + ( 30 0) 2 = = = ft heck your calculations and rounding. Set up the yard on a coordinate plane so that the apple tree is at the origin. Then use the distance formula to find the distance. Set up the yard on a coordinate plane so that the apple tree is at the origin. Then use the distance formula to find the distance. PTS: 1 IF: verage REF: 19599fb df-9c7d f0d2ea OJ: pplication LO: MTH TOP: 1-6 Midpoint and istance in the oordinate Plane KEY: coordinate geometry distance MS: OK 1 3

15 I: 6. NS: The transformation is a 90 rotation with center of rotation at point O. To be a reflection, each point and its image are the same distance from a line of reflection. To be a translation, each point of moves the same distance in the same direction. What happens to one of the segments in the triangle? Is '' an image of after a rotation of 45 degrees? The transformation is not a reflection because each point and its image are not the same distance from a line of reflection. The transformation is not a translation because each point of the triangle does not move the same distance in the same direction. PTS: 1 IF: verage REF: 195c020e df-9c7d f0d2ea OJ: Identifying Transformations ST: OH.OHS.MTH LO: MTH TOP: 1-7 Transformations in the oordinate Plane KEY: coordinate geometry transformation rotation MS: OK 1 7. NS: ollinear points are points that lie on the same line. R, G, and N are three collinear points. ollinear points are points that lie on the same line. ollinear points are points that lie on the same line. Points R, P, and G are noncollinear. PTS: 1 IF: asic REF: 19bb877e df-9c7d f0d2ea OJ: Naming Points, Lines, and Planes LO: MTH TOP: 1-1 Understanding Points, Lines, and Planes MS: OK 1 4

16 I: 8. NS: onditional: If a figure is a square, then it is a rectangle. True. onverse: If a figure is a rectangle, then it is a square. False. rectangle does not necessarily have four congruent sides. ecause the converse is false, the biconditional is false. For a biconditional statement to be true, both the conditional statement and its converse must be true. ll rectangles have four 90-degree angles as well. rectangle does have four 90-degree angles, but does it have four congruent sides? PTS: 1 IF: asic REF: 19d823ce df-9c7d f0d2ea OJ: nalyzing the Truth Value of a iconditional Statement LO: MTH.P MTH.P TOP: 2-4 iconditional Statements and efinitions KEY: biconditional truth value MS: OK 1 9. NS: 4x 6 = 34 Given equation [1] ddition Property of Equality 4x = 40 Simplify. 4x = [2] ivision Property of Equality x = 10 Simplify. heck the properties. heck the properties. heck the properties. PTS: 1 IF: asic REF: 19dce df-9c7d f0d2ea OJ: Solving an Equation in lgebra NT: NT.SS.MTH REI.1 LO: MTH.P MTH TOP: 2-5 lgebraic Proof KEY: algebraic proof proof MS: OK 2 5

17 I: 10. NS: m JKL = m JKM + m MKL [1] ngle ddition Postulate 100 = (6x + 8) + (2x 4) Substitution Property of Equality 100 = 8x + 4 Simplify. 96 = 8x Subtraction Property of Equality 12 = x [2] ivision Property of Equality x = 12 Symmetric Property of Equality heck the properties. heck the justifications. The Segment ddition Postulate refers to segments, not angles. PTS: 1 IF: verage REF: 19e1862e df-9c7d f0d2ea OJ: Solving an Equation in Geometry NT: NT.SS.MTH REI.1 LO: MTH.P MTH MTH TOP: 2-5 lgebraic Proof KEY: algebraic proof proof MS: OK NS: The Transitive Property of ongruence states that if figure figure and figure figure, then figure figure. The Reflexive Property of ongruence states that figure is congruent to figure. The Substitution Property of Equality states that if a = b, then b can be substituted for a in any expression. The Symmetric Property of ongruence states that if figure is congruent to figure, then figure is congruent to figure. PTS: 1 IF: asic REF: 19e1ad3e df-9c7d f0d2ea OJ: Identifying Properties of Equality and ongruence LO: MTH.P TOP: 2-5 lgebraic Proof KEY: congruence properties reflexive symmetric transitive MS: OK 1 6

18 I: 12. NS: Step 1 reate an equation The angles are supplements and their sum equals 180. (2x 2 + 3x 5) + (x x 7) = 180 Step 2 Solve the equation 3x x 12 = 180 3x x 192 = 0 (3x + 32)(x 6) = 0 x = 32 3 or 6. When x = 32 3, the measurement of the second angle is x x 7 = ngles cannot have negative measurements, so x = 6. Step 3 Solve for the required values The measurement of the first angle is 2x 2 + 3x 5 = 2(6) 2 + 3(6) 5 = 85. The measurement of the second angle is x x 7 = (6) (6) 7= 95. The angles are supplements. Use the definition of supplements to solve for x. heck for algebra mistakes. When x equals 5, the second angle is not 120 degrees. The angles are supplements. Use the definition of supplements to solve for x. PTS: 1 IF: dvanced REF: 19e8ad df-9c7d f0d2ea TOP: 2-6 Geometric Proof KEY: supplementary angles MS: OK 2 7

19 I: 13. NS: Two-column proof: Statements Reasons 1. is a right angle Given 2. m = efinition of a right angle 3. m = m 1 + m 2 3. ngle ddition Postulate 4. m 1 + m 2 = Substitution 5. m 1 = m 3 5. efinition of congruent angles 6. m 3 + m 2 = Substitution 7. 2 and 3 are complementary. 7. efinition of complementary angles In a paragraph proof, statements and reasons appear together. In a paragraph proof, statements and reasons appear together. In a paragraph proof, statements and reasons appear together. PTS: 1 IF: verage REF: 19ed71fa df-9c7d f0d2ea OJ: Reading a Paragraph Proof LO: MTH.P MTH.P TOP: 2-7 Flowchart and Paragraph Proofs KEY: paragraph proof two column proof MS: OK NS: To determine which line is the transversal for a given angle pair, locate the line that connects the vertices. orresponding angles lie on the same side of the transversal l, on the same sides of lines n and m. lternate interior angles lie on opposite sides of the transversal, between two lines. To find which line is the transversal for a given angle pair, locate the line that connects the vertices. To find which line is the transversal for a given angle pair, locate the line that connects the vertices. PTS: 1 IF: verage REF: 1a21e5e df-9c7d f0d2ea OJ: Identifying ngle Pairs and Transversals LO: MTH MTH TOP: 3-1 Lines and ngles KEY: corresponding angles transversal MS: OK 1 8

20 I: 15. NS: (x) = (3x 70) orresponding ngles Postulate 0 = 2x 70 Subtract x from both sides. 70 = 2x dd 70 to both sides. 35 = x ivide both sides by 2. m = 3x 70 m = 3(35) 70 = 35 Substitute 35 for x. Simplify. If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Use the orresponding ngles Postulate. First, set the measures of the corresponding angles equal to each other. Then, solve for x and substitute in the expression (3x 70). PTS: 1 IF: verage REF: 1a24483e df-9c7d f0d2ea OJ: Using the orresponding ngles Postulate LO: MTH TOP: 3-2 ngles Formed by Parallel Lines and Transversals KEY: corresponding angles parallel lines MS: OK NS: Substitute 10 for x in each expression: 3x + 4 = 3(10) + 4 = 34 4x 6 = 4(10) 6 = 34 The angles are alternate interior angles, and they are congruent, so the lanes are parallel by the onverse of the lternate Interior ngles Theorem. The lanes are parallel by the onverse of the lternate Interior ngles Theorem. The angles are alternate interior angles. The angles are alternate interior angles. PTS: 1 IF: verage REF: 1a30340a df-9c7d f0d2ea OJ: pplication LO: MTH MTH TOP: 3-3 Proving Lines Parallel KEY: parallel lines proof MS: OK 2 9

21 I: 17. NS: slope of = = 5 2 = 5 2 slope of = 6 10 = 4 = The lines have different slopes, so they are not parallel. The product of the slopes is 5 2 = 1, not 1, so the slopes are not perpendicular. 2 5 The lines are coplanar, so they cannot be skew. The product of the slopes is 1, not 1. So the slopes are not perpendicular. The slopes are different so they are not parallel. PTS: 1 IF: verage REF: 1a39e48a df-9c7d f0d2ea OJ: etermining Whether Lines are Parallel, Perpendicular, or Neither NT: NT.SS.MTH G.GPE.5 LO: MTH MTH TOP: 3-5 Slopes of Lines KEY: slope perpendicular parallel MS: OK 1 10

22 I: 18. NS: Solve the first equation for y to find the slope-intercept form. ompare the slopes and y-intercepts of both equations. 12x + 3y = 3 3y = 12x + 3 y = 4x +1 The slope of the first equation is 4 and the y-intercept is 1. y = 4x + 1 The slope of the second equation is 4 and the y-intercept is 1. The lines have different slopes, so they intersect. Write both lines in slope-intercept form and compare. Write both lines in slope-intercept form and compare. PTS: 1 IF: verage REF: 1a40e48e df-9c7d f0d2ea OJ: lassifying Pairs of Lines LO: MTH MTH TOP: 3-6 Lines in the oordinate Plane KEY: parallel perpendicular coordinate geometry MS: OK NS: E = + E Segment ddition Postulate 6x = ( 4x + 8) + 27 Substitute 6x for E and 4x + 8 for. 6x = 4x + 35 Simplify. 2x = 35 Subtract 4x from both sides. 2x 2 = 35 2 ivide both sides by 2. x = 35 or Simplify. E = 6x = 6( 17.5) = 105 You found the value of x. Find the length of the specified segment. You found the length of a different segment. heck your equation. Make sure you are not subtracting instead of adding. PTS: 1 IF: verage REF: 1935dc4e df-9c7d f0d2ea OJ: Using the Segment ddition Postulate LO: MTH TOP: 1-2 Measuring and onstructing Segments KEY: segment addition postulate MS: OK 2 11

23 I: 20. NS: y the Protractor Postulate, m O = m O m O. First, measure O and O. m O = m O m O = = 90 Thus, O is a right angle. To find the measure of angle O, subtract the measure of angle O from the measure of angle O. The sum of the measure of angle O and the measure of angle O is equal to the measure of angle O. Use the Protractor Postulate. PTS: 1 IF: verage REF: 193ac df-9c7d f0d2ea OJ: Measuring and lassifying ngles LO: MTH MTH TOP: 1-3 Measuring and onstructing ngles KEY: angle measure protractor degrees MS: OK NS: Step 1 Solve for x. m = m efinition of angle bisector. (7x 1) = (4x + 8) Substitute 7x 1 for and 4x + 8 for. 7x = 4x + 9 dd 1 to both sides. 3x = 9 Subtract 4x from both sides. x = 3 ivide both sides by 3. Step 2 Find m. m = 7x 1 = 7(3) 1 = 20 heck your simplification technique. Substitute this value of x into the expression for the angle. This answer is the entire angle. ivide by two. PTS: 1 IF: verage REF: 193f65be df-9c7d f0d2ea OJ: Finding the Measure of an ngle LO: MTH MTH TOP: 1-3 Measuring and onstructing ngles KEY: angle bisector angle addition postulate MS: OK 2 12

24 I: 22. NS: Subtract from 90 and simplify = 58.9 Find the measure of a complementary angle, not a supplementary angle. omplementary angles are angles whose measures have a sum of 90 degrees. The measures of complementary angles add to 90 degrees. PTS: 1 IF: asic REF: df-9c7d f0d2ea OJ: Finding the Measures of omplements and Supplements LO: MTH TOP: 1-4 Pairs of ngles KEY: complement complementary angles MS: OK NS: Subtract from 180 and simplify. 180 (8z + 10) = 180 8z 10 = (170 8z) The measures of supplementary angles add to 180 degrees. Supplementary angles are angles whose measures have a sum of 180 degrees. Find the measure of a supplementary angle, not a complementary angle. PTS: 1 IF: verage REF: 19468cd df-9c7d f0d2ea OJ: Finding the Measures of omplements and Supplements LO: MTH TOP: 1-4 Pairs of ngles KEY: supplement supplementary angles MS: OK 2 13

25 I: 24. NS: Step 1 Let the coordinates of N equal (x, y). Step 2 Use the Midpoint Formula. Ê Ê Ë Á 1, 2 ˆ = x + x 1 2 y 1 + y ˆ 2 Ê, 2 2 Ë Á = 6 + x 6 + yˆ, 2 2 Ë Á Step 3 Find the x- and y-coordinates. 1 = 6 + x 2 = 6 + y 2 2 Set the coordinates equal. Ê 2( 1) = x ˆ Ê 2( 2) = y ˆ Ë Á 2 2 Ë Á Multiply both sides by 2. 2 = 6 + x 4 = 6 + y Simplify. x = 8 y = 10 Solve for x or y, as appropriate. The coordinates of N are (8, 10). Let the coordinates of N be (x, y). Substitute known values into the Midpoint Formula to solve for x and y. This is the midpoint of line segment M. If M is the midpoint of line segment N, what are the coordinates of N? Let the coordinates of N be (x, y). Substitute known values into the Midpoint Formula to solve for x and y. PTS: 1 IF: verage REF: 1954dafa df-9c7d f0d2ea OJ: Finding the oordinates of an Endpoint LO: MTH TOP: 1-6 Midpoint and istance in the oordinate Plane KEY: coordinate geometry midpoint MS: OK 2 14

26 I: 25. NS: onditional: p q If an animal is a bird, then it has two eyes. onverse: q p If an animal has two eyes, then it is a bird. Inverse: p q If an animal is not a bird, then it does not have two eyes. ontrapositive: q p If an animal does not have two eyes, then it is not a bird. Given that the conditional statement is true, the contrapositive will also be true because the two are logically equivalent. It is easy to find a counterexample for the converse, and since the converse and inverse are logically equivalent, they will both be false. To find the contrapositive, exchange and negate the hypothesis and the conclusion. To find the inverse, negate the hypothesis and the conclusion. To find the converse, exchange the hypothesis and the conclusion. PTS: 1 IF: verage REF: 19cc10f df-9c7d f0d2ea OJ: pplication LO: MTH.P MTH.P MTH.P TOP: 2-2 onditional Statements KEY: inverse converse contrapositive conditional MS: OK NS: Let p and q represent the following. p: It is a right triangle. q: a 2 + b 2 = c 2. The given conditional is p q. The converse is q p. If a 2 + b 2 = c 2, then the figure is a right triangle with sides a, b, and c. The biconditional is p q. figure is a right triangle with sides a, b, and c if and only if a 2 + b 2 = c 2. Find the converse, not inverse. Find the converse, not the contrapositive. Find the converse, not the contrapositive. PTS: 1 IF: verage REF: 19d7fcbe df-9c7d f0d2ea OJ: Writing a iconditional Statement LO: MTH.P MTH.P TOP: 2-4 iconditional Statements and efinitions KEY: biconditional MS: OK 2 15

27 I: 27. NS: P = 2l + 2w Given equation 26 = 2(8) + 2w [1] Substitution Property of Equality 26 = w Simplify. 16 = = 2w Subtraction Property of Equality Simplify = 2w 2 [2] ivision Property of Equality 5 = w Simplify. w = 5 Symmetric Property of Equality The variables P and l are substituted, not simplified. Use the Substitution Property. heck the properties. heck the justifications. PTS: 1 IF: verage REF: 19df23d df-9c7d f0d2ea OJ: Problem-Solving pplication NT: NT.SS.MTH REI.1 LO: MTH.P MTH TOP: 2-5 lgebraic Proof KEY: algebraic proof proof MS: OK NS: orresponding angles lie on the same side of a transversal, on the same sides of the two lines the transversal crosses. So, 8 and 4 are corresponding angles. ngle 4 and angle 1 are supplementary angles, not corresponding angles. orresponding angles lie on the same side of a transversal, on the same sides of two lines. ngle 5 and angle 7 are vertical angles, not corresponding angles. PTS: 1 IF: asic REF: 1a1f df-9c7d f0d2ea OJ: lassifying Pairs of ngles LO: MTH TOP: 3-1 Lines and ngles KEY: corresponding angles transversal MS: OK 1 16

28 I: 29. NS: m 1 = 3(20) + 30 = 90 ; m 2 = 5(20) 10 = 90 m 1 = m 2 = 90 l Ä m Substitute 20 for x. Substitution Property of Equality onverse of the lternate Interior ngles Theorem ngles 1 and 2 are alternate interior angles and are congruent. ngles 1 and 2 are alternate interior angles and are congruent. ngles 1 and 2 are alternate interior angles and are congruent. PTS: 1 IF: verage REF: 1a2b df-9c7d f0d2ea OJ: etermining Whether Lines are Parallel LO: MTH MTH TOP: 3-3 Proving Lines Parallel KEY: parallel alternate interior angles MS: OK NS: 3 ( 5) slope of = 2 4 = 2 6 = 1 3 slope of = y ( 2) = y x 6 x, x 6 y x = 1 Parallel lines have the same slope. Write an equation comparing the 3 slopes of and. 3(y + 2) = 1(6 x) ross multiply. 3y 6 = 6 x istribute. 3y = 12 x Simplify. y = 1 x 4 3 Ï The set of possible values for x and y is Ê Ë Áx, yˆ y = 1 x 4 Ô Ì, x 6 Ô 3 ÓÔ Ô. heck the constraints of x. In the slope formula, the difference of the y-values is in the numerator and the difference of the x-values in the denominator. First, find the slopes of the segments. Then, set the slopes equal to each other and cross multiply and simplify. PTS: 1 IF: dvanced REF: 1a3c1fd df-9c7d f0d2ea NT: NT.SS.MTH G.GPE.5 ST: OH.OHS.MTH LO: MTH MTH TOP: 3-5 Slopes of Lines KEY: slope parallel MS: OK 3 17