Unit 1 and 2
Practice Questions
1. What in the inverse of the statement below?
If today is Wednesday, then tomorrow is Thursday.
If today is not Wednesday, then tomorrow is not Thursday. The inverse is formed by negating both
pieces.
2. What is the ”if” portion of a conditional statement called?
The if portion is called the hypothesis and the then portion is called the conclusion
3. Write a statement is logically equivalent to the statement below?
If A, then B.
If not A, then not B. This is logically equivalent because the original and the converse are logically
equivalent.
4. If you write the converse of an inverse, what will be the result?
The result will be the contrapositive of the original statement because the contrapositive has not only the
reversal from the converse, but also the negation from the inverse.
5. What is the converse of the statement below?
If you study, then you will do well on the exam.
If you do well on the exam, then you studied. The converse is formed by reversing the two pieces.
6. If you sleep in, then you will miss school is the inverse of what statement?
If you do not sleep in, then you will not miss school. Writing the inverse of an inverse creates the
original statement again.
7. What must be true in order to write a bi-conditional statement?
All versions of the statement must be true.
8. Change the definition below into a biconditional statement.
Triangle-a shape with 3 sides.
It is a triangle if and only if it is a shape with 3 sides. A biconditional statement always included if and
only if in the center.
9. What is the contrapositive of the statement below?
If you are a student, then you are in school.
If you are not in school, then you are not a student. Writing the contrapositive of a statement involves
reversing the pieces and negating each of them.
10. Can the statement above be written as a biconditional statement?
No, because the converse is not true, teachers are in school and they are not students. If it were possible
the biconditional would be: You are a student if and only if you are in school.
11. A line segment AB, with a midpoint at (3, 5), is locates in the standard (x, y) coordinate plane. If point
A is (2, 7), what are the coordinates of point B?
(4, 3) This is because 3 is the average of 2 and 4 and 5 is the average of 7 and 3.
12. Point B lies between A and C on segment AC. If AC=12 cm and AB=5 cm, how long is BC?
7 cm By the segment addition property if the entire segment is 12 and a portion of it is 5, we can find
the remainder by subtracting 5 from 12.
1
13. Points B and C lie on AD as shown below. The length of AC is 16 units long; and BD is 20 units long.
How many units long, if it can be determined, is BC?
It cannot be determined because not enough information was provided. Had the length of AD been given
we would have been able to find the length.
14. If X is the midpoint of segment YZ, what is true about segments XY and XZ?
XY and XZ must be congruent, because by definition a midpoint divides a segment into to two pieces of
equal length.
15. What is the midpoint of the line segment whose endpoints are (5, -2) and (-3, 7)?
(1, 2.5) This can be found by averaging the two x coordinates and the two y coordinates.
16. If (2, b) is the midpoint of a segment with endpoints (1, 6) and (3, b+3), what is the value of b?
9. Since b+3 is three greater than b it follows that for b to be the midpoint it must be three more than
six and is therefore nine.
17. A line contains the points A, B, C, and D. Point B is between points A and C. Point D is between
points C and B. Write three inequalities that must be true about the lengths of the segments?
BD < AC, BD < AD, BC < AC, or any other statement where the smaller segment is contained inside
of the larger segment.
18. If Q is the midpoint of segment RS, what is true about segments QR and RS?
QR is half the length of RS. This is because a midpoint by definition divides a segment in half.
19. What is the midpoint of a line segment with endpoints (0, 0) and (a, b)?
( a2 , 2b ). This can be found by averaging the two x coordinates and the two y coordinates.
20. Point B lies between A and C on segment AC. If AB=6 cm and BC=5 cm, how long is AC?
11 cm. Based on the segment addition property if I want to find the entire segment length I need only
add up all of the pieces.
2
Unit 3
Practice Questions
1. What single transformation could be used to place line segment AB directly on top of line segment CD?
A translation down 8 or a reflection over the line y=-1.
2. ∆ABC is dilated about the origin by a scale factor of 3. If the original coordinates of A are (2, -3),
what are the coordinates of A?
(6, -9) When dilation about the origin we multiply the coordinates by the scale factor.
3. A reflection over the y-axis has what effect on a coordinate point?
It negates the x value and leaves the y value alone. For example (2, 3) would become (-2, 3).
4. Describes a translation left 2 and up 5 in coordinate notation.
(x-2, y+5) The x component controls right and left movement, while the y component controls up and
down movement.
5. What will the coordinates of the endpoints of P Q be if the segment is flipped across the line y = x?
P’ (1, 2) and Q’(-3, 4) Reflecting over the line y=x has the effect of switching the x and y coordinates.
6. Write a rule to describe a dilation by a scale factor of 2 about the origin.
(2x, 2y) Dilation involves multiplying the coordinates by the scale factor as is shown in this rule.
7. What transformation is represented by the rule (−x, y)?
A reflection over the y-axis, see question 3.
8. A triangle, ∆ABC is reflected across the x-axis to have the image ∆A0 B 0 C 0 in the standard coordinate
plane. The coordinates of point A are (c, d). What are the coordinates of A0 ?
(c, -d) Reflecting over the x axis has the effect of negating the y coordinate and leaving the x alone.
9. A compositions of reflections over two parallel lines is that same as a single what?
Translation. Flipping a shape along the same axis twice returns the shape to its original orientation and
shapes only maintain their orientation in a translation.
10. Describe the translation represented by (x − 1, y + 3).
Left 1 and Up 3. The x component controls right and left movement, while the y component controls up
and down movement.
3
11. Which transformation(s) could be used to place triangle ABC on top of triangle DEF ?
Rotating 180◦ about the origin or reflecting first over one axis and then over the other.
12. If a direction is not specified, in what direction do we generally rotate?
Counterclockwise, this is the way the quadrants of the coordinate plan are numbered.
13. A line segment in a standard coordinate plane with endpoints at (-3, 2) and (7, -12) is reflected across the
x-axis and then across the y-axis. What are the coordinates of the endpoints of the new line segment?
(3, -2) and (-7, 12) Reflecting across the x-axis negates the y coordinate and then reflecting across the
y-axis negates the x coordinate so after both reflections both coordinates have changed sign.
14. A 360 degree rotation has what effect on a figure?
None. This turns the shape in a complete circle and it ends up exactly where it started.
15. What will be the vertices of triangle ABC (below) if it is flipped across the y-axis and then the x-axis?
A’(-3, -6), B’(-3, -1), C’(-6, -1). See question 13 for further explanation.
16. What will be the vertices of triangle ABC (above) if it is translated down 3 and left 5?
A’(-2, 3), B’ (-2, -2), C’(1, -2) 5 was subtracted from each x and 3 was subtracted from each y.
17. A compositions of reflections over two intersecting lines is that same as a single what?
A rotation, see question 11 for an example where the two lines are the x and y axis.
18. Which composition of transformations could be used to place ∆ABC on top of ∆F ED?
Rotate 90◦ and then translate down 3. There are other possible correct solutions.
19. Which transformation can be described as a slide? A translation.
20. Which transformation can be described as a turn? A rotation.
4
Unit 4
Practice Questions
1. Suppose the kites F LY A and KIT E are similar. What is the length of LY ?
F
8
K
12
A
E
L
I
x
30
Y
T
20. There are several ways to solve this, one way is to consider that 12 × 2.5 = 30 and then do 8 × 2.5
to get the missing value.
2. What is true about the diagonals a kite?
They are perpendicular.
3. Evaluate the truth value of the following statement.
If it is a rhombus, then it is a kite
True. A rhombus is a special kite, but the fact that it has all sides congruent means that the adjacent
sides are congruent so it is a kite.
4. What is the area of the kite shown below?
39. This can be found by multiplying the diagonals (which measure 13 and 9) and then dividing by 2.
5. Name three things that are true about the diagonals of a square.
They are congruent and perpendicular and they bisect each other.
6. In the rectangle ACDF below, AC is three times as long as BC.
What is the area of ABEF ?
4 in2 . If AC is three times as long as BC then AC is 3 in so it follows that AB is 2 in. Then we can
find the area of ABEF by multiplying base and height of 2 × 2.
7. A rectangular lot that measures 150 ft by 200 ft is completely fenced. What is the approximate length,
in feet, of the fence?
700 ft. The length of the fence is the perimeter of the rectangle, which can be found by adding up all of
the sides (150+200+150+200).
5
8. In the figure shown below, each pair of intersecting line segments meets at a right angle and all the
lengths given are in inches. What is the perimeter, in inches of the figure?
80. The perimeter of this shape is the same as the perimeter of the rectangle with the same base and
height. I can see from the bottom that the base is 26 and adding up the left side gives me a height of 14
so I simply add 26+14+26+14.
9. List the three special types of parallelograms
A square, a rectangle, and a rhombus.
10. Which of the parallelograms are similar?
50
30
B
A
50
70
40
C
56
B and C. The ratio between the sides and the bases is the same and is 0.8 in each case. The is not true
of any other pair.
11. Parallelogram ABCD, with dimensions in inches, is shown in the diagram below. What is the area of
the parallelogram, in square inches?
36. The base of the shape is the 3 and the 6 together making for a base of 9 and the height is 4. Therefore
base × height gives 9×4 or 36.
12. Your friend shows you a scale drawing of her apartment. The drawing of the apartment is a rectangle
4 inches by 6 inches. Your friend wants to know the length of the shorter side of her apartment. If she
knows the length of the longer side is 30 feet, how many feet long is the shorter side?
20. As illustrated in question 10 similar figures have the same ratio for each pair of sides. The ratio
from 6 to 30 is to multiply by 5 so 4 × 5 = 20.
13. A square is circumscribed about a circle of 7-foot radius. What is the area of the square in square feet?
196 feet2 . If the circle has a radius of 7 then it had a diameter of 14. This makes the base and the height
of the square both 14 and that makes an area of 196.
14. In the standard coordinate plane, the points (0, 0), (10, 0), (13, 6) and (3, 6) are the vertices of a
parallelogram. What is the area, in square coordinate units, of the parallelogram? 60. A quick sketch of
this show that it has a base of 10 and a height of 6. From there it is simple to calculate the area of 60.
6
−−→
15. In the figure below, ABCD is a trapezoid, E lies on AD, and angles measures are as marked. What is
the measure of 6 BDC?
45◦ . 6 CBD ∼
= 6 ABD because they are alternate interior angles and 6 ADB + 6 BDC + 6 CDE = 180◦
so we can substitute (30+x+105=180) and solve to find 6 BDC.
16. Given quadrilateral ABCD where 6 A = 100◦ , 6 B = 75◦ , and 6 D = 65◦ , what is the measure of 6 C?
120◦ . The angles in a quadrilateral add up to 360◦ so if we subtract what was given from 360 we will
have our remaining angle.
17. ABDC is a parallelogram. BD is
2
3
the size of AC. If AE = 9 how long is BE?
6 Because the diagonals of a parallelogram bisect each other we know that if AE=9 then AC=18. Then
BD is 23 of 18 or 12, which makes BE=6.
18. True or False If a quadrilateral is a square, then it is a rectangle.
True, a square has four right angles so it is also a rectangle.
19. ABF G is a rectangle, BCDF is a square, BC = 6 inches, and E is the midpoint of F D. The length of
AB is twice the length of BC. What is the area of trapezoid ABEG?
81 inches2 . AB=12 since it is twice BC. Then GF=15 since it is the same as AB plus half of BC.
BF=6 since BCDF is a square. Then the two bases are 12 and 15 and the height is 6 so the area is
1
2 (12 + 15)(6) = 81.
20. True or False If a quadrilateral is a kite, then it is a parallelogram.
False, a kite is only a parallelogram in the special case where it is also a rhombus.
7
Unit 5
Practice Questions
1. mkn and both are cut by transversal t If the measure of 6 4 = 40◦ , what is the measure of 6 8?
8 = 40◦ because they are corresponding angles.
6
2.
θ and 6 α are vertical angles. If 6 θ = 30◦ , what is the measure of 6 α?
α = 30◦ because vertical angles are congruent.
6
6
3. Are right angles congruent or supplementary?
They are both congruent and supplementary.
4. Create a complete list of all the angles that are supplementary to 6 11 given that lkm and pkq.
6
2, 6 4, 6 5, 6 7, 6 10, 6 12, 6 13, 6 15
5. When two parallel lines are cut by a transversal, the corresponding angles are what?
Congruent
6. If 6 A = 3x + 12 and 6 B = 4x + 8 are complementary, then what is the measure of 6 A?
42◦ . 3x + 12 + 4x + 8 = 90 combine like terms 7x + 20 = 90 subtract 20 7x = 70 divide by 7 x = 10
substitute back in 3(10)+12=42
7. In the figure below lkm and both line are cut by the transversal t. Give a complete list of all the angles
congruent to 6 8.
6
1, 6 4, 6 5, 6 8
8. If B is in the interior of 6 XY Z and 6 XY Z = 68◦ and 6 XY B = 42◦ , what is the measure of 6 BY Z?
26◦ . By the angle addition property we must subtract the part from the whole to get the remainder.
6
9.
6
θ and 6 α are a linear pair. If 6 θ = 30◦ , what is the measure of 6 α?
α = 150◦ because angles in a linear pair are supplementary and 180-30=150.
10. When two parallel lines are cut by a transversal, the same side exterior angles are what?
Supplementary.
8
11. lkm and s and t are traversals that cross l and m and intersect with line m at point C. The the measure
of 6 BAC = 70◦ and the measure of 6 1 = 30◦ what is the measure of 6 2?
80◦ By looking at the corresponding angles for both
180. Then 70 + 30 = 100 so 6 2 = 80◦ .
6
1 and
6
BAC we can see that they plus
6
2 make
12. When two parallel lines are cut by a transversal, the alternate interior angles are what?
Congruent.
13. Which angles form a linear pair?
6
5
4
3
1
2
3 and 5, 5 and 6, 3 and 4, 4 and 6
14. If 6 A is complementary to 6 B and 6 B is complementary to 6 C, then what can be said about 6 A and
6 C?
They must be congruent.
15. If 6 ABC and 6 CBD are a linear pair, find the measure of 6 ABC.
C
13x + 15
A
12x + 15
B
D
◦
93 . First we need to set our equation equal to 180 because linear pairs are supplementary. This gives us
13x + 15 + 12x + 15 = 180. Then combining like terms leaves us with 25x + 30 = 180. Subtracting 30 and
dividing by 25 tells us that x = 6. We then must plug that back into 6 ABC to find that 13(6) + 15 = 93.
16. Name three types of angle pairs that are supplementary.
linear pairs, same side interior angles, same side exterior angles.
17. What is the degree measure of the smaller of the two angles formed by the line and the ray shown in
the figure below?
58◦ , see question 15 for setup.
18. Name three types of angle pairs that are congruent.
Vertical angles, alternate interior angles, alternate exterior angles, corresponding angles
9
19. If A is in the interior of 6 BCD and 6 ACD = 55◦ and 6 BCA = 30◦ , what is the measure of 6 BCD?
85◦ , we must add the two parts together to find the whole.
−−→ −−→
20. Given the figure below with BEkAD, what is m6 CBE?
6
50◦ , 6 ABE = 120◦ because it is same side interior with
ABC we are left with 50◦ .
10
6
BAC. Then if we take away the 70 from