Download Geom EOC Review Silverdale

Geom. EOCC review
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1. Show that the conjecture is false by finding a counterexample.
If
____
____
, then
.
a.
,
c.
,
b.
,
d.
,
2. Write a conditional statement from the statement.
A horse has 4 legs.
a. If it has 4 legs then it is a horse.
c. If it is a horse then it has 4 legs.
b. Every horse has 4 legs.
d. It has 4 legs and it is a horse.
3. Fill in the blanks to complete the two-column proof.
Given:
and
are supplementary. m
= 135
1
Prove: m
2
= 45
Proof:
1.
and
2. [1]
3. m
+m
4. 135 + m
5. m
= 45
____
Statements
are supplementary.
= 180
= 180
a. [1] m
= 135
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
b. [1] m
= 135
[2] Definition of supplementary angles
[3] Substitution Property
c. [1] m
= 135
[2] Definition of supplementary angles
[3] Subtraction Property of Equality
d. [1] m
= 135
[2] Definition of complementary angles
[3] Subtraction Property of Equality
4. Give an example of corresponding angles.
Created by Ellen Kraft Silverdale WA Reasons
1. Given
2. Given
3. [2]
4. Substitution Property
5. [3]
3 4
2 1
7 8
6
____
5
a.
and
c.
b.
and
d.
5. Identify the transversal and classify the angle pair
n
and
and
and
.
m
1
2
3
4
9
l
____
5
6
8
7
10
12
11
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.
6. Find m
.
R
U
(4x – 24)º
a. m
b. m
T
S (3x)º
>>
>>
V
=
=
Created by Ellen Kraft Silverdale WA c. m
d. m
=
=
____
7. Use the information
show that
.
, and the theorems you have learned to
l
1
2
m
a. By substitution,
and
By the Substitution Property of Equality,
.
By the Converse of the Alternate Interior Angles Theorem,
b. By substitution,
and
Since
and
are alternate interior angles,
By the Converse of the Same-Side Interior Angles Theorem,
c. By substitution,
and
Since
and
are same-side interior angles,
By the Converse of the Same-Side Interior Angles Theorem,
d. Since
and
are same-side interior angles,
.
By substitution,
.
By the Converse of the Alternate Interior Angles Theorem,
____
8. Write and solve an inequality for x.
D
2x + 4
A
____
8
C
B
a.
c.
b.
d.
9. Use the slope formula to determine the slope of the line.
Created by Ellen Kraft Silverdale WA .
.
.
.
.
.
.
.
and
.
y
10
8
6
4
2
–10 –8
–6
–4
–2
–2
2
4
6
8
10
x
–4
–6
A
–8
B
–10
a. 0
b.
c.
3
−2
d. undefined
2
−3
____ 10. Use slopes to determine whether the lines are parallel, perpendicular, or neither.
a. neither
c. parallel
b. perpendicular
____ 11. One of the acute angles in a right triangle has a measure of
angle?
a.
c.
b.
d.
____ 12. Find
and
, given
,
. What is the measure of the other acute
, and
N
F
D
E
P
M
a.
,
c.
b.
,
d.
____ 13. Given:
Identify all pairs of congruent corresponding parts.
Created by Ellen Kraft Silverdale WA ,
,
.
A
M
B
C
a.
b.
c.
d.
O
,
,
,
,
N
,
,
,
,
____ 14. Given that
and m
,
,
,
,
,
,
,
,
= 23°, find m
,
,
,
,
.
E
23º
A
D
C
B
a. m
b. m
= 77°
= 67°
____ 15. Given:
c. m
d. m
,
,
= 23°
= 113°
. T is the midpoint of
.
R
S
T
U
Prove:
Complete the proof.
Proof:
Statements
1.
2.
3.
4.
5.
6.
and
are right angles.
Created by Ellen Kraft Silverdale WA Reasons
1. Given
2. [1]
3. Right Angle Congruence Theorem
4. Given
5. [2]
6. Given
7. T is the midpoint of
8.
7. Given
8. Definition of midpoint
.
9. [3]
10. Definition of congruent triangles
9.
10.
a. [1] Definition of right angles
[2] Third Angles Theorem
[3] Transitive Property of Congruence
b. [1] Definition of perpendicular lines
[2] Third Angles Theorem
[3] Reflexive Property of Congruence
c. [1] Definition of perpendicular lines
[2] Vertical Angles Theorem
[3] Symmetric Property of Congruence
d. [1] Definition of perpendicular lines
[2] Third Angles Theorem
[3] Symmetric Property of Congruence
____ 16. Tom is wearing his favorite bow tie to the school dance. The bow tie is in the shape of two triangles.
Given:
,
,
,
Prove:
B
D
C
A
E
Complete the proof.
Proof:
Statements
1.
2.
3.
4.
5. [3]
,
,
a. [1] Reflexive Angles Theorem
[2] Third Angles Theorem
[3]
b. [1] Third Angles Theorem
[2] Vertical Angles Theorem
[3]
____ 17. Given the lengths marked on the figure and that
Created by Ellen Kraft Silverdale WA Reasons
1. Given
2. Given
3. [1]
4. [2]
5. Definition of congruent triangles
c. [1] Vertical Angles Theorem
[2] Third Angles Theorem
[3]
d. [1] Vertical Angles Theorem
[2] Third Angles Theorem
[3]
bisects
, use SSS to explain why
.
4 cm
E
A
3 cm
3 cm
D
4 cm
C
B
a.
b.
c.
d. The triangles are not congruent.
____ 18. The figure shows part of the roof structure of a house. Use SAS to explain why
.
R
||
S
||
T
U
Complete the explanation.
It is given that [1]. Since
and
are right angles, [2] by the Right Angle Congruence Theorem.
By the Reflexive Property of Congruence, [3]. Therefore,
by SAS.
a. [1]
c. [1]
[2]
[2]
[3]
[3]
b. [1]
d. [1]
[2]
[2]
[3]
[3]
____ 19. Show
for
.
6a - 2
A
D
a+7
4a - 2
B
C
16
Complete the proof.
.
.
by the Reflexive Property of Congruence. So
Created by Ellen Kraft Silverdale WA by [5].
a. [1]
[2]
[3] 16
[4] 16
[5] SAS
c. [1]
[2]
[3] 16
[4] 16
[5] SAS
b. [1]
[2]
[3] 26
[4] 26
[5] SSS
d. [1]
[2]
[3] 16
[4] 16
[5] SSS
____ 20. Use AAS to prove the triangles congruent.
Given:
Prove: ΔABC
,
ΔHGF
,
G
>
>> F |
A
H
C
>
| > >
B
Complete the flowchart proof.
Proof:
Given
1.
ΔABC
Given
2.
Given
a. 1. Alternate Exterior Angles Theorem
2. Alternate Interior Angles Theorem
b. 1. Alternate Interior Angles Theorem
2. Alternate Exterior Angles Theorem
c. 1. Alternate Exterior Angles Theorem
2. Alternate Exterior Angles Theorem
d. 1. Alternate Interior Angles Theorem
2. Alternate Interior Angles Theorem
Created by Ellen Kraft Silverdale WA ΔHGF
AAS
____ 21. Determine if you can use the HL Congruence Theorem to prove ΔACD
need to know.
P
A
B
|
^
^
|
C
a.
b.
c.
d.
D
Yes.
No. You do not know that
No. You do not know that
No. You do not know that
____ 22. Given:
Prove:
ΔDBA. If not, tell what else you
Q
and
are right angles.
.
.
,
bisects
F
B
)
C
)
A
D
G
Complete the flowchart proof.
Proof:
Given
1.
2.
bisects
Given.
Definition of
angle bisector.
3.
Created by Ellen Kraft Silverdale WA ΔACB
ΔACD
4.
5.
a. 1. Congruent Complements Theorem
2.
3. Transitive Property of Congruence
4. CPCTC
5. AAS
b. 1. Congruent Supplements Theorem
2.
3. Transitive Property of Congruence
4. AAS
5. CPCTC
____ 23. Find CA.
c. 1. Congruent Supplements Theorem
2.
3. Reflexive Property of Congruence
4. AAS
5. CPCTC
d. 1. Congruent Complements Theorem
2.
3. Reflexive Property of Congruence
4. CPCTC
5. AAS
A
)
s+2
)
)
C
2 s − 10
B
a. CA = 10
b. CA = 12
c. CA = 14
d. Not enough information. An equiangular triangle is not necessarily equilateral.
____ 24. Find the value of x. Express your answer in simplest radical form.
x
3
6
a. x =
b. x =
c. x =
d. x =
____ 25. The size of a TV screen is given by the length of its diagonal. The screen aspect ratio is the ratio of its width
to its height. The screen aspect ratio of a standard TV screen is 4:3. What are the width and height of a 27"
TV screen?
Created by Ellen Kraft Silverdale WA height
27"
width
a. width: 21.6 in., height: 16.2 in.
c. width: 21.6 in., height: 5.4 in.
b. width: 16.2 in., height: 21.6 in.
d. width: 5.4 in., height: 21.6 in.
____ 26. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.
25
20
a. The missing side length is 15. The side lengths form a Pythagorean triple because they are
nonzero whole numbers that satisfy the equation
.
b. The missing side length is 32.02. The side lengths do not form a Pythagorean triple
because one of them is not a nonzero whole number.
c. The missing side length is 5. The side lengths form a Pythagorean triple because they are
nonzero whole numbers that satisfy the equation
.
d. The missing side length is 32.02. The side lengths form a Pythagorean triple because they
satisfy the equation
.
____ 27. Find the value of x. Express your answer in simplest radical form.
5
x
x
a.
x=
b. x =
c.
d.
x=
x=
____ 28. Find the values of x and y. Express your answers in simplest radical form.
Created by Ellen Kraft Silverdale WA 30º
y
24
60º
a.
x
,
c.
,
b.
d.
,
,
____ 29. The Yield sign has a shape of an equilateral triangle with side length of 36 inches. What is the height of the
sign? Will a rectangular metal sheet of 36
32 inches be big enough to make one sign?
a. The Yield sign is about 33.7 inches tall. So the rectangular metal sheet will not be big
enough to make one sign.
b. The Yield sign is about 31.2 inches tall. So the rectangular metal sheet will be big enough
to make one sign.
c. The Yield sign is about 25.5 inches tall. So the rectangular metal sheet will be big enough
to make one sign.
d. The Yield sign is about 50.9 inches tall. So the rectangular metal sheet will not be big
enough to make one sign.
____ 30. Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides.
a. polygon, decagon
c. polygon, dodecagon
b. polygon, hexagon
d. not a polygon
____ 31. Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.
a. regular and concave
b. irregular and concave
Created by Ellen Kraft Silverdale WA c. regular and convex
d. irregular and convex
____ 32. MNOP is a parallelogram. Find MP.
M
N
5x
3x+12
P
O
a. MP = 25
c. MP = 20
b. MP = 30
d. MP = 6
____ 33. Write the trigonometric ratio for cos X as a fraction and as a decimal rounded to the nearest hundredth.
Y
15
9
X
a.
b.
Z
12
c.
cos X =
d.
cos X =
____ 34. Use a special right triangle to write tan 60
a.
b.
cos X =
cos X =
as a fraction.
c.
d.
____ 35. Use your calculator to find the trigonometric ratios sin 79 , cos 47 , and tan 77 . Round to the nearest
hundredth.
a. sin 79 = –0.99, cos 47 = –0.44, tan 77 = –32.27
b. sin 79 = –0.44, cos 47 = –0.99, tan 77 = –32.27
c. sin 79 = 0.68, cos 47 = 0.98, tan 77 = 4.33
d. sin 79 = 0.98, cos 47 = 0.68, tan 77 = 4.33
____ 36. Use the trigonometric ratio
to determine which angle of the triangle is
.
1
1.3 cm
2
0.5 cm
3
1.2 cm
a.
b.
2
1
Created by Ellen Kraft Silverdale WA c.
3
d. No solution.
____ 37. Use your calculator to find the angle measures
a degree.
a.
= 44.4°,
= 72.5°,
b.
= 0.8°,
= 1.3°,
c.
= 1.3°,
d.
= 72.5°,
= 0.8°,
= 44.4°,
Created by Ellen Kraft Silverdale WA to the nearest tenth of
= 88.5°
= 1.5°
= 1.5°
= 88.5°
Geom. EOCC review
Answer Section
MULTIPLE CHOICE
1. ANS: A
Pick values for a and b that follow the condition
if the conjecture holds.
Values of a and b
Let
and
Let
and
Let
and
. Then substitute them into the second inequality to see
a>b
.
Conclusion
The conjecture holds.
.
The conjecture holds.
.
The conjecture is false.
and
is a counterexample.
The conjecture is false when a is positive and b is negative.
Feedback
A
B
C
D
Correct!
In this case, a/b is greater than zero, so it is not a counterexample.
In this case, a is not greater than b. The counterexample should have a > b and a/b less
than or equal to 0.
In this case, a is not greater than b. a > b is the condition of the conjecture. The
counterexample should have a > b and a/b less than or equal to 0.
PTS: 1
KEY: inductive reasoning | counterexample
2. ANS: C
Identify the hypothesis and conclusion.
Hypothesis
A horse
If it is a horse,
Conclusion
has 4 legs.
then it has 4 legs.
Feedback
A
B
C
D
Identify the hypothesis and conclusion.
A conditional statement should have a hypothesis and a conclusion.
Correct!
A conditional statement should have a hypothesis and a conclusion.
PTS: 1
3. ANS: C
Proof:
KEY: conditional statement | if-then | hypothesis | conclusion
Statements
Created by Ellen Kraft Silverdale WA Reasons
1.
and
are supplementary.
2. m
= 135
3. m
+m
= 180
4. 135 + m
= 180
5. m
= 45
1. Given
2. Given
3. Definition of supplementary angles
4. Substitution Property
5. Subtraction Property of Equality
Feedback
A
B
C
D
Check to the given information.
To get from Step 4 to Step 5, use subtraction, not substitution.
Correct!
The angles are supplementary, not complementary.
PTS: 1
NAT: 12.3.5.a | 12.5.5.a
KEY: geometric proof | proof
4. ANS: A
Corresponding angles lie on the same side of a transversal, on the same sides of the two lines the transversal
crosses. So,
and
are corresponding angles.
Feedback
A
B
C
D
Correct!
Angle 4 and angle 1 are supplementary angles, not corresponding angles.
Corresponding angles lie on the same side of a transversal, on the same sides of two
lines.
Angle 5 and angle 7 are vertical angles, not corresponding angles.
PTS: 1
NAT: 12.3.3.e
KEY: corresponding angles | transversal
5. ANS: A
To determine which line is the transversal for a given angle pair, locate the line that connects the vertices.
Corresponding angles lie on the same side of the transversal l, on the same sides of lines n and m.
Feedback
A
B
C
D
Correct!
Alternate 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
6. ANS: D
NAT: 12.3.3.e
KEY: corresponding angles | transversal
Alternate Exterior Angles Theorem
Subtract
from both sides.
Divide both sides by
.
m
Substitute 24 for
Feedback
Created by Ellen Kraft Silverdale WA .
A
B
C
D
Find the measure of angle RST, not the supplement.
Find the measure of angle RST, not the value of x.
After finding x, substitute to find the angle measure.
Correct!
PTS: 1
NAT: 12.2.1.d | 12.3.3.e
KEY: alternate exterior angles | parallel lines
7. ANS: A
;
Substitute 20 for x.
Substitution Property of Equality
Converse of the Alternate Interior Angles Theorem
Feedback
A
B
C
D
Correct!
Angles 1 and 2 are alternate interior angles and are congruent.
Angles 1 and 2 are alternate interior angles and are congruent.
Angles 1 and 2 are alternate interior angles and are congruent.
PTS: 1
8. ANS: A
NAT: 12.3.3.e | 12.3.5.a
is the shorter segment.
Substitute
for
and 8 for
Subtract 4 from both sides.
Divide both sides by 2 and simplify.
KEY: parallel | alternate interior angles
.
Feedback
A
B
C
D
Correct!
Change the inequality sign.
Check your simplification methods.
Do not change the inequality sign when subtracting 4 or dividing by 2.
PTS: 1
NAT: 12.3.3.g
KEY: distance from a point to a line | inequality | perpendicular
9. ANS: B
Substitute (6, –7) for (
) and (9, –9) for (
) in the slope formula.
Feedback
A
B
C
D
The slope is the difference in y-values divided by the difference in x-values.
Correct!
Use the slope formula.
The slope is the difference in y-values divided by the difference in x-values.
PTS: 1
10. ANS: A
KEY: slope formula
Created by Ellen Kraft Silverdale WA y
15
(6, 15)
(–2, 7)
(3, 5)
(10, 5)
10
x
slope of
slope of
The lines have different slopes, so they are not parallel.
The product of the slopes is
, not
, so the slopes are not perpendicular.
The lines are coplanar, so they cannot be skew.
Feedback
A
B
C
Correct!
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
NAT: 12.3.4.a
KEY: slope | perpendicular | parallel
11. ANS: C
Let the acute angles be
and
, with m
.
m
m
The acute angles of a right triangle are complementary.
m
Substitute
for m
.
m
Subtract
from both sides.
Feedback
A
B
C
D
The two acute angles in a right triangle are complementary.
This is the measure of the given angle. Find the measure of the other acute angle.
Correct!
The measure of the other acute angle is less than 90 degrees.
PTS: 1
12. ANS: A
NAT: 12.2.1.d | 12.3.3.e
Third Angles Theorem
Definition of congruent angles
Substitute
for
and
Subtract
from both sides.
Divide both sides by –3.
So
Since
.
,
.
Created by Ellen Kraft Silverdale WA KEY: triangle sum theorem
for
.
Feedback
A
B
C
D
Correct!
Use the Third Angles Theorem.
The Third Angles Theorem states that if two angles of one triangle are congruent to two
angles of another triangle, then the third pair of angles are congruent.
These are the measures of angles F and P, not angles E and N.
PTS: 1
NAT: 12.2.1.d | 12.3.3.f
KEY: third angles theorem | triangle sum theorem
13. ANS: A
Corresponding angles and corresponding sides are parts which lie in the same position in the triangles.
Corresponding angles:
,
,
Corresponding sides:
,
,
Feedback
A
B
C
D
Correct!
The corresponding angles should be in the same position in triangle ABC and triangle
MNO.
The corresponding sides should be in the same position in triangle ABC and triangle
MNO.
Check that the corresponding angles and sides are congruent.
PTS: 1
14. ANS: B
NAT: 12.3.3.e
KEY: correspondence | corresponding parts
Triangle Sum Theorem
Substitution.
Simplify.
Subtract 113 from both sides.
Corresponding parts of congruent triangles are
congruent.
Definition of congruent angles
Corresponding parts of congruent triangles are
congruent.
Feedback
A
B
C
D
The sum of all angle measures in a triangle is equal to 180 degrees.
Correct!
Check which angles are corresponding angles.
Check your calculations.
PTS: 1
NAT: 12.2.1.d | 12.3.3.e
KEY: triangle sum theorem | congruent triangles | corresponding parts
15. ANS: B
Proof:
Statements
Reasons
1.
Given
1.
2.
and
are right angles.
2. Definition of perpendicular lines
Created by Ellen Kraft Silverdale WA 3.
4.
5.
6.
7. T is the midpoint of
8.
9.
10.
.
3. Right Angle Congruence Theorem
4. Given
5. Third Angles Theorem
6. Given
7. Given
8. Definition of midpoint
9. Reflexive Property of Congruence
10. Definition of congruent triangles
Feedback
A
B
C
D
Use the definition of perpendicular lines to show that the lines intersect to form right
angles.
Correct!
Angle S and angle U are not vertical angles. Use a different justification for Reason 5.
Use the correct property to show that the part is congruent to itself.
PTS: 1
16. ANS: D
Proof:
NAT: 12.3.3.e | 12.3.5.a
Statements
1.
2.
3.
4.
5.
,
,
KEY: proof | congruent triangles
Reasons
1. Given
2. Given
3. Vertical Angles Theorem
4. Third Angles Theorem
5. Definition of congruent triangles
Feedback
A
B
C
D
There is no Reflexive Angles Theorem. Find another reason for angle BCA being
congruent to angle DCE.
In order to use the Third Angles Theorem, first establish that two angles of one triangle
are congruent to two angles of the other triangle.
It is given that angle ABC is congruent to angle EDC. You are trying to prove that
triangle ABC is congruent to triangle EDC.
Correct!
PTS: 1
NAT: 12.3.3.e | 12.3.5.a
KEY: proof | congruent triangles
17. ANS: A
It is given that
,
, and
bisects
. By the definition of segment bisector,
. All three pairs of corresponding sides of the triangles are congruent. Therefore, ΔABC
ΔDEC
by SSS.
Feedback
A
B
C
Correct!
Use the fact that segment AD bisects segment BE.
The corresponding sides need to belong to different triangles. Use the fact that segment
Created by Ellen Kraft Silverdale WA D
AD bisects segment BE.
The corresponding sides of the triangles are congruent. Use the fact that segment AD
bisects segment BE.
PTS: 1
NAT: 12.3.2.e | 12.3.5.a
KEY: SSS | congruent triangles
18. ANS: C
It is given that
. Since
and
are right angles,
by the Right Angle
Congruence Theorem. By the Reflexive Property of Congruence,
. Therefore,
by
SAS.
Feedback
A
B
C
D
Check the figure to see what is given.
Angle SRT and angle URT are not right angles.
Correct!
Segment SU being congruent to itself does not help in proving the triangles congruent.
PTS: 1
19. ANS: D
NAT: 12.3.5.a
.
.
KEY: proof | SAS | congruent triangles
by the Reflexive Property of Congruence. So
by SSS.
Feedback
A
B
C
D
Substitute 3 for a.
Check the measures of segment AD and segment CB.
Use the correct postulate.
Correct!
PTS: 1
20. ANS: B
1.
and
.
2.
and
Theorem,
NAT: 12.3.2.e | 12.3.5.a
KEY: proof | SSS | congruent triangles
are alternate interior angles and
. Thus by the Alternate Interior Angles Theorem,
are alternate exterior angles and
.
. Thus by the Alternate Exterior Angles
Feedback
A
B
C
D
You switched the definitions of alternate interior and alternate exterior angles.
Correct!
If line segment AB is parallel to line segment GH, are angle B and angle G alternate
exterior angles or alternate interior angles?
If line AC is parallel to line FG, are angle ACB and angle HFG alternate interior angles
or alternate exterior angles?
PTS: 1
NAT: 12.3.3.e | 12.3.5.a
Created by Ellen Kraft Silverdale WA KEY: proof | congruent triangles | AAS
21. ANS: A
is given. In addition, by the Reflexive Property of Congruence,
. Since
and
, by the Perpendicular Transversal Theorem
. By the definition of right angle,
is a
right angle. Similarly,
is a right angle. Therefore, ΔABD
ΔDCA by the HL Congruence Theorem.
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A
B
C
D
Correct!
Since line segment AC is parallel to line segment BD, what does the Perpendicular
Transversal Theorem tell you about line segment BD and line segment PB?
What do you know about the other pair of legs of the right triangles ABD and DCA?
What do you know about line segments AB and CD?
PTS: 1
NAT: 12.3.2.e | 12.3.3.e | 12.3.5.a
KEY: proof | congruent triangles | HL
22. ANS: C
1a. By the Linear Pair Theorem,
and
are supplementary and
and
are
supplementary.
1b. Given
, by the Congruent Supplements Theorem,
.
2.
by the definition of an angle bisector.
3.
by the Reflexive Property of Congruence
4. Two angles and a nonincluded side of ΔACB and ΔACD are congruent. By AAS,
ΔACB
ΔACD.
5. Since ΔACB
by CPCTC.
ΔACD,
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A
B
C
D
For reason 1, check whether the linear pairs are complementary or supplementary.
Find the correct property that states that a line segment is congruent to itself.
Correct!
For statement 2, use the fact that line segment AC bisects angle A, not angle C.
PTS: 1
NAT: 12.3.2.e | 12.3.5.a
KEY: congruent triangles | corresponding parts | CPCTC | flowchart proof
23. ANS: C
Equiangular triangles are equilateral.
ΔABC is equilateral.
Definition of equilateral triangle.
Subtract s and add 10 to both sides of the equation.
Substitute 12 for s in the equation for AB.
Simplify.
Definition of equilateral triangle.
Substitute 14 for AB.
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Created by Ellen Kraft Silverdale WA A
B
C
D
Equiangular triangles are equilateral. Use AB = BC to solve for s, and then use AC = AB
or AC = BC to find AC.
This is s. Substitute s in the original equation to find AC.
Correct!
By a corollary to the Isosceles Triangle Theorem, equiangular triangles are equilateral.
Use AB = BC to solve for s, and then use AC = AB or AC = BC to find AC.
PTS: 1
24. ANS: A
NAT: 12.3.3.f
KEY: equilateral triangle | side length
Pythagorean Theorem
Substitute 3 for a, 6 for b, and x for c.
Simplify.
Find the positive square root.
Simplify the radical.
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A
B
C
D
Correct!
Simplify the square root of 45 correctly.
Apply the Pythagorean Theorem by substituting 3 for a, 6 for b, and x for c.
You reversed the order of the radicand and the number outside the radical sign.
PTS: 1
NAT: 12.3.3.d
KEY: Pythagorean Theorem | side length
25. ANS: A
Let 3x be the height in inches. Then 4x is the width of the TV screen.
Pythagorean Theorem
Substitute 4x for a, 3x for b, and 27 for c.
Multiply and combine like terms.
Divide both sides by 25.
Find the positive square root.
in.
Width:
Height:
in.
in.
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A
B
C
D
Correct!
The screen aspect ratio is the ratio of its width to its height (4:3), not the reverse.
The screen aspect ratio is not 4:3 in this case. The height is represented by 3x, not x.
The screen aspect ratio is the ratio of its width to its height (4:3). The height is
represented by 3x and the width is represented by 4x.
PTS: 1
26. ANS: A
NAT: 12.3.3.c | 12.3.3.d
Pythagorean Theorem
Substitute 20 for a and 25 for c.
Created by Ellen Kraft Silverdale WA KEY: Pythagorean Theorem | side length
Multiply and subtract 400 from both sides.
Find the positive square root.
The side lengths are nonzero whole numbers that satisfy the equation
Pythagorean triple.
, so they form a
Feedback
A
B
C
D
Correct!
Apply the Pythagorean Theorem by substituting 20 for a and 25 for c.
Apply the Pythagorean Theorem by substituting 20 for a and 25 for c.
Apply the Pythagorean Theorem by substituting 20 for a and 25 for c. If one of the side
lengths is not a nonzero whole number, the side lengths cannot form a Pythagorean
triple.
PTS: 1
NAT: 12.3.3.d
KEY: Pythagorean Theorem | side length | Pythagorean triple
27. ANS: A
The triangle is an isosceles right triangle, which is a 45°–45°–90° triangle. The length of the hypotenuse is 5.
Hypotenuse
Divide both sides by
.
Rationalize the denominator.
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A
B
C
D
Correct!
In a 45°–45°–90° triangle, both legs are congruent, and the length of the hypotenuse is
the length of a leg times the square root of 2.
In a 45°–45°–90° triangle, both legs are congruent, and the length of the hypotenuse is
the length of a leg times the square root of 2.
In a 45°–45°–90° triangle, both legs are congruent, and the length of the hypotenuse is
the length of a leg times the square root of 2.
PTS: 1
NAT: 12.3.3.b | 12.3.3.d
KEY: special right triangle | 45-45-90 | isosceles right triangle
28. ANS: A
Hypotenuse
Divide both sides by 2.
Feedback
A
B
Correct!
If two angles of a triangle are not congruent, the shorter side lies opposite the smaller
angle.
Created by Ellen Kraft Silverdale WA C
D
In a 30°–60°–90° triangle, the length of the longer leg is the length of the shorter leg
times the square root of 3.
In a 30°–60°–90° triangle, the length of the hypotenuse is 2 times the length of the
shorter leg, and the length of longer leg is the length of the shorter leg times the square
root of 3.
PTS: 1
NAT: 12.3.3.b | 12.3.3.d
KEY: special right triangles | 30-60-90
29. ANS: B
Step 1 Divide the equilateral triangle into two 30°-60°-90° triangles. The height of the Yield sign is the length
of the longer leg.
x
60º
h
36 in.
30º
Step 2 Find the length x of the shorter leg.
Hypotenuse
Divide both sides by 2.
Step 3 Find the length h of the longer leg.
The Yield sign is about 31.2 inches tall. So the rectangular metal sheet will be big enough to make one sign.
Feedback
A
B
C
D
Divide the equilateral triangle into two 30°–60°–90° triangles. The height of the Yield
sign is the length of the longer leg.
Correct!
The longer leg in a 30°–60°–90° triangle equals the product of the shorter leg times
square root of 3.
The longer leg in a 30°–60°–90° triangle equals the product of the shorter leg times
square root of 3.
PTS: 1
NAT: 12.3.3.b | 12.3.3.c | 12.3.3.d
KEY: special right triangles | equilateral triangle | 30-60-90
30. ANS: A
A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. A
polygon with 10 sides is called a decagon.
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A
B
C
D
Correct!
A hexagon has 6 sides.
A dodecagon has 12 sides.
A polygon is a closed plane figure formed by three or more segments that intersect only
Created by Ellen Kraft Silverdale WA at their endpoints.
PTS: 1
NAT: 12.3.3.e
KEY: classify polygons
31. ANS: B
If the shape were regular all the sides would be congruent and all the angles would be congruent. Since
neither the sides nor the angles are congruent, the shape is irregular. Because some of the diagonals of this
shape contain points in the exterior of the polygon, the shape is concave.
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A
B
C
D
In regular shapes, all sides and all angles are congruent.
Correct!
In regular shapes, all sides and all angles are congruent.
In convex shapes, no diagonals contain points in the exterior.
PTS: 1
32. ANS: B
MP = NO
MP = 5x = 5(6) = 30
NAT: 12.3.3.e
KEY: classify polygons | concave | convex
Opposite sides of a parallelogram are congruent.
Definition of congruent segments
Substitute.
Simplify and solve.
Substitute and solve for entire segment measure.
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A
B
C
D
Opposite sides of a parallelogram are congruent.
Correct!
Set the expressions for opposite sides of the parallelogram equal to each other.
Substitute the solution for x back into the original expression.
PTS: 1
33. ANS: C
cos X =
NAT: 12.3.3.f
The cosine of an
KEY: parallelogram | opposite sides
is
.
Feedback
A
B
C
D
Cosine is the ratio of the adjacent side to the hypotenuse.
Sine is the ratio of the opposite side to the hypotenuse.
Correct!
Tangent is the ratio of the opposite to the adjacent side.
PTS: 1
NAT: 12.2.3.c
34. ANS: A
Draw and label a 30 –60 –90 triangle.
Created by Ellen Kraft Silverdale WA KEY: trigonometric ratio | trigonometry | cosine
The tangent of an angle is
.
30°
2s
60°
s
Feedback
A
B
C
D
Correct!
The tangent of an angle is the ratio opposite leg/ adjacent leg.
Use the ratios of the sides of a 30°–60°–90° triangle to create the fraction.
The tangent of an angle is the ratio opposite leg/ adjacent leg.
PTS: 1
NAT: 12.2.3.c
KEY: trigonometric ratio | trigonometry | tangent | special right triangles | 30-60-90
35. ANS: D
Make sure your calculator is in degree mode.
sin 79 = 0.98, cos 47 = 0.68, tan 77 = 4.33
Feedback
A
B
C
D
Change your calculator to degree mode.
Change your calculator to degree mode.
Switch the first and second answers.
Correct!
PTS: 1
36. ANS: A
KEY: trigonometric ratio | trigonometry | cosine | sine | tangent
Sine is the ratio of the opposite leg to the hypotenuse.
1.2 is the length of the leg opposite
1.3 is the length of the hypotenuse.
0.5 is the length of the leg adjacent
1.3 is the length of the hypotenuse.
Since
,
2 is
.
.
A.
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A
B
C
D
Correct!
The sine of an angle is the length of the opposite leg divided by the length of the
hypotenuse.
The sine of an angle is the length of the opposite leg divided by the length of the
hypotenuse.
The sine of an angle is the length of the opposite leg divided by the length of the
hypotenuse.
PTS: 1
37. ANS: A
NAT: 12.2.3.c
Created by Ellen Kraft Silverdale WA KEY: trigonometric ratio | trigonometry
Change your calculator to degree mode.
Use the inverse trigonometric functions on your calculator to find each angle measure.
Feedback
A
B
C
D
Correct!
Change your calculator to degree mode.
Change your calculator to degree mode.
Switch the first and second answer.
PTS: 1
KEY: trigonometric ratio | trigonometry | inverse trigonometric ratio
Created by Ellen Kraft Silverdale WA