Ch 5 Study Guide Geometry

Name:

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

Lines s, t, and u are perpendicular bisectors of the sides of

and meet at J. If ,

and

, find x, y, and z.

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

is an altitude, , and

. Find .

a.	34	c.	18
b.	32	d.	31

is an angle bisector, ,

, and

. Find . Is

an altitude?

a.	50; no	c.	47; yes
b.	32; no	d.	17.3; no

Determine the relationship between the measures of the given angles.

a.		c.
b.

a.		c.
b.

Determine the relationship between the lengths of the given sides.

a.		c.	cannot be determined
b.		d.

a.		c.
b.

Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. Explain.

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.

9.2, 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?

a.		c.
b.		d.

Short Answer

Write a two-column proof.

12.

and

, then

13.

is a median of isosceles

, then

14.

is a median and a perpendicular bisector of

, then

Write an indirect proof.

15.

Given:

Prove:

16.

Given:

Prove:

is positive.

17.

and

, then

18.

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

19.

If , then

20.

If is an integer, then n is even.

21.

Given:

is an even number.
Prove: n is an even number.

22.

Given:

Prove:

23.

Given:

Prove:

24.

Given:

Prove:

25.

Given:

;

Prove:

26.

Given:

;

Prove:

27.

Given:

Prove:

28.

Given: ; A is the midpoint of BF and CD.
Prove:

29.

Given:

is a perpendicular bisector of equilateral triangle

.
Prove:

30.

Given:

is a perpendicular bisector of equilateral triangle

is a median of

.
Prove:

31.

Given: Square GHJK
Prove:

32.

Given:

with angle measures as shown
Prove:

33.

Given:

is a perpendicular bisector of

.
Prove:

is isosceles

34.

Given:

Prove:

Write a two-column proof.

35.

Given:

Prove:

36.

Given:

;

bisects

.
Prove:

37.

Given:

Prove:

38.

Given:

Prove:

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.

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,

is an angle bisector of

and

is a perpendicular bisector of

R is the point of intersection of

and

Show that

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.

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.

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.

45.

Triangle ABC has vertices

and

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?

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.

48.

In the figure below, the length of BF is given by

and the length of AF is given by

Write an inequality relating x and y. Then solve the inequality for y in terms of x.

49.

Prove that

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,

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

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

and

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:

Prove:

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?

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?

62.

Use an indirect proof to prove the SAS Inequality Theorem.
Given:

Prove:

63.

Let

represent the legs of a tripod. Assume the legs are congruent and

Compare the lengths of