64 Chapter 1 Foundations for Geometry

1. Draw and label plane N containing two lines that intersect at B.

Use the figure to name each of the following.

2. four noncoplanar points 3. line containing B and E

4. The coordinate of A is -3, and the coordinate of B is 0.5. Find AB.

5. E, F, and G represent mile markers along a straight highway. Find EF.

6. J is the midpoint of −

HK . Find HJ, JK, and HK.

Classify each angle by its measure.

7. m∠LMP = 70° 8. m∠QMN = 90° 9. m∠PMN = 125°

10. ��� TV bisects ∠RTS. If the m∠RTV = (16x - 6) ° and m∠VTS = (13x + 9) °, what is the m∠RTV?

11. An angle’s measure is 5 degrees less than 3 times the measure of its supplement. Find the measure of the angle and its supplement.

Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.

12. ∠2 and ∠3 13. ∠4 and ∠5 14. ∠1 and ∠4

15. Find the perimeter and area of a rectangle with b = 8 ft and h = 4 ft.

Find the circumference and area of each circle to the nearest tenth.

16. r = 15 m 17. d = 25 ft 18. d = 2.8 cm

19. Find the midpoint of the segment with endpoints (-4, 6) and (3, 2) .

20. M is the midpoint of −

LN . M has coordinates (-5, 1) , and L has coordinates (2, 4) . Find the coordinates of N.

21. Given A (-5, 1) , B (-1, 3) , C (1, 4) , and D (4, 1) , is −

AB � −

CD ? Explain.

Identify each transformation. Then use arrow notation to describe the transformation.

22. 23.

24. A designer used the translation (x, y) → (x + 3, y - 3) to transform atriangular-shaped pin ABC. Find the coordinates and draw the image of �ABC.

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134 Chapter 2 Geometric Reasoning

Find the next item in each pattern.

1. 2. 405, 135, 45, 15, …

3. Complete the conjecture “The sum of two even numbers is −−− ? . ”

4. Show that the conjecture “All complementary angles are adjacent” is false by finding a counterexample.

5. Identify the hypothesis and conclusion of the conditional statement “The show is cancelled if it rains.”

6. Write a conditional statement from the sentence “Parallel lines do not intersect.”

Determine if each conditional is true. If false, give a counterexample.

7. If two lines intersect, then they form four right angles.

8. If a number is divisible by 10, then it is divisible by 5.

Use the conditional “If you live in the United States, then you live in Kentucky” for Items 9–11. Write the indicated type of statement and determine its truth value.

9. converse 10. inverse 11. contrapositive

12. Determine if the following conjecture is valid by the Law of Detachment.Given: If it is colder than 50°F, Tom wears a sweater. It is 46°F today.Conjecture: Tom is wearing a sweater.

13. Use the Law of Syllogism to draw a conclusion from the given information.Given: If a figure is a square, then it is a quadrilateral. If a figure is a

quadrilateral, then it is a polygon. Figure ABCD is a square.

14. Write the conditional statement and converse within the biconditional “Chad will work on Saturday if and only if he gets paid overtime.”

15. Determine if the biconditional “B is the midpoint of −−

AC iff AB = BC” is true. If false, give a counterexample.

Solve each equation. Write a justification for each step.

16. 8 - 5s = 1 17. 0.4t + 3 = 1.6 18. 38 = -3w + 2

Identify the property that justifies each statement.

19. If 2x = y and y = 7, then 2x = 7. 20. m∠DEF = m∠DEF

21. ∠X � ∠P, and ∠P � ∠D. So ∠X � ∠D. 22. If −−

ST � −−

XY , then −−

XY � −−

ST .

Use the given plan to write a proof in each format.

Given: ∠AFB � ∠EFD Prove:

�� FB bisects ∠AFC.

Plan: Since vertical angles are congruent, ∠EFD � ∠BFC. Use the Transitive Property to conclude that ∠AFB � ∠BFC. Thus

�� FB bisects ∠AFC by the definition of angle bisector.

23. two-column proof 24. paragraph proof 25. flowchart proof

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206 Chapter 3 Parallel and Perpendicular Lines

Identify each of the following.

1. a pair of parallel planes

2. a pair of parallel segments

3. a pair of skew segments

Find each angle measure.

4. 5. 6.

Use the given information and the theorems and postulates you have learned to show f ‖ g.

7. m∠4 = (16x + 20) °, m∠5 = (12x + 32) °, x = 3

8. m∠3 = (18x + 6) °, m∠5 = (21x + 18) °, x = 4

Write a two-column proof.

9. Given: ∠1 � ∠2, n ⊥ �

Prove: n ⊥ m

Use the slope formula to determine the slope of each line.

10. 11. 12.

13. Greg is on a 32-mile bicycle trail from Elroy, Wisconsin, to Sparta, Wisconsin. He leaves Elroy at 9:30 A.M. and arrives in Sparta at 2:00 P.M. Graph the line that represents Greg’s distance from Elroy at a given time. Find and interpret the slope of the line.

14. Graph � �� QR and � �� ST for Q (3, 3) , R (6, -5) , S (-4, 6) , and T (-1, -2) . Use slopes to determine whether the lines are parallel, perpendicular, or neither.

15. Write the equation of the line through (-2, -5) with slope - 3 _ 4

in point-slope form.

16. Determine whether the lines 6x + y = 3 and 2x + 3y = 1 are parallel, intersect, or coincide.

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288 Chapter 4 Triangle Congruence

1. Classify �ACD by its angle measures.

Classify each triangle by its side lengths.

2. �ACD 3. �ABC 4. �ABD

5. While surveying the triangular plot of land shown, a surveyor finds that m∠S = 43°. The measure of ∠RTP is twice that of ∠RTS. What is m∠R?

Given: �XYZ � �JKL Identify the congruent corresponding parts.

6. −−

JL � −−−− ? 7. ∠Y � −−−− ? 8. ∠L � −−−− ? 9. −−

YZ � −−−− ?

10. Given: T is the midpoint of −−

PR and −−

SQ . Prove: �PTS � �RTQ

11. The figure represents a walkway with triangular supports. Given that

−− GJ bisects

∠HGK and ∠H � ∠K, use AAS to prove �HGJ � �KGJ

12. Given: −−

AB � −−

DC , 13. Given: −−

PQ ‖ −−

SR ,

−− AB ⊥

−− AC , ∠S � ∠Q

−−

DC ⊥ −−

DB Prove: −−

PS ‖ −−

QR Prove: �ABC � �DCB

14. Position a right triangle with legs 3 m and 4 m long in the coordinate plane. Give the coordinates of each vertex.

15. Assign coordinates to each vertex and write a coordinate proof.

Given: Square ABCDProve:

−− AC �

−− BD

Find each value.

16. y 17. m∠S

18. Given: Isosceles �ABC has coordinates A (2a, 0) , B (0, 2b) , and C (-2a, 0) . D is the midpoint of

−− AC , and E is the midpoint of

−− AB .

Prove: �AED is isosceles.

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370 Chapter 5 Properties and Attributes of Triangles

Find each measure.

1. KL 2. m∠WXY 3. BC

4. −−−

MQ , −−−

NQ , and −−

PQ are the 5. −−

EG and −−

FG are angle 6. In �XYZ, XC = 261, perpendicular bisectors of bisectors of �DEF. and ZW = 118. �RST. Find RS and RQ. Find m∠GEF and the Find XW, BW, and BZ. distance from G to

−− DF .

7. Find the orthocenter of �JKL with vertices J (-5, 2) , K (-5, 10) , and L (1, 4) .

8. In �GHJ at right, find PR, GJ, and m∠GRP.

9. Write an indirect proof that two obtuse angles cannot form a linear pair.

10. Write the angles of 11. Write the sides of �BEH in order from �RTY in order from smallest to largest. shortest to longest.

12. The distance from Arville to Branton is 114 miles. The distance from Branton to Camford is 247 miles. If the three towns form a triangle, what is the range of distances from Arville to Camford?

13. Compare m∠SPV 14. Find the range of and m∠ZPV. values for x.

15. Find the missing side length in the triangle. Tell if the side lengths form a Pythagorean triple. Explain.

16. Tell if the measures 18, 20, and 27 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

17. An IMAX screen is 62 feet tall and 82 feet wide. What is the length of the screen’s diagonal? Round to the nearest inch.

Find the values of the variables. Give your answers in simplest radical form.

18. 19. 20.

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442 Chapter 6 Polygons and Quadrilaterals

Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

1. 2.

3. The base of a fountain is in the shape of a quadrilateral, as shown.Find the measure of each interior angle of the fountain.

4. Find the sum of the interior angle measures of a convex nonagon.

5. Find the measure of each exterior angle of a regular 15-gon.

6. In �EFGH, EH = 28, HZ = 9, and 7. JKLM is a parallelogram.m∠EHG = 145°. Find FH and m∠FEH. Find KL and m∠L.

8. Three vertices of �PQRS are P (-2, -3) , R (7, 5) , and S (6, 1) . Find the coordinates of Q.

9. Show that WXYZ is 10. Determine if CDGH a parallelogram for must be a parallelogram. a = 4 and b = 3. Justify your answer.

11. Show that a quadrilateral with vertices K (-7, -3) , L (2, 0) , S (5, -4) , and T (-4, -7) is a parallelogram.

12. In rectangle PLCM, 13. In rhombus EHKN, LC = 19, and LM = 23. m∠NQK = (7z + 6) °, andFind PT and PM. m∠ENQ = (5z + 1) °. Find m∠HEQ and m∠EHK.

Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

14. Given: −−

NP � −−

PQ � −−−

QM � −−−

MN 15. Given: −−

NP � −−−

MQ , −−−

NM � −−

PQ , −−−

NQ � −−−

MP Conclusion: MNPQ is a square. Conclusion: MNPQ is a rectangle.

Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.

16. A (-5, 7) , C (3, 6) , E (7, -1) , G (-1, 0) 17. P (4, 1) , Q (3, 4) , R (-3, 2) , S (-2, -1)

18. m∠JFR = 43°, and 19. PV = 61.1, and m∠JNB = 68°. YS = 24.7. Find m∠FBN. Find MY.

20. Find HR.

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508 Chapter 7 Similarity

1. Two points on � are A (-6, 4) and B (10, -6) . Write a ratio expressing the slope of �.

2. Alana has a photograph that is 5 in. long and 3.5 in. wide. She enlarges it so that its length is 8 in. What is the width of the enlarged photograph?

Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement.

3. �ABC and �MNP 4. rectangle DEFG and rectangle HJKL

5. Given: RSTU 6. Derrick is building a skateboard ramp Prove: �RWV ∼ �SWT as shown. Given that BD = DF = FG = 3 ft, find CD and EF to the nearest tenth.

Find the length of each segment.

7. −−

PR 8. −−−

YW and −−−

WZ

9. To find the height of a tree, a student 10. The plan for a living room uses the scale of measured the tree’s shadow and her 1.5 in. : 30 ft. Use a ruler and find the length of own shadow. If the student’s height the actual room’s diagonal

−− AB .

is 5 ft 8 in., what is the height of the tree?

11. Given: A (6, 5) , B (3, 4) , C (6, 3) , D (-3, 2) , and E (6, -1) Prove: �ABC ∼ �ADE

12. A quilter designed this patch for a quilt but needs a larger version for a different project. Draw the quilt patch after a dilation with scale factor 3 __

2 .

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576 Chapter 8 Right Triangles and Trigonometry

Find x, y, and z.

1. 2. 3.

Use a special right triangle to write each trigonometric ratio as a fraction.

4. cos 60° 5. sin 45° 6. tan 60°

Find each length. Round to the nearest hundredth.

7. PR 8. AB 9. FG

10. Nate built a skateboard ramp that covers a horizontal distance of 10 ft. The ramp rises a total of 3.5 ft. What angle does the ramp make with the ground? Round to the nearest degree.

11. An observer at the top of a skyscraper sights a tour bus at an angle of depression of 61°. The skyscraper is 910 ft tall. What is the horizontal distance from the base of the skyscraper to the tour bus? Round to the nearest foot.

Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree.

12. m∠B 13. RS 14. m∠M

Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth.

15. ⟨1, 3⟩ 16. ⟨-4, 1⟩ 17. ⟨2, -3⟩

Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree.

18. The velocity of a plane is given by the vector ⟨3, 5⟩.

19. A wind velocity is given by the vector ⟨4, 1⟩.

20. Kate is rowing across a river. She sets out at a bearing of N 40° E and paddles at a constant rate of 3.5 mi/h. There is a 2 mi/h current moving due east. What are Kate’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree.

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644 Chapter 9 Extending Perimeter, Circumference, and Area

Find each measurement.

1. the height h of a triangle in which A = 12 x 2 y ft 2 and b = 3x ft

2. the base b 1 of a trapezoid in which A = 161.5 cm 2 , h = 17 cm, and b 2 = 13 cm

3. the area A of a kite in which d 1 = 25 in. and d 2 = 12 in.

4. Find the circumference and area of �A with diameter 12 in. Give your answers in terms of π.

5. Find the area of a regular hexagon with a side length of 14 m. Round to the nearest tenth.

Find the shaded area. Round to the nearest tenth, if necessary.

6. 7.

8. The diagram shows a plan for a pond. Use a composite figure to estimate the pond’s area. The grid has squares with side lengths of 1 yd.

9. Draw and classify the polygon with vertices A (1, 5) , B (2, 3) , C (-2, 1) , and D (-3, 3) . Find the perimeter and area of the polygon.

Find the area of each polygon with the given vertices.

10. E (-3, 4) , F (1, 1) , G (0, -4) , H (-4, 1) 11. J (3, 4) , K (4, -1) , L (-2, -4) , M (-3, 3)

Describe the effect of each change on the perimeter or circumference and area of the given figure.

12. The base and height of a triangle with base 10 cm and height 12 cm are multiplied by 3.

13. The radius of a circle with radius 12 m is multiplied by 1 _ 2

.

14. A circular garden plot has a diameter of 21 ft. Janelle is planning a new circular plot with an area 1 __

9 as large. How will the circumference of the new plot compare to the

circumference of the old plot?

A point is chosen randomly on −−

NS . Find the probability of each event.

15. The point is on −−−

NQ .

16. The point is not on −−

QR .

17. The point is on −−−

NQ or −−

RS .

18. A shuttle bus for a festival stops at the parking lot every 18 minutes and stays at the lot for 2 minutes. If you go to the festival at a random time, what is the probability that the shuttle bus will be at the parking lot when you arrive?

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