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Ch 5 Study Guide Geometry



Multiple Choice
Identify the choice that best completes the statement or answers the question.
 

 1. 

Lines s, t, and u are perpendicular bisectors of the sides of mc001-1.jpg and meet at J. If mc001-2.jpg, mc001-3.jpg, mc001-4.jpg and mc001-5.jpg, find x, y, and z.
mc001-6.jpg
a.
x = 1, y = 5, z = 5
c.
x = 5, y = 1, z = 5
b.
x = 2.5, y = 2, z = 2.3
d.
x = 0, y = 6, z = 2.3
 

 2. 

mc002-1.jpg is an altitude, mc002-2.jpg, and mc002-3.jpg. Find mc002-4.jpg.
mc002-5.jpg
a.
34
c.
18
b.
32
d.
31
 

 3. 

mc003-1.jpg is an angle bisector, mc003-2.jpg, mc003-3.jpg, and mc003-4.jpg. Find mc003-5.jpg. Is mc003-6.jpg an altitude?
mc003-7.jpg
a.
50; no
c.
47; yes
b.
32; no
d.
17.3; no
 
 
Determine the relationship between the measures of the given angles.
 

 4. 

mc004-1.jpg
mc004-2.jpg
a.
mc004-3.jpg
c.
mc004-5.jpg
b.
mc004-4.jpg
 

 5. 

mc005-1.jpg
mc005-2.jpg
a.
mc005-3.jpg
c.
mc005-5.jpg
b.
mc005-4.jpg
 
 
Determine the relationship between the lengths of the given sides.
 

 6. 

mc006-1.jpg
mc006-2.jpg
a.
mc006-3.jpg
c.
cannot be determined
b.
mc006-4.jpg
d.
mc006-5.jpg
 

 7. 

mc007-1.jpg
mc007-2.jpg
a.
mc007-3.jpg
c.
mc007-5.jpg
b.
mc007-4.jpg
 
 
Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain.
 

 8. 

3, 9, 10
a.
Yes; the third side is the longest.
b.
No; the sum of the lengths of two sides is not greater than the third.
c.
No; the first side is not long enough.
d.
Yes; the sum of the lengths of any two sides is greater than the third.
 

 9. 

9.2, 14.5, 17.1
a.
Yes; the third side is the longest.
b.
No; the first side is not long enough.
c.
Yes; the sum of the lengths of any two sides is greater than the third.
d.
No; the sum of the lengths of two sides is not greater than the third.
 

 10. 

An isosceles triangle has a base 9.6 units long. If the congruent side lengths have measures to the first decimal place, what is the shortest possible length of the sides?
a.
4.9
c.
4.7
b.
19.3
d.
9.7
 

 11. 

Which segment is the shortest possible distance from point D to plane P?
mc011-1.jpg
a.
mc011-2.jpg
c.
mc011-4.jpg
b.
mc011-3.jpg
d.
mc011-5.jpg
 

Short Answer
 
 
Write a two-column proof.
 

 12. 

If sa012-1.jpg and sa012-2.jpg, then sa012-3.jpg.
sa012-4.jpg
 

 13. 

If sa013-1.jpg is a median of isosceles sa013-2.jpg, then sa013-3.jpg.
sa013-4.jpg
 

 14. 

If sa014-1.jpg is a median and a perpendicular bisector of sa014-2.jpg, then sa014-3.jpg.
sa014-4.jpg
 
 
Write an indirect proof.
 

 15. 

Given: sa015-1.jpg
Prove: sa015-2.jpg
 

 16. 

Given: sa016-1.jpg
Prove: sa016-2.jpg is positive.
 

 17. 

If sa017-1.jpg and sa017-2.jpg, then sa017-3.jpg.
 

 18. 

If n is a multiple of 9, then it is a multiple of 3.
 

 19. 

If sa019-1.jpg, then sa019-2.jpg.
 

 20. 

If sa020-1.jpg is an integer, then n is even.
 

 21. 

Given: sa021-1.jpg is an even number.
Prove: n is an even number.
 

 22. 

Given: sa022-1.jpg
Prove: sa022-2.jpg
 

 23. 

Given: sa023-1.jpg
Prove: sa023-2.jpg
 

 24. 

Given: sa024-1.jpg
Prove: sa024-2.jpg
 

 25. 

Given: sa025-1.jpg; sa025-2.jpg
Prove: sa025-3.jpg
sa025-4.jpg
 

 26. 

Given: sa026-1.jpg; sa026-2.jpg
Prove: sa026-3.jpg
sa026-4.jpg
 

 27. 

Given: sa027-1.jpg
Prove: sa027-2.jpg
sa027-3.jpgsa027-4.jpg
 

 28. 

Given: sa028-1.jpg; A is the midpoint of BF and CD.
Prove: sa028-2.jpg
sa028-3.jpg
 

 29. 

Given: sa029-1.jpg is a perpendicular bisector of equilateral triangle sa029-2.jpg.
Prove: sa029-3.jpg
sa029-4.jpg
 

 30. 

Given: sa030-1.jpg is a perpendicular bisector of equilateral triangle sa030-2.jpg. sa030-3.jpg is a median of sa030-4.jpg.
Prove: sa030-5.jpg
sa030-6.jpg
 

 31. 

Given: Square GHJK
Prove: sa031-1.jpg
sa031-2.jpg
 

 32. 

Given: sa032-1.jpg with angle measures as shown
Prove: sa032-2.jpg
sa032-3.jpg
 

 33. 

Given: sa033-1.jpg is a perpendicular bisector of sa033-2.jpg.
Prove: sa033-3.jpg is isosceles
sa033-4.jpg
 

 34. 

Given: sa034-1.jpg
Prove: sa034-2.jpg
sa034-3.jpgsa034-4.jpg
 
 
Write a two-column proof.
 

 35. 

Given: sa035-1.jpg
Prove: sa035-2.jpg
sa035-3.jpg
 

 36. 

Given: sa036-1.jpg; sa036-2.jpg bisects sa036-3.jpg.
Prove: sa036-4.jpg
sa036-5.jpg
 

 37. 

Given: sa037-1.jpg
Prove: sa037-2.jpg
sa037-3.jpg
 

 38. 

Given: sa038-1.jpg, sa038-2.jpg
Prove: sa038-3.jpg
sa038-4.jpg
 

 39. 

A circular swimming pool needs to be designed for a triangular lawn surrounded by apartment buildings at the three corners, as shown in the figure below. The swimming pool should be located such that the center of the pool is equidistant from the buildings at the corners. Describe the position of the swimming pool.
sa039-1.jpg
 

 40. 

Three children are playing a game on the playground. They stand on the playground facing each other. They want to place a ball on the playground so that it is equidistant from each of the three children. Describe the position of the ball.
 

 41. 

In the figure below, sa041-1.jpg is an angle bisector of sa041-2.jpg and sa041-3.jpg is a perpendicular bisector of sa041-4.jpg R is the point of intersection of sa041-5.jpg and sa041-6.jpg Show that sa041-7.jpg
sa041-8.jpg
 

 42. 

In the figure below, rectangle ABCD represents a wall. The following clues are given about the position of a painting on the wall.

(1)  The painting is as far from point A as it is from point B.
(2)  If you measure the distance from point D to the painting or from point C to the painting, you would measure the same distance.

Describe the position of the painting.
sa042-1.jpg
 

 43. 

Samantha and her friends are playing carrom, a board game in which players slide pieces to hit other pieces. Samantha hits a piece called the queen. The new position of the queen is given by the following clues

(1)  The queen is as far from point P as it is from point Q.
(2)  If you moved straight along SP to the queen or along SR to the queen, you would move the same distance.

Describe the position of the queen.

sa043-1.jpg
 

 44. 

A salesperson travels from city A to city B and then to city C. From city C, the salesperson travels directly back to city A as shown in the diagram below. Write the lengths of the legs of the trip in order from least to greatest.
sa044-1.jpg
 

 45. 

Triangle ABC has vertices sa045-1.jpgand sa045-2.jpg List the angles in order from the greatest to the least measure.
 

 46. 

A triangle is formed by three kitchen appliances as shown in the figure. The distances shown are measured in feet. What is wrong with the labels on the triangle?

sa046-1.jpg
 

 47. 

A tree 44 meters high cast a shadow 60 meters long, as shown below. Write an inequality
relating x and y. Then solve the inequality for x in terms of y.
sa047-1.jpg
 

 48. 

In the figure below, the length of BF is given by sa048-1.jpg and the length of AF is given by sa048-2.jpg Write an inequality relating x and y. Then solve the inequality for y in terms of x.
sa048-3.jpg
 

 49. 

Prove that sa049-1.jpg has no more than one obtuse angle.
 

 50. 

Samantha and Nick have rectangular rooms with the same perimeter. Samantha’s room is
9 feet by 12 feet. The length of Nick’s room is 8 feet. Prove that the width of Nick’s room
is less than 14 feet.
 

 51. 

On a field trip, sa051-1.jpg of the students have jackets. If there are 20 students on a field trip, verify
that 15 students have jackets.
 

 52. 

Lauren, Rose, and Keith worked at the face-painting booth at the Fall Festival. Lauren painted
13 faces, Rose painted 29 faces, and Keith painted 15 faces. Has Rose painted more faces than the total number of faces painted by Lauren and Keith?
 

 53. 

Megan and Sara took part in a 500-meter race on sports day at the school. As the race finished, Megan claimed that she was the winner of the race. The teacher said that according to the stopwatch, Megan took 10 minutes to complete the race, whereas Sara completed the race in 9 minutes. So, Megan was not the winner of the race. Explain whether this is an example of indirect reasoning.
 

 54. 

Two sides of a triangle are 3 feet and 4 feet long. Let x represent the measure of the third side of the triangle. Suppose x is whole number such that sa054-1.jpg List the measures of the sides of the triangles that are possible.
 

 55. 

One side of a triangle is 5 feet long. Let x represent the measure of the second side and let y represent the measure of the third side. Suppose x and y are whole numbers and that sa055-1.jpg and sa055-2.jpg List the measures of the sides of the triangles that are possible.
 
 
John has five straws. He wishes to use the straws to make a triangular design. The straws measure
6 centimeters, 2 centimeters, 7 centimeters, 8 centimeters, and 13 centimeters.
 

 56. 

How many different triangles could John make with the straws?
 

 57. 

How many different triangles with even perimeters could John make?
 

 58. 

The two sides of a triangle are 3 feet and 6 feet long. Let b represents the measure of the third side. List a possible range for b.
 

 59. 

Use an indirect proof to prove the SSS Inequality Theorem.
Given: sa059-1.jpg
Prove: sa059-2.jpg
sa059-3.jpg
 

 60. 

Measure the distance between the handles of a pair of scissors as shown in the picture below. Now, apply force to the handles of the scissors to cut a piece of paper and measure again. How do the measures compare?

sa060-1.jpg
 

 61. 

The figure below shows the position of two planes, plane 1 and plane 2, after traveling 170 miles in different directions from the airport marked by P. Which plane is farther from the airport?

sa061-1.jpg
 

 62. 

Use an indirect proof to prove the SAS Inequality Theorem.
Given: sa062-1.jpg
Prove: sa062-2.jpg
sa062-3.jpg
 

 63. 

Let sa063-1.jpg represent the legs of a tripod. Assume the legs are congruent and sa063-2.jpg Compare the lengths of sa063-3.jpg

sa063-4.jpg
 



 
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