Download Day 1 Review - Worksheet 12 18 ab and ab 30 24

Secondary II
Day 1 Review - Worksheet
Name ___________________________
GCF, LCM, AND FACTORING
Write the prime factorization for each number.
1. 12
2. 30
3. 24
4. 40
5. 60
9. 24 & 40
10. 4 & 14 & 8
Find the LCM for each set of numbers:
6. 6 & 9
7. 4 & 7
8. 12 & 30
11. Ronna has soccer practice every 4 days. She also has violin lessons every 10 days. Ronna has both
activities today after school. When will she have both activities again in the same day? (Hint: find LCM)
12. Emilio's family volunteers at the local soup kitchen every 30 days. Emilio has swimming lessons
every 9 days. He has both activities this Saturday. When will he have both activities again on the same day?
Find the GCF for each set of numbers:
13. 25 & 45
14. 48 & 20
15. 15 & 16
16. 6 & 9
17. 14 & 28 & 49
18. Kim is creating treat bags for her birthday party guests. She has 32 packs of gum, 24 bracelets, and
16 lip glosses. What is the greatest number of treat bags she can make if she wants to use all of the
items and have the same number of each treat in each bag? How many of each treat will be in a
bag? (Hint: first find GCF)
19. Bryan is dividing students into groups for a nature hike. He wants to divide the boys and girls so that each
group has the same number of both boys and girls. There are 21 boys and 56 girls signed up for the hike.
Into how many groups can the students be divided? How many boys and how many girls will be in each
group?
Identify the greatest common factor between each set of terms.
2 3
3
20. 12a b and 18a b
21. 30hp
5
and 24h 4 p 3
22. 20 y m x
2
3 5
and 12 y 6 mx 4
1
Secondary II
Day 2 Review – Worksheet
Name ___________________________
Lines and System of Equations
Find the slope between each pair of points.
1. (4,5) & (-2,3)
2. (4,-6) & (-1,-6)
3. (-3,7) & (-3,10)
Convert each equation from standard form to slope-intercept form.
4. 3x  5 y  30
5. 6 x  2 y  52
6.  y  9 x  12
Convert each equation from slope-intercept form to standard form.
7. y  4 x  2
8. y 
2
x6
3
9. y  
5
x2
2
Determine the x and y intercepts of each equation.
10. y  3x  6
11.
2
x  3y  6
5
Determine the x and y intercepts of each equation. Graph each equation and label your graph.
12. y  2 x  5
13. 4 x  5 y  20
14. x  3
15. A movie theater sells tickets for matinee showings for $7 and evening shows for $10. Write an expression
that represents the total amount the theater can earn selling tickets.
16. The basketball booster club runs the concession stand during a weekend tournament. They sell
hamburgers for $2.50 each and hot dogs for $1.50 each. They hope to earn $900 during the tournament. Find
an equation to represent the total amount the booster club hopes to earn.
a. If the club sells 315 hamburgers during the tournament, how many hot dogs must they sell to reach
their goal?
b. If the club sells 0 hot dogs during the tournament, how many hamburgers must they sell to reach their
goal?
17. Mattie sells heads of lettuce for $1.99 each from a roadside farmer’s market stand. Each week she loses 2
heads of lettuce due to spoilage. Write a linear function that represents the total amount Mattie earns each
week selling heads of lettuce taking into account the value of the lettuce she loses due to spoilage.
2
Secondary II
Write the equation of a line with the given information.
18. Passes through the points (3,-5) and (-1,2)
19. Passes through the points (4,-9) and (4,3)
20. Passes through the point (2,-6) with a slope of -4
21. Parallel to y  2 x  5 that passes through ( 3 , - 2 )
22. Perpendicular to y  3x  1 that passes through point ( 3 , - 2 )
23. Write a system of linear equations to represent each problem. Define each variable. Graph the system
of equations, label the axes
Eric sells model cars from a booth at a local flea market. He purchases each model car from a distributor
for $12 and the flea market charges him a booth fee of $50. Eric sells each model car for $20.
24. Find the x- and y- intercepts of the equation, and graph.
a. 𝑥 = −3
b. 3𝑦 + 2𝑥 = 10
c. 𝑦 − 3𝑥 = 0
3
Secondary II
Day 3 Review – Worksheet
Name ___________________________
RADICALS AND EXPONENTS
1. Write a system of linear equations to represent each problem. Define each variable. Graph the system
of equations, label the axes
Eric sells model cars from a booth at a local flea market. He purchases each model car from a distributor
for $12 and the flea market charges him a booth fee of $50. Eric sells each model car for $20.
2. Solve each system of equations by substitution. Determine if the system is consistent or inconsistent.
 y  2x  3
2 x  y  5
 y  3x  2
 y  3x  4
b) 
a) 
3. Solve each system of equations by elimination. Determine whether the system is consistent or
inconsistent.
a)
4 x  y  2
2 x  2 y  26
10 x  6 y  6

5 x  5 y  5
b) 
4. Write a system of equations to represent each problem situation. Solve the system and explain what
your solution means in context of the problem.
a) The high school band is selling fruit baskets as a fundraiser. They sell a large basket containing 10 apples
and 15 oranges for $20. They sell a small basket containing 5 apples and 6 oranges for $8.50. How much is
the band charging for each apple and each orange?
b) Taylor and Natsumi are making block towers out of large and small blocks. They are stacking the
blocks on top of each other in a single column. Taylor uses 4 large blocks and 2 small blocks to make
a tower 63.8 inches tall. Natsumi uses 9 large blocks and 4 small blocks to make a tower 139.8 inches
tall. How tall is each large block and each small block?
5. Simplify each expression without a calculator.
a.
3
8
b.
3
3
c.
64
125
d.
5
32
e.
3
54
6. Write each radical as a power.
a.
y
b.
3
c.
5
5
x
d.
5
84
7. Write each power as a radical.
1
2
2
a. 12 3
b. 7 5
c. c 3
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Secondary II
2.1 - Worksheet
Name ___________________________
Define each term:
1. Induction: ______________________________________________________________
2. Deduction:_____________________________________________________________
3. Propositional Form:______________________________________________________
Identify the specific information, the general information, and the conclusion for each problem. (Hint: If
there is none write there is none and not leave the problem blank.)
4. You read an article in the paper that says a high-fat diet increases a person’s risk of heart disease.
You know your father has a lot of fat in his diet, so you worry that he is at higher risk of heart
disease.
Specific information: __________________________________________________________
General Information: __________________________________________________________
Conclusion: ___________________________________________________________________
5. Janice tells you that she has been to the mall three times in the past week, and every time there
were a lot of people there. “It’s always crowed at the mall,” she says.
Specific information: __________________________________________________________
General Information: __________________________________________________________
Conclusion: ___________________________________________________________________
6. Ava read an article that said eating too much sugar can lead to tooth decay and cavities. Ava
noticed that her little brother Phillip eats a lot of sugar. She concludes that Phillip’s teeth will
decay and develop cavities.
Specific information: __________________________________________________________
General Information: __________________________________________________________
Conclusion: ___________________________________________________________________
5
Secondary II
Determine whether inductive reasoning or deductive reasoning is used in each situation. Then determine
whether the conclusion is correct and explain your reasoning.
7. Jason sees a line of 10 school buses and notices that each is yellow. He concludes that all school
buses must be yellow.
8. Caitlyn has been told that every taxi in New York is yellow. When she sees a red car in New York
City, she concludes that it cannot be a taxi.
9. Carlos is told that all garter snakes are not venomous. He sees a garter snake in his backyard and
concludes that it is not venomous.
10. Isabella sees 5 red fire trucks. She concludes that all fire trucks are red.
Write each statement in propositional form.
11. Three points are all located on the same line. So, the points are collinear points.
12. Two angles are supplementary angles if the sum of their angle measures is equal to 180ᵒ.
13. A ray divides an angle into two congruent angles. So, the ray is an angle bisector.
Identify the hypothesis and the conclusion of each conditional statement.
14. If the sum of two angles is 180ᵒ, then the angles are supplementary.
Hypothesis: __________________________________________________________________
Conclusion:___________________________________________________________________
15. If the sum of two angle measures is equal to 90ᵒ, then the angles are complementary angles.
Hypothesis: __________________________________________________________________
Conclusion:___________________________________________________________________
For each conditional statement write the hypothesis as the “Given” and the conclusion as the “Prove.”
16.
Given: ______________________________________________
Prove:_______________________________________________
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Secondary II
17.
Given: ______________________________________________
Prove:_______________________________________________
Simplify the expression by hand.
18.
19.
4
625
72
Identify the greatest common factor between each set of terms.
20. 12𝑎 2 𝑏3 𝑎𝑛𝑑 18𝑎 3 𝑏
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Secondary II
2.2 - Worksheet
Name ___________________________
1. Draw a figure to illustrate each term:
a) Vertical Angles
b) Linear Pair
c) Adjacent Angles
Define the following
2. Supplementary Angles:____________________________________________________________
3. Complementary Angles:___________________________________________________________
4. Linear Pair Postulate:______________________________________________________________
5. Segment Addition Postulate:________________________________________________________
6. Angle Addition Postulate:__________________________________________________________
Solve for x.
7.
8.
X=________________
9.
X=________________
10.
X=_______________
X=_______________
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Secondary II
Use the given information to determine the measure of the angles in each pair.
11. The measure of the complement of an angle is three times the measure of the angle. What is the
measure of each angle?
12. The measure of the supplement of an angle is twice the measure of the angle. What is the
measure of each angle?
For each diagram, determine whether angles 1 and 2 form adjacent angles
13.a)
b)
For each diagram, determine whether angles 1 and 2 form a linear pair
14.a)
b)
Name each pair of vertical angles then find the given values
15.
Vertical angles:____________________________________
Given m<1=110ᵒ Find m<6 and m<2
m<6=________________
m<2=________________
16.
Vertical angles:____________________________________
Given m<7=63ᵒ Find m<3 and m<8
m<3___________________
m<8___________________
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Secondary II
Complete the statement. Then write the postulate used.
17.
19.
18.
20. Write in propositional form: The measure of m<A and m<B add up to 90ᵒ, so they are complementary.
21.
22. Find the greatest common factor: 25𝑎 3 𝑏, 5𝑎𝑏, 15𝑎 4 𝑏3
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Secondary II
2.3 - Worksheet
Name ________________________
Identify the property demonstrated in each example
1.
4.
2.
5.
3.
Rewrite each conditional statement by separating the hypothesis and conclusion. The hypothesis
becomes the “Given” information and the conclusion becomes the “Prove” information.
6. Conditional Statement: 𝐼𝑓 < 2 ≅< 1, 𝑡ℎ𝑒𝑛 < 2 ≅< 3
Given:
Prove:
7. Conditional Statement: 𝐴𝐵 + 𝑅𝑆 = 𝐶𝐷 + 𝑅𝑆, 𝑖𝑓 𝐴𝐵 = 𝐶𝐷
Given:
Prove:
8. Rewrite the flow chart proof as a two column proof.
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Secondary II
9. Rewrite the two-column proof as a paragraph proof
10. Fill in the missing information
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Secondary II
11. Fill in the missing information
12. Prove using any method
13. Given the diagram find: (Use correct notation)
a) One pair of vertical angles.
b) One pair of supplementary angles.
c) One pair of complementary angles.
d) One pair of adjacent angles that are neither complementary nor supplementary.
e) A linear pair of angles.
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Secondary II
2.4/2.5 - Worksheet
Name ___________________________
1. Given the following figure answer all parts.
a) Vertical angles:_________________________________
b) Alternate Interior angles:_________________________
c) Alternate Exterior angles:___________________________
d) Same-side exterior angles:______________________________
e) Same-side interior angles:_____________________________________________________
f) Corresponding Angles:________________________________________________________
g) Linear Pairs:_________________________________________________________________
Determine what kind of angles are given.
2.
4.
3.
5.
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Secondary II
Use the following picture to solve for the given values.
6. If the m<1=80ᵒ, find
m<2=_____________________
m<3=_____________________
m<5=_____________________
m<6=_____________________
7. If the m<1=x+5 and m<8 = x, find
x=_______________
m<8=_______________
m<7=______________
Write the theorem that is illustrated by each statement and diagram
8.
10.
9.
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Secondary II
11.
Write the converse of each conditional statement. Then, determine whether the converse is true.
a. If two points are collinear, then they are on the same lines.
b. If a triangle has two sides with equal lengths, then it is an isosceles triangle.
c. If the lengths of the sides of a triangle are 3 cm, 4 cm, and 5 cm, then the triangle is a right
triangle.
Write the inverse of each conditional statement. Then, determine whether the inverse is true.
a. If a triangle is a right triangle, then the sum of the measures of its acute angles is 90º
12.
b. If a polygon is a triangle, then the sum of its exterior angles is 360º.
c. If two angles are complementary, then the sum of their measures is 90º
13.
Write the contrapositive of each conditional statement. Then, determine whether the contrapositive is
true.
a. If one of the acute angles of a right triangle measures 30º, then it is a 30º - 60º - 90º triangle.
b. If a quadrilateral is an isosceles trapezoid, then it has two pairs of congruent base angles.
c. If two angles are supplementary, then the sum of their measures is 180º.
Use the diagram to write the “Given” and “Prove” statements for each theorem.
14.
Given:___________________________________________________________________
Prove:___________________________________________________________________
15.
Given:____________________________________________________________________
Prove:____________________________________________________________________
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Secondary II
Fill in the missing parts of the given proof
16.
17.
17
Secondary II
Review - Chapter 2
Name ___________________________
Identify the specific information, the general information, and the conclusion for each problem situation.
1- You hear from your teacher that spending too much time in the sun without sunblock increases the risk
of skin cancer. Your friend Susan spends as much time as she can outside working on her tan without
sunscreen, so you tell her that she in increasing her risk of skin cancer when she is older.
Specific Information:_________________________________________________________________
General Information:_________________________________________________________________
Conclusion:________________________________________________________________________
2- Janice tells you that she has been to the mall three times in the past week, and every time there was a
lot of people there. "It's always crowded at the mall," she says.
Specific Information:_________________________________________________________________
General Information:_________________________________________________________________
Conclusion:_________________________________________________________________________
Determine whether inductive reasoning or deductive reasoning is used in each situation.
3- Isabella sees 5 red fire trucks. She concludes that all fire trucks are red.
4- Miriam has been told that lightning never strikes twice in the same place. During a lightning storm,
she sees a tree struck by lightning and goes to stand next to it, convinced that it is the safest place to
be.
For problems 5 and 6:
a. Write the conditional statement in propositional form (If-Then form).
b. Identify the hypothesis and the conclusion of the conditional statement.
c. Write a converse for part a
d. Write an inverse for part a
e. Write a contrapositive for part a
5- The measure of an angle is 90°. So, the angle is a right angle.
a.
b.
c.
d.
e.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
6- A ray divides an angle into two congruent angles. So, the ray is an angle bisector.
a.
b.
c.
d.
e.
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
_________________________________________________________________________
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Secondary II
7- For each conditional statement, draw a diagram and then write the hypothesis as the "Given" and the
conclusion as the "Prove."
8- If ∠𝑄𝑅𝑆 and ∠𝑆𝑅𝑇 are complementary angles, then m∠𝑄𝑅𝑆 + m∠𝑆𝑅𝑇 = 90°.
Given:
Prove:
9- If ⃗⃗⃗⃗⃗
𝑃𝐺 bisects ∠𝐹𝑃𝐻, then ∠𝐹𝑃𝐺 ≅ ∠𝐺𝑃𝐻
Given:
Prove:
9- Draw a figure to illustrate each term then define the term.
a. Supplementary angles
d. Complementary angles
b.
Adjacent angles
c.
Vertical angles
e. Linear pair
Solve for x
10-
11-
12- Define the following properties (make sure you know and understand them)
a. Transitive Property:______________________________________________________________
b. Reflexive Property:______________________________________________________________
c. Substitution Property:____________________________________________________________
d. Addition Property:_______________________________________________________________
e. Subtraction Property:_____________________________________________________________
13- Identify the types of angles given the following picture
a)
Alternate interior angles
d) Alternate exterior angles
b)
Corresponding Angles
e) Same side interior angles
c)
Vertical Angles
f) Linear Pair
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Secondary II
14- Two angles are complementary. The smaller angle is 15⁰ less than the larger angle. What is the
measure of the larger angle?
15- Two angles are supplementary. The larger angle is 28⁰ greater than the smaller angle. What is the
measure of the smaller angle?
16- Write the converse, inverse and contrapositive for the Alternate Interior Angle Theorem: If a
transversal intersects two parallel lines, then the alternate interior angles form are congruent.
17- Use the given information to determine the measures of each of the
numbered angles.
p ∥ 𝑞 and m∠1 = 54°
m<2 = _________
m<3 = __________
m<4 =___________
m<5 = _________
m<6 = __________
m<7 = ___________
m<8 = __________
18- Use the figure to find all angles given m<5 = 95⁰.
m<1=____________
m<2 = __________
m<3 = ___________
m<4 = __________
m<6 = ___________
m<7 = __________
m<8 = ___________
For the following proofs fill in the missing pieces
19-
20
Secondary II
20-
Review Problems
21. Write the prime factorization for 24
_____________________
22. Find the slope between the 2 points (2, -1) and (8, 4)
________________________
23. Find the x – intercept and y-intercept or the following line. Y = 4x + 8
24. Simplify √24
________ _____________
_________________________
25. Solve the system by substitution
𝑦 = 2𝑥 + 1
{
𝑦=3
________________________
21
Secondary II
Worksheet – After Chapter 2 Test
Name __________________________
Simplifying Radical Expressions
22
Secondary II
3.1/3.2 - Worksheet
Name ___________________________
Determine the measure of the missing angle in each triangle.
1.
2.
3.
List the side lengths from shortest to longest for each diagram.
4.
5.
6.
Identify the interior angles, the exterior angle, and the remote interior angles of each triangle.
7.
8.
9.
Solve for x in each diagram.
10.
13.
11. 12.
14.
23
Secondary II
Use the given information for each triangle to write two inequalities that you would need to prove the Exterior Angle
Inequality Theorem.
15.
16.
Without measuring the angles, list the angles of each triangle in order from least to greatest measure.
17.
18.
19.
Determine whether it is possible to form a triangle using each set of segments with the given measurements. Explain
your reasoning.
20. 8, 8, 8
21. 10, 5, 21
22. 4, 5.1, 12.5
23. 112, 300, 190
Write an inequality the expresses the possible lengths of the unknown side of each triangle.
24.
25.
26.
27. a. Give an appropriate name for the angle pair 6 and 8.
b. If the measure of angle 6 can be expressed by (13x - 5)
degrees and the measure of angle 8 can be expressed by
(4x + 40) degrees. Solve for x.
c. Then record the measure of all angles in the diagram.
24
Secondary II
3.3/3.4 - Worksheet
Name ___________________________
1. Determine the length of the hypotenuse of each triangle. Write your answer as a radical in simplest form.
a.
b.
2. Determine the lengths of the legs of each triangle. Write your answer as a radical in simplest form.
a.
b.
3. Determine the measure of the indicated interior angle.
a. 𝑚∠𝐷𝐹𝐸
b. 𝑚∠𝐻𝐴𝐾
c. 𝑚∠𝑇𝑅𝐴
4. Given the short leg, determine the lengths of the long leg and the hypotenuse of each triangle. Write your answer
as a radical in simplest form.
a.
b.
5. Given the hypotenuse, determine the lengths of the two legs of each triangle. Write your answer as a radical in
simplest form.
a.
b.
6. Given the long leg, determine the lengths of the short leg and the hypotenuse of each triangle. Write your answer
as a radical in simplest form.
a.
b.
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Secondary II
7. Determine the area of each triangle.
a.
b.
8. Soren is flying a kite on the beach. The string forms a 45º angle with the ground. If he has let out 16
meters of line how high above the ground is the kite?
9. The perimeter of the square is 32 centimeters. Calculate the length of its diagonal.
10. Find the slope of the line through the points (5, -4) and (-9, -2). Write the equation of the line.
11. Multiply and simplify: 5√8𝑥 ∙ 3√2𝑥 5
12. Calculate the distance between the points (8, -7) and (20, 9).
Sketch and label a triangle for each row, and fill in the missing information.
A
13.
45
14.
30
16.
60
17.
b
Sketch & Label
Triangle
12
6
10
45
8√3
18
30
60
c
8
60
19.
20.
a
45
15.
18.
B
16
24
26
Secondary II
4.1/4.2 - Worksheet
Name ___________________________
1.
2. Given the image and pre-image, determine the scale factor.
a.
b.
3. Use quadrilateral ABCD shown on the grid to complete part (a) through part (c).
a. On the grid, draw the image of quadrilateral
ABCD dilated using a scale factor of 3 with the center
of dilation at the origin. Label the image JKLM.
b. On the grid, draw the image of quadrilateral ABCD
dilated using a scale factor of 0.5 with the center of
dilation at the origin. Label the image WXYZ.
c. Identify the coordinates of the vertices of
quadrilaterals JKLM and WXYZ.
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Secondary II
4. The vertices of trapezoid WXYZ are W(-1, 2), X(-3, -1), Y(5, -1), and Z(3, 2). Without drawing the
figure, determine the coordinates of the vertices of the image of trapezoid WXYZ dilated using a scale
factor of 5 with the center of the dilation at the origin. Explain your reasoning.
5. The vertices of triangle ABC are A(-6, 15), B(0, 5), and C(3, 10). Without drawing the figure,
1
determine the coordinates of the vertices of the image ABC dilated using a scale factor of 3 with
center of dilation at the origin. Explain your reasoning.
6. The vertices of hexagon PQRSTV are P(-5, 0), Q(-5, 5), R(0, 7), S(5, 2), T(5, -2), and V(0,-5).
Without drawing the figure, determine the coordinates of the vertices of the image of hexagon
PQRSTV dilated about the origin using a scale factor of 4.2. Explain your reasoning.
7. Give an example of each term. Include a sketch with each example.
a. Angle-Angle Similarity Theorem
b. Side-Side-Side Similarity Theorem
c. Side-Angle-Side Similarity Theorem
d. Included angle
e. Included side
8. Explain how you know that the triangles are similar.
a.
b.
c.
9. What information would you need to use the Side-Angle-Side Similarity Theorem to prove that the
triangles are similar?
10. What information would you need to use the Angle-Angle Similarity Theorem to prove that the
triangles are similar?
11. What information would you need to use the Side-Side-Side Similarity Theorem to prove that the
triangles are similar?
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Secondary II
12. Determine whether each pair of triangles is similar. Explain your reasoning.
a.
b.
c.
d.
Sketch and label a triangle for each row, and fill in the missing information.
A
13
14
B
60
16
30
17
30
20
Sketch & Label Triangle
14√3
2.6
8√3
45
53√2
60
60
c
20
30
19
b
4√3
45
15
18
a
25
15√3
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Secondary II
4.3/4.4 - Worksheet
Name ___________________________
Match each definition to its corresponding term.
1.
2.
3.
4.
5.
Angle Bisector/Proportional Side Theorem
Triangle Proportionality Theorem
Converse of the Triangle Proportionality Theorem
Proportional Segments Theorem
Triangle Midsegment Theorem
__________
__________
__________
__________
__________
a. If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides
proportionally.
b. A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in
the same ratio as the lengths of the sides adjacent to the angle.
c. If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
d. The midsegment of a triangle is parallel to the third side of the triangle and half the measure of the
third side of the triangle
e. If three parallel lines intersect two transversals, then they divide the transversals proportionally.
Calculate the length of the indicated segment in each figure. #6 - 10
6.
bisects
. Calculate
7.
HF.
9.
bisects
XW.
bisects
. Calculate
AD.
. Calculate
10.
bisects
. Calculate
SP.
bisects
FD.
8.
. Calculate
11. Use the Triangle
Proportionality Theorem
and the Proportional
Segments Theorem to
determine the missing
value.
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Secondary II
Use the Triangle Proportionality Theorem and the Proportional Segments Theorem to determine the missing value.
12.
13.
14. Jimmy is hitting a golf ball towards the hole. The line from Jimmy
to the hole bisects the angle formed by the lines from Jimmy to the
oak tree and from Jimmy to the sand trap. The oak tree is 200 yards
from Jimmy, the sand trap is 320 yards from Jimmy, and the hole is
250 yards from the sand trap. How far is the hole from the oak tree?
15. The road from Central City on the map shown bisects the angle
formed by the roads from Central
City to Minville and from Central City to Oceanview. Central City is
12 miles from Oceanview, Minville is 6 miles from the beach, and
Oceanview is 8 miles from the beach. How far is Central City from
Minville?
Use the diagram and given information to write two statements that can be justified using the Triangle Midsegment
Theorem. Hint: parallel lines, length of bases. Give the side length asked for.
16.
17.
If DE is 4.3 cm, AC = ?
If RV = 7, RS = 22, RW = ?
Given: RST is a triangle
Given: ABC is a triangle
V is the midpoint of
D is the midpoint of
W is the midpoint of
E is the midpoint of BC
31
Secondary II
18. The sides of triangle LMN have midpoints
, and
Use each similarity statement to write the corresponding
sides of the triangles as proportions.
. Compare
19.
the length of
to the length of
.
20.
21. Solve for x.
23. Solve for x.
22. Solve for x.
24. Solve for x, y, and z.
25. Solve for x, y, and z.
26. Marsha wants to walk from the parking lot
through the forest to the clearing, as shown in
the diagram. She knows that the forest ranger
station is 154 feet from the flag pole and the
flag pole is 350 feet from the clearing. How
far is the parking lot from the clearing?
32
Secondary II
27. Discuss the similarities and differences in the graphs of the two exponential functions. Use complete
sentences.
y  5 x and y  5x  2  3
28. Given:
List the Following:
a) 2 pairs of corresponding angles
b) 2 pairs of vertical angles
c) 1 pair of alternate exterior angles
d) 1 pair of same side exterior angles
e) If the lines are parallel and 𝑚∠2 = 51°, record the
measure of all the angles in the diagram.
29. Given the diagram:
30. Would following side lengths make a triangle?
21, 13, 33
If yes, give a name for the type of triangle.
𝑚∠𝑊𝑋𝑍 = 131°, and 𝑚∠𝑌 = 42°, Find the
measure of angle Z. Show your work.
31. Do the following side lengths form a right triangle? Justify your answer/show work!
12, 13, 5
33
Secondary II
4.5/4.6 - Worksheet
Name ___________________________
Explain how you know that each pair of triangles is similar.
1.
2.
3.
4. Elly and Jeff are on opposite sides of a canyon that runs east to west,
according to the graphic. They want to know how wide the canyon is.
Each person stands 10 feet from the edge. Then, Elly walks 24 feet west,
and Jeff walks 360 feet east. What is the width of the canyon?
5. Zoe and Ramon are hiking on a glacier. They become separated by a
crevasse running east to west. Each person stands 9 feet from the edge.
Then, Zoe walks 48 feet east, and Ramon walks 12 feet west. What is the
width of the crevasse?
6. Minh wanted to measure the height of a statue. She lined herself up with
the statue’s shadow so that the tip of her shadow met the tip of the statue’s
shadow. She marked the spot where she was standing. Then, she measured
the distance from where she was standing to the tip of the shadow,
and from the statue to the tip of the shadow. What is the height of the statue?
6. Dimitri wants to measure the height of a palm tree. He lines himself up
with the palm tree’s shadow so that the tip of his shadow meets the tip of the
palm tree’s shadow. Then, he asks a friend to measure the distance from
where he was standing to the tip of his shadow and the distance from the palm
tree to the tip of its shadow. What is the height of the palm tree?
7. Andre is making a map of a state park. He finds a small bog, and he wants to
measure the distance across the widest part. He first marks the points A, C, and
E. Andre measures the distances shown on the image. Andre also marks point B
along AC and point D along AE, such that BD is parallel to CE. What is the
width of the bog at the widest point?
34
Secondary II
8. Shira finds a tidal pool while walking on the beach. She wants to know the maximum
width of the tidal pool. Using indirect measurement, she begins by marking the points A,
C, and E. Shira measures the distances shown on the image. Next, Shira marks point B
along AC and point D along AE, such that BD is parallel to CE. What is the distance
across the tidal pool at its widest point?
9. Keisha is visiting a museum. She wants to know the height of one of the
sculptures. She places a small mirror on the ground between herself and the
sculpture, then she backs up until she can see the top of the sculpture in
the mirror. What is the height of the sculpture?
10. Micah wants to know the height of his school. He places a small
mirror on the ground between himself and the school, then he backs
up until he can see the highest point of the school in the mirror.
What is the height of Micah’s school?
Review.
11. Marsha wants to walk from the parking lot through the forest to the
clearing, as shown in the diagram. She knows that the forest
ranger station is 154 feet from the flag pole and the flag pole is
350 feet from the clearing. How far is the parking lot from the
clearing?
13. Use the Right Triangle/Altitude
Similarity Theorem to write three
14. Given XU = 9, XY = 18, ZT = 7.5, ZY = 15, TU = 10.5,
ZX = ?
similarity statements involving the
triangles in the diagram.
35
Secondary II
15. Find the missing value.
17.
has vertices J(6, 2), K(1, 3), and
16.
bisects
. Calculate YW.
18. Given: xy = 10. Find all the missing side lengths.
L(7, 0). What are the vertices of the image
y
after a dilation with a scale factor of 12
60
using the origin as the center of dilation?
45
x
z
w
19. Given 𝑚∠𝑌 = 47°, 𝑚∠𝑍 = 69°.
20. Would it be possible to form a triangle with segments that
are 4 cm, √3 cm, and √7 cm? Justify with work.
21. Would the side lengths above form a right triangle?
Justify with work.
What side length of the triangle is the longest?
Give the measure of angle ZXW.
22. Factor: 9a3b5  6a 2b4
24. Given: 𝑚∠6 = 122° and line P is parallel to line R. Find
the measures of all remaining angles in the diagram.
23. Multiply: 2 4  5 4
25. Prove the Pythagorean Theorem using similar triangles.
Given: ∆ABC with right angle C
Place side a along the diameter of a circle of radius c so that B is at the
center of the circle.
36
Secondary II
Review - Chapter 3/4
Name __________________________
1. Determine the measure of the missing angle in each triangle.
a.
b.
w
y
z
2. List the side lengths from shortest to longest for each diagram.
a.
b.
3. What is the order from least to greatest of the measures of angles of the triangle shown?
4. Identify the interior angles, the exterior angle, and the remote interior angles of each triangle.
5. Solve for x in each diagram.
a.
b.
6. Determine whether it is possible to form a triangle using each set of segments with the given
measurements. Explain your reasoning
a.
8 feet, 9 feet, 11 feet
b. 4 meters, 5.1 meters, 12.5 meters
c. 10 yards, 5 yards, 21 yards
37
Secondary II
7. Determine the length of the missing sides for each special right triangle. Write your answer as a
radical in simplest form.
a.
b.
c.
d.
8. Prospect Park is a square with side lengths of 512 meters. One of the paths through the park runs diagonally
from the northeast corner to the southwest corner, and it divides the park into two 45 0 – 45 0 – 900 triangles.
How long is that path?
9. Determine the area the triangle.
10. Universal Sporting Goods sells pennants in the shape of 30º– 60º– 90º triangles. The length of the longest side
of each pennant is 8 inches. What is the area of the pennant?
11. Given the image and the pre-image, determine the scale factor.
a.
b.
38
Secondary II
12. ∆GHI has vertices G(0, 5), H(4, 2), and I(3, 3). What are the vertices of the image after dilation with a scale
factor of 9 using the origin as the center of dilation?
13. Explain how you know that the triangles are similar.
a.
b.
14. What information would you need to use the Angle-Angle Similarity Theorem
to prove that the triangles are similar?
15. What information would you need to use the Side-Angle-Side
Similarity Theorem to prove that the triangles
are similar?
16. What information would you need to use the Side-Side-Side Similarity Theorem to prove that the triangles are
similar?
17. Determine whether each pair of triangles is similar. Explain your reasoning.
a.
b.
c.
39
Secondary II
18. Calculate the length of the indicated segment in the figure.
a.
b.
c.
19. On the map shown, Willow Street bisects the angle formed by Maple Avenue and South Street. Mia’s house is
5 miles from the school and 4 miles from the fruit market. Rick’s house is 6 miles from the fruit market. How
far is Rick’s house from the school?
20. Explain how you know how the pair of triangles is similar.
21. You want to measure the height of a tree at the community park. You stand in the tree’s shadow so that they
tip of your shadow meets the tip of the tree’s shadow on the ground, 2 meters from where you are standing.
The distance from the tree to the tip of the tree’s shadow is 5.4 meters. You are 1.25 meters tall. Draw a
diagram to represent the situation. Then, determine the height of the tree.
40
Secondary II
Worksheet - After Chapter 3/4 - TEST
Name __________________________________
Parallel Lines with Transversals, Factoring & Simplifying
1. Given two parallel lines crossed by a transversal, identify the relationship of each pair of marked angles.
a.
b.
c.
2. Find the measure of each angle indicated.
a.
c.
3. Solve for x.
a.
b.
d.
b.
41
Secondary II
4. Find the measure of the angle indicated in bold.
a.
c.
b.
d.
5. Write the prime factorization of each. Do not use exponents.
a.
b.
c.
d.
e.
f.
6. Find the GCF of each.
a.
7. Factor by finding the GCF.
a.
b.
c.
b.
c.
d.
e.
f.
42
Secondary II
5.1-5.6 - Worksheet
Name ___________________________
1. List the corresponding sides and angles, using congruence symbols, for each pair of triangles
represented by the given congruence statement.
a. ∆𝐴𝐸𝑈 ≅ ∆𝐵𝐶𝐷
b. ∆𝐽𝐾𝐿 ≅ ∆𝑅𝑆𝑇
2. Determine whether each pair of given triangles are
congruent by SSS. Use the Distance Formula and a
protractor when necessary.
3. Perform the transformation described. Then, verify that the
triangles are congruent by SSS. Use the Distance
Formula and a protractor when necessary.
Rotate ∆𝐷𝐸𝐹 180° clockwise about the origin to form
∆𝑄𝑅𝑆. Verify that ∆𝐷𝐸𝐹 ≅ ∆𝑄𝑅𝑆 by SSS.
4.
43
Secondary II
5.
6. Translate ∆𝐷𝐸𝐹 11 units to the left and 10 units down to
form ∆𝑄𝑅𝑆. Verify that ∆𝐷𝐸𝐹 ≅ ∆𝑄𝑅𝑆 by SAS.
7. Determine the angle measure or side measure that is needed in order to prove that each set of
triangles are congruent by SAS.
a.
b.
c.
d.
8. Determine whether there is enough information to prove that each pair of triangles are congruent by SSS or
SAS. Write the congruence statements to justify your reasoning.
a.
b.
44
Secondary II
c.
d.
9.
10.
11.
45
Secondary II
12. Determine the angel measure or side measure that is needed in order to prove that each set of
triangles are congruent by ASA.
a.
b.
c.
13.
d.
14.
15.
46
Secondary II
16. Determine the angle measure or side measure that is needed in order to prove that each set of
triangles are congruent by AAS.
a.
b.
c.
d.
17. Determine whether there is enough information to prove that each pair of triangles are congruent
by ASA or AAS. Write the congruence statements to justify your reasoning.
a.
b.
c.
d.
47
Secondary II
5.7 - Worksheet
Name _______________________________
1. Given: ̅̅̅̅
𝐴𝐵 ⊥ ̅̅̅̅
𝐶𝑅 , ∠𝐴 ≅ ∠𝐵
Prove: ∆𝐶𝐴𝑅 ≅ ∆𝐶𝐵𝑅
4. Given: 𝐸 is the midpoint of ̅̅̅̅
𝐴𝐺 and ̅̅̅̅̅
𝑀𝑇
Prove: ∆𝐴𝑇𝐸 ≅ ∆𝐺𝑀𝐸
M
B
A
C
A
Statements
E
R
T
Reasons
Statements
Reasons
5. Given: 𝐶 is the midpoint of ̅̅̅̅
𝐵𝐸 and ∠𝐵 ≅ ∠𝐸
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐶
E
̅̅̅̅ , 𝑁 is the midpoint of ̅̅
̅̅̅
2. Given: ̅̅̅̅
𝐵𝐸 ≅ ̅𝐾𝐸
𝐵𝐾
Prove: ∆𝐵𝐸𝑁 ≅ ∆𝐾𝐸𝑁
E
A
B
C
D
K
N
Statements
G
Reasons
B
Statements
̅̅̅̅ ≅ ̅̅̅̅̅
̅̅̅̅ bisects ∠𝐴𝐾𝐷
3. Given: ̅𝐴𝐾
𝐷𝐾 , ̅𝑅𝐾
Prove: ∆𝐴𝑅𝐾 ≅ ∆𝐷𝑅𝐾
A
R
D
Reasons
̅̅̅̅, 𝑀𝐶
̅̅̅̅̅ ≅ 𝐸𝐶
̅̅̅̅
6. Given: ̅̅̅̅̅
𝐴𝑀 ≅ 𝐴𝐸
Prove: ∆𝑀𝐴𝐶 ≅ ∆𝐸𝐴𝐶
A
M
E
K
Statements
C
Reasons
Statements
Reasons
48
Secondary II
7. Given: ̅̅̅̅
𝐵𝐷 bisects ∠𝐴𝐵𝐶 and ∠𝐴𝐷𝐶
Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐷
B
10. Given: ̅̅̅
𝐾𝐽 ≅ ̅̅̅̅̅
𝑁𝑀
Prove: ∆𝐽𝐾𝐿 ≅ ∆𝑀𝑁𝐿
K
A
C
L
D
Statements
M
J
Reasons
N
Statements
̅̅̅̅, ̅̅̅̅
8. Given: ̅̅̅̅
𝑋𝑌 ≅ 𝑅𝑇
𝑋𝑍 ≅ ̅̅̅
𝑅𝑆̅
̅̅̅̅ and ̅̅̅̅
11. Given: ̅𝐷𝐹
𝐸𝐹 are legs of an isosceles ∆
∠𝐷𝐹𝐺 ≅ ∠𝐸𝐹𝐺
Prove: ∆𝑋𝑌𝑍 ≅ ∆𝑅𝑇𝑆
T
Reasons
Z
X
Prove: ∆𝐷𝐹𝐺 ≅ ∆
D
F
E
G
R
S
Statements
Y
̅̅̅̅ ≅ 𝐶𝐸
̅̅̅̅
9. Given: ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐷 ≅ 𝐶𝐵
̅̅̅̅ and 𝐵𝐷
̅̅̅̅ are two legs of an isosceles ∆
𝐵𝐸
Prove: ∆𝐴𝐵𝐷 ≅ ∆𝐶𝐵𝐸
C
B
E
Statements
Statements
Reasons
Reasons
P
Q
R
S
A
D
Reasons
12. Given: ̅̅̅̅
𝑃𝑄 ∥ ̅̅̅̅
𝑆𝑅 , ̅̅̅̅
𝑃𝑄 ⊥ ̅̅̅̅
𝑄𝑆 , ̅̅̅̅
𝑃𝑅 ⊥ ̅̅̅̅
𝑅𝑆
Prove: ∆𝑃𝑄𝑆 ≅ ∆𝑆𝑅𝑃
Statements
Reasons
49
Secondary II
̅̅̅̅ ≅ ̅̅̅̅
̅̅̅̅ ⊥ ̅̅̅̅
13. Given: ̅𝐷𝐸
𝐸𝐺 , ̅̅̅̅
𝐹𝐺 ⊥ ̅̅̅̅
𝐸𝐺 , ̅𝐷𝐸
𝐹𝐺
Prove: ∆𝐷𝐸𝐺 ≅ ∆𝐹𝐺𝐸
E
15. Given: ̅̅̅̅̅
𝐾𝑀 bisects ∠𝐽𝑀𝐿
Prove: ∆𝐽𝐾𝑀 ≅ ∆𝐿𝐾𝑀
D
K
J
G
L
F
Statements
Reasons
M
Statements
14. Given: ∠𝑁 ≅ ∠𝐷, ∠𝑁𝑀𝑍 ≅ ∠𝐷𝑍𝑀
Prove: ∆𝑁𝑀𝑍 ≅ ∆𝐷𝑍𝑀
M
D
Reasons
̅̅̅̅ ≅ 𝐷𝐵
̅̅̅̅, 𝐵𝐶
̅̅̅̅ ≅ 𝐵𝐸
̅̅̅̅ , ∠𝐴𝐵𝐶 ≅ ∠𝐷𝐵𝐸
16. Given: 𝐴𝐵
Prove: ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐵𝐸
A
B
N
Z
C
E
Statements
Reasons
D
Statements
Reasons
50
Secondary II
6.1/6.2 Worksheet
Name _________________________
Choose the diagram that models each right triangle congruence theorem.
1. Hypotenuse-Leg (HL) Congruence Theorem
__________
2. Leg-Leg (LL) Congruence Theorem
__________
3. Hypotenuse-Angle (HA) Congruence Theorem
__________
4. Leg-Angle (LA) Congruence Theorem
__________
5. Mark the appropriate sides to make each congruence statement true by the Hypotenuse-Leg Congruence
Theorem.
a.
b.
6. Mark the appropriate sides to make each congruence statement true by the Leg-Leg Congruence Theorem.
a.
b.
7. Mark the appropriate sides and angles to make each congruence statement true by the Hypotenuse-Angle
Congruence Theorem.
a.
b.
51
Secondary II
8. Mark the appropriate sides and angles to make each congruence statement true by the Leg-Angle
Congruence Theorem.
a.
b.
9. For each figure, determine if there is enough information to prove that the two triangles are congruent. If
so, name the congruence theorem used.
a.
b.
c.
10. An auto dealership displays one of their cars by driving
it up a ramp onto a display platform. Later they will
drive the car off the platform using a ramp on the
opposite side. Both ramps form a right triangle with
the ground and the platform. Is there enough
information to determine whether the two ramps
have the same length? Explain.
52
Secondary II
11. Two ladders resting on level ground are leaning against the side of a house. The bottom of
each ladder is exactly 2.5 feet directly out from the base of the house. The point at which
each ladder rests against the house is 10 feet directly above the base of the house. Is there
enough information to determine whether the two ladders have the same length? Explain.
12. Create a two-column proof to prove each statement.
a.
Statement
Reason
Statement
Reason
Statement
Reason
Statement
Reason
b.
c.
d.
53
Secondary II
13. Use the given information to answer each question.
a. Calculate MR given that the perimeter of ∆HMR is 60 centimeters.
b. Greta has a summer home on Lake Winnie. Using the diagram, how
wide is Lake Winnie?
c. Given rectangle ACDE, calculate the measure of ∠𝐶𝐷𝐵.
14. Are the two triangles congruent by SSS? Use the distance formula to prove.
15. Determine the length of the missing sides for each special right triangle. Write your answer as a radical in
simplest form.
a.
b.
c.
54
Secondary II
16. Identify the types of angles given the following picture. Determine the
angle relationships.
a. Alternate interior angles
b. Alternate exterior angles
c. Corresponding Angles
d. Same side interior angles
17. Solve and graph the inequality  6x  7  2x  17
18. Tell whether ordered pair (4, 1) is a solution of the system. Show your work.
f(x) = {
x  2y  6
3 x  y  11
2𝑥 − 3𝑦 ≤ 12
19. Sketch the solution to the system of inequalities 𝑓(𝑥) = {
𝑥 + 5𝑦 < 20
55
Secondary II
6.3/6.4 - Worksheet
Name _________________________
1. Choose the term from the box that best completes each sentence.
a. A(n) __________________________________________________is the angle formed by the two
congruent legs in an isosceles triangle.
b. In an isosceles triangle, the altitudes to the congruent sides are congruent, as stated in the
__________________________________________________.
c. In an isosceles triangle, the angle bisectors to the congruent sides are congruent, as stated in
the__________________________________________________.
d. The__________________________________________________ states that the altitude from the vertex
angle of an isosceles triangle is the perpendicular bisector of the base.
e. The __________________________________________________states that the altitude to the base of
an isosceles triangle bisects the base.
f. The altitude to the base of an isosceles triangle bisects the vertex angle, as stated in the
__________________________________________________.
2. Write the theorem that justifies the truth of each statement.
a.
b.
3. Determine the value of x in each isosceles triangle.
a.
b.
4. Use the given information to answer each question.
a. When building a house, rafters are used to support the roof. The rafter shown in the diagram has the
shape of an isosceles triangle. In the diagram, 𝑁𝑃 ⊥ 𝑅𝑄, 𝑁𝑅 ≅ 𝑁𝑄, NP =12 feet, and RP=16 feet. Use this
̅̅̅̅. Explain.
information to determine the length of 𝑁𝑄
56
Secondary II
b. While growing up, Nikki often camped out in her back yard in a pup tent.
A pup tent has two rectangular sides made of canvas, and a front and
back in the shape of two isosceles triangles also made of canvas. The
zipper in front, represented by ̅̅̅̅̅
𝑀𝐺    in the diagram, is the height of the
pup tent and the altitude of isosceles ∆𝐸𝑀𝐻. If the length of̅̅̅̅̅
𝐸𝐺   is 2.5
̅̅̅̅ ? Explain.
feet, what is the length of 𝐻𝐺
5. For each pair of triangles, use the Hinge Theorem or its converse to write a conclusion using an inequality.
a.
b.
6. Which of the following is not a geometric sequence?
a. 424, 106, 26.5, 6.625, . . .
b.
5
2
, 5, 10, 20, . . .
c. 8, -16, 32, -64, . . .
d. -7, -8, -9, -10 . . .
7. Solve each system of equations by elimination.
a.
3x  5 y  8

2 x  5 y  22
10 x  6 y  6
5 x  5 y  5
b. 
8. Solve.
a. –3x + 7 = 31
b. 3(5  4 x  4 x)  9(2 x  8)  21
9. Graph the solution of the inequality −𝑦 + 15 ≥ −3 − 2𝑦 ?
<---------------------------------------------->
10. Write a statement that indicates that the triangles are congruent
11. Solve for x in each diagram.
a.
b.
c.
57
Secondary II
Review - Chapter 5/6
Name ___________________________
1. Which set of congruence statements show that ∆𝑅𝑇𝑆 ≅ ∆𝑉𝑋𝑊 by the AAS Congruence Theorem?
2.
3. List the corresponding sides and angles, using congruence symbols, for each pair of triangles if
4. Are the two triangles congruent by SSS? Use the distance formula to prove.
58
Secondary II
5. Are the two triangles congruent by SAS? Use the
distance formula and a protractor.
6. Are the two triangles congruent by ASA? Use the distance
formula and a protractor.
For the following problems, prove the triangles are congruent by SSS, SAS, AAS, ASA.
̅̅̅̅
7. Given: ̅̅̅̅
𝐵𝐸 ≅ ̅̅̅̅
𝐾𝐸 ; 𝑁 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐾
8.
Prove: ∆𝐵𝐸𝑁 ≅ ∆𝐾𝐸𝑁
59
Secondary II
9. Given: RS bisects ∠𝑄𝑅𝑇 𝑎𝑛𝑑 ∠𝑄𝑆𝑇
Prove: ∆QRS ≅ ∆TRS
11.
Prove: ∆𝐴𝐶𝐵 ≅ ∆𝐷𝐶𝐸
Triangle BUN is isosceles with ̅̅̅̅
𝐵𝑈 ≅ ̅̅̅̅
𝐵𝑁 . What additional given information is needed to prove
∆𝐵𝑈𝐺 ≅ ∆𝐵𝑁𝐴 by ASA?
a. ∠𝑈𝐵𝐴
b. ∠𝑈𝐵𝐺
c. ∠𝐵𝐺𝑈
d. ∠𝐵𝑈𝐺
12.
10. Given: C is the midpoint of BE; ∠𝐴 ≅ ∠𝐷
≅ ∠𝑁𝐵𝐺
≅ ∠𝑁𝐵𝐴
≅ ∠𝐵𝐴𝑁
≅ ∠𝐵𝑁𝐴
̅̅̅̅ ≅ 𝑅𝑆
̅̅̅̅. Which of the following is not true?
In the figure shown, 𝐷𝑇
a. ∠𝑆 ≅ ∠𝑇
̅̅ ≅ 𝐺𝑇
̅̅̅̅
b. ̅̅
𝐺𝑆
c. ∆𝐺𝑆𝑇 is not isosceles.
d. ∠𝐷𝐺𝑇 ≅ ∠𝑅𝐺𝑆
Matching
Choose the diagram that models each right triangle congruence theorem.
13. Hypotenuse-Leg (HL) Congruence Theorem
14. Leg-Leg (LL) Congruence Theorem
15.
Hypotenuse-Angle (HA) Congruence Theorem
16. Leg-Angle (LA) Congruence Theorem
60
Secondary II
Free Response
17. In the figure shown, ̅̅̅̅
𝑄𝑋 ⊥ ̅̅̅̅
𝑋𝑃, ̅̅̅̅
𝑅𝑃 ⊥ ̅̅̅̅
𝑋𝑃, ̅̅̅̅
𝑄𝑇 ≅ ̅̅̅̅
𝑅𝑆, 𝑎𝑛𝑑 ̅̅̅̅
𝑋𝑆 ≅ ̅̅̅̅
𝑇𝑃 . Determine whether ∆𝑋𝑄𝑇
is congruent to ∆𝑃𝑅𝑆. Explain your reasoning.
18. Using the figure shown, determine whether ∆𝐻𝐴𝑊 ≅ ∆𝑇𝐴𝑊. Explain your reasoning.
19.
Statement
20.
Reason
Mark the appropriate sides to make each congruence statement true by the Hypotenuse-Leg
Congruence Theorem.
a.
b.
21. Mark the appropriate sides to make each congruence statement true by the Leg-Leg Congruence
Theorem.
a.
b.
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Secondary II
22. Mark the appropriate sides and angles to make each congruence statement true by the HypotenuseAngle Congruence Theorem.
a.
b.
23. Mark the appropriate sides and angles to make each congruence statement true by the Leg-Angle
Congruence Theorem.
a.
b.
24. For each figure, determine if there is enough information to prove that the two triangles are
congruent. If so, name the congruence theorem used.
a.
b.
Determine if the following triangles are congruent. If so, determine if they are congruent by SSS, SAS, ASA,
or AAS. Write the congruence statements to justify your reasoning.
62
Secondary II
31.
32. Determine the value of x in the isosceles triangle.
33.
.
63
Secondary II
Review
1. For each pair of triangles, tell if you are using the Hinge Theorem or its converse, then write a
conclusion using an inequality,
a.
b.
2. Write a Given statement and state the theorem that proves the triangles are congruent. Then, write a
congruence statement.
Given: _____________________________________________
Theorem: __________________________________________
Statement: _________________________________________
3. Determine the relationship between ∠4 𝑎𝑛𝑑 ∠8 and wriite a postulate or theorem that justifies your
answer.
_________________________________________________
________________________________________________
Use the graph for question 4.
4.
a. Rotate ∆𝐴𝐵𝐶, about the origin, 270° clockwise,
to create ∆𝑋𝑌𝑍.
b. Translate ∆𝐴𝐵𝐶 up 12 and left 14 to create ∆𝑅𝑆𝑇.
c. Reflect ∆𝐴𝐵𝐶 across the y-axis to create ∆𝐷𝐸𝐹
5. Find the distance and the midpoint of A (-1, 5) and B (11, -3). Do not use decimals.
64
Secondary II
Worksheet – After Chapter 5/6 Test
Name _____________________
1.
2.
3.
4.
5. Graph the image of the figure using the transformation given.
a.
b.
6. Find the GCF of each.
a.
b.
c.
d.
7. Each pair of figures is similar. Find the missing side.
a.
b.
65
Secondary II
7.1/7.2/7.3 - Worksheet
Name ___________________________
Use the given statements and the Perpendicular/Parallel Line Theorem to identify the pair of parallel lines in each
figure.
1.
2.
3.
Complete each statement for square GKJH.
Complete each statement for rectangle TMNU.
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Secondary II
Complete each statement for parallelogram MNPL.
Complete each statement for rhombus UVWX.
Determine the missing statement needed to prove each quadrilateral is a parallelogram by the
Parallelogram/Congruent-Parallel Side Theorem.
Complete each statement for kite PRSQ.
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Secondary II
Write the term from the box that best completes each statement.
26. The ________________________ are either pair of angles of a trapezoid that share a base as a common side.
27. A(n) ________________________ is a trapezoid with congruent non-parallel sides.
28. A(n) _________________________ is a statement that contains if and only if.
29. The _________________________ of a trapezoid is a segment formed by connecting the midpoints of the legs
of a trapezoid.
Complete each statement for trapezoid UVWX.
Use the given figure to answer each question.
33.
34.
35.
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Secondary II
36.
37.
38.
39.
40. Simon connected a square and two congruent right triangles together to form an isosceles trapezoid. Draw a
diagram to represent the isosceles trapezoid.
69
Secondary II
7.4/7.5 - Worksheet
Name ___________________________
Draw all possible diagonals from vertex A for each polygon. Then write the number of triangles formed by the
diagonals.
4.
5.
6.
Calculate the sum of the interior angle measures of each polygon.
7. A polygon has 8 sides.
8. A polygon has 16 sides.
9. A polygon has 25 sides.
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Secondary II
The sum of the measures of the interior angles of a polygon is given. Determine the number of sides for each
polygon.
10. 1800°
11. 1260°
12. 6840°
For each regular polygon, calculate the measure of each of its interior angles.
Calculate the number of sides for each polygon.
16. The measure of each angle of a regular polygon is 156°.
17. The measure of each angle of a regular polygon is 162°.
18. The measure of each angle of a regular polygon is 165.6°.
Extend each vertex of the polygon to create one exterior angle at each vertex.
Calculate the sum of the measures of the exterior angles for each polygon.
22. hexagon
23. Nonagon
24. 150-gon
Given the measure of an interior angle of a polygon, calculate the measure of the adjacent exterior angle.
25. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 120°?
26. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that
measures 135°?
Given the regular polygon, calculate the measure of each of its exterior angles.
27. What is the measure of each exterior angle of a regular pentagon?
28. What is the measure of each exterior angle of a regular octagon?
29. What is the measure of each exterior angle of a regular 12-gon?
Calculate the number of sides of the regular polygon given the measure of each exterior angle.
30. 90°
31. 60°
32. 30°
71
Secondary II
7.6 - Worksheet
Name ___________________________
List all of the quadrilaterals that have the given characteristics.
1. diagonals congruent
2. no parallel sides
3. diagonals bisect each other
4. all angles congruent
5. two pairs of parallel sides
Identify all the terms from the following list that apply to each figure: quadrilateral, parallelogram, rectangle, square,
trapezoid, rhombus, kite.
6.
7.
8.
9.
10.
Give the most specific name that best describes each quadrilateral. Explain your answer.
11.
12.
14.
13.
15.
Tell whether each statement is true or false. If false, explain why.
16. A square is also a rhombus.
17. Diagonals of a rectangle are perpendicular.
18. A parallelogram has exactly one pair of opposite angles congruent.
19. A square has diagonals that are perpendicular and congruent.
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Secondary II
20. All quadrilaterals have supplementary consecutive angles.
21. Calculate the sum of the interior angles of a polygon that has 13 sides.
22. How many sides does a polygon have if the sum of the measures of the interior angles is 3240°?
23. Calculate the measure of the interior angles for the given polygon.
24. Calculate the number of sides a regular polygon has if the measure of each angle is 160°.
25. Calculate the sum of the measures of the exterior angles of a 20-gon.
26. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 108°?
27. What is the measure of each exterior angle of a regular decagon?
28. Calculate the number of sides a regular polygon has if each exterior angle measures 40°.
73
Secondary II
Review - Chapter 7
1. Quadrilateral VWXY is a square
a)
b)
c)
d)
e)
Name all parallel segments.
Name all congruent segments.
Name all right angles.
Name all congruent angles.
Name all congruent triangles.
Name ___________________________
2. Quadrilateral PQRS is a rectangle
a) Name all parallel segments.
b) Name all congruent segments.
c) Name all right angles.
d) Name all congruent angles.
e) Name all congruent triangles.
3. Quadrilateral PLGM is a parallelogram
4. Quadrilateral RHMB is a rhombus
a) If m∠𝑃𝐿𝐺 = 124°, what is 𝑚∠𝐺𝑀𝑃?
a) If 𝑚∠𝐻𝑅𝐵 𝑖𝑠 70°, what is 𝑚∠𝐻𝑀𝐵?
b) If 𝑚∠𝐿𝑃𝑀 = 56°, what is 𝑚∠𝐿𝐺𝑀?
b) If 𝑚∠𝑅𝐻𝐵 = 55°, what is ∠𝑀𝐻𝐵?
̅̅̅̅ = 20 meters, what is 𝑀𝑃
̅̅̅̅̅?
c) If 𝐿𝐺
̅̅̅̅ = 25 𝑓𝑒𝑒𝑡, what is 𝐻𝑅?
̅̅̅̅̅̅
c)If 𝑅𝐵
d) If ̅̅̅̅
𝑃𝑅 = 12 𝑖𝑛𝑐ℎ𝑒𝑠, what is ̅̅̅̅
𝐺𝑅?
d) If ̅̅̅̅
𝐻𝑆 = 18 𝑐𝑚, what is ̅̅̅̅
𝑆𝐵?
e) What is 𝑚∠𝑅𝑆𝐵?
5. Quadrilateral ABCD is a kite
6. Quadrilateral WXYZ is an isosceles trapezoid
a) If 𝑚∠𝐴𝐵𝐶 = 95°, what is 𝑚∠𝐴𝐷𝐶?
a) If 𝑚∠𝑋𝑊𝑍 = 66°, what is 𝑚∠𝑌𝑍𝑊?
b) If 𝑚∠𝐵𝐶𝐸 = 34°, what is 𝑚∠𝐸𝐵𝐶?
̅̅̅̅̅ = 10 𝑖𝑛𝑐ℎ𝑒𝑠, what is 𝑍𝑋
̅̅̅̅ ?
b) If 𝑊𝑌
c) If ̅̅̅̅
𝐴𝐵 = 16 𝑓𝑒𝑒𝑡, what is ̅̅̅̅
𝐴𝐷?
c) If ̅̅̅̅̅
𝑊𝑋 = 7 𝑖𝑛𝑐ℎ𝑒𝑠, what is ̅̅̅̅
𝑍𝑌 ?
d) If ̅̅̅̅
𝐵𝐷 = 25 𝑓𝑒𝑒𝑡, what is ̅̅̅̅
𝐸𝐷 ?
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Secondary II
7. Determine each measure of a regular nonagon.
8. Determine each measure of a regular 15-gon.
a) The sum of the interior angles.
a) The sum of the interior angles.
b) The measure of an interior angle.
b) The measure of an interior angle.
c) The measure of an exterior angle.
c) The measure of an exterior angle.
Determine the measure of each missing angle in each figure.
9.
10.
11. Use the figure to answer each question.
a) What is the sum of the measures of the interior angles?
b) What is the value of x?
c) What is the measure of ∠𝑅𝑄𝑃?
12. Suppose that the measure of each interior angle of a regular polygon is 157.5°. Classify the polygon.
13. Suppose that the measure of each exterior angle of a regular polygon is 22.5°. Classify the polygon.
14. Find the measure of the midsegment.
15. Find the value of x.
List all types of quadrilaterals with the given characteristics.
16. The quadrilateral has four congruent sides.
17. Exactly one pair of opposite sides of the quadrilateral is parallel.
18. Opposite angles of the quadrilateral are congruent.
19. Exactly two pairs of adjacent sides are congruent.
20. The sum of the measures of the exterior angles of the quadrilateral is 360°.
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Secondary II
21. The diagonals of the quadrilateral do not bisect each other.
22. Quadrilateral ABCD has congruent diagonals that are perpendicular to each other.
23. Quadrilateral JKLM has consecutive vertex angles that are supplementary but not congruent. If the diagonals
bisect vertex angles, what type of quadrilateral is JKLM?
24.
25.
26.
27.
28.
29.
30.
76
Secondary II
Worksheet – After Chapter 7 Test
Name ______________________
1. Solve each equation.
a.
d.
b.
2. Evaluate each using the values given.
a.
3. Sketch the graph of each line.
a.
c.
e.
b.
b.
4. Classify each angle as acute, obtuse, right, or straight.
a.
b.
c.
d.
5. Name each angle in four ways.
a.
b.
d.
6. Find the measure of each angle indicated.
a.
b.
c.
c.
77
Secondary II
8.1 - Worksheet
Name ___________________________
1. Use the Pythagorean Theorem to find the missing length.
a.
b.
c.
e.
f.
g.
d.
h.
2. Do the following lengths forma right triangle?
a.
b.
c.
d. a = 6.4, b = 12, c = 12.2
3. Find the value of each trigonometric ratio.
a.
b.
c.
d.
e.
f.
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Secondary II
g.
h.
i.
3. For the following, draw and label a right triangle that contains the given information. Find the trig ratios for
the indicated angles.
a.
𝑎 = 21, 𝑏 = 20, 𝑐 = 29, ∠𝐶 = 90°
b. 𝑥 = 12, 𝑦 = 35, 𝑧 = 37, ∠𝑍 = 90°
sin A=_____, cos A= _____, tan A=_____
sin X=_____, cos X= _____, tan X= _____
sin B=_____, cos B= _____, tan B=_____
sin Y=_____, cos Y= _____, tan Y= _____
79
Secondary II
8.2 - Worksheet
Name ___________________________
Rationalize the denominator. Simplify.
1
1.
2.
2
15
3.
5
8
18
4.
3 2
27
Find the missing sides using the Special Right Triangle Ratios.
A
B
5. 6.
7.
60
1
8.
1
45
C
B
C
A
9. Find the missing side of the triangle.
a.
b.
c.
d.
e.
f.
10. Find all missing acute angles for the given right triangles. You may need to use trig ratios (sin, cos or tan)
a.
b.
c.
∠A = __________
∠A = __________
∠A = __________
∠C = __________
∠C = __________
∠C = __________
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Secondary II
11. If (5, 12) is on the graph, find sin, cos, and tan.
12. A ladder 20 feet long is leaning against a building at a point 15 feet above the ground. What angle
does the ladder make with the ground? Round to the nearest degree.
13. Ben is flying a kite with 125 meters of string out. The string makes an angle of 39º with the level of
the ground. How high is the kite to the nearest meter?
14. You are standing 40 feet away from a building. What is the angle of elevation from the ground to the
top of the building if the height of the building is 115 ft?
15. During the construction of a house, a 6-foot-long board is used to support a wall. The board has an
angle of elevation from the ground to the wall of
. How far is the base of the wall from the board?
16. Use a calculator to solve. Round to the nearest hundredth.
a. cos 73°
b. tan 81°
c. sin 33°
d. tan 57°
17. Use your calculator to find the value of the following angles. Round to the nearest hundredth.
a. tan 𝑋 = 0.47
b. sin 𝐶 = 0.34
c. cos 𝜃 = 0.81
18. Assume a triangle has remote interior angles of 57° and 68°, what is the measure of the exterior
angle?
19. Calculate the sum of the interior angles of a heptagon. (Show your work!)
20. Calculate the measure of each interior angle of a regular heptagon. (Show your work!)
21. Calculate the sum of the exterior angles of a heptagon. (Justify your answer)
22. Calculate the measure of each exterior angle of a regular heptagon. (Show your work!)
81
Secondary II
8.3 - Worksheet
Name ___________________________
1. Draw the angles in standard position:
a. 23º
b. - 46º
c. 720º
d. - 216º
2. State the reference angle for each angle. (Hint: Sketch the angle in standard position)
a. -160°
b. 320°
c. 57°
d. 142°
3. Find the trig ratios without using a calculator. Draw the following angles in standard position, draw
the reference triangle, label the sides, and give the trig ratio. Rationalize answers.
a. sin 60º
b. cos (-120º)
c. tan 225º
d. tan 330º
e. cos 150º
f. tan (-240º)
g. sin 90º
h. cos (-180º)
i. sin 45º
j. cos (-45º)
k. tan (-90º)
l. tan 300º
4. If an angle is in standard position and contains the point ( -2, 5) , then cos  =______
5. You are standing 40 ft away from a building. The angle of elevation from the ground to the top of the building
is 57˚. What is the height of the building?
6. Jerome is flying a kite on the beach. The kite is attached to a 100-foot string and is flying 45 feet above the
ground. Calculate the measure of the angle formed by the string and the ground.
7. Quadrilateral ABCD is a kite.
a) If
, what is
? Explain.
b) If
, what is
? Explain.
c) If the length of
is 16 feet, what is AD? Explain.
d) If the length of
is 25 feet, what is ED? Explain.
8. What is the measure of each exterior angle in a regular nonagon?
9. Find the measure of angle β. Show your work!
84
2x-22
x+3

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Secondary II
Review - Chapter 8
Make sure you understand how to correctly complete EVERY problem on the review!!!
There will be a Calculator and a NON Calculator part of the test, make sure that you are following the
instructions for each problem.
1. Find the exact values of x and y. NO Calculators!
a.
b.
c.
2. Find the missing sides or angles. Calculator allowed. Round to 3 decimal places.
a.
b.
c.
d.
3. Find the third side of the triangle, and then evaluate the six trigonometric functions of
angle 𝜃. NO Calculator.
4. Draw the angles in standard position and state the reference angle for each angle.
a. 68°
b. - 130°
c. 760°
d. - 215°
e. 333°
5. If an angle is in standard position and contains the point ( 3, 6) , then cos 𝜃 =______ . NO Calculator.
6. Find the trig ratios without using a calculator. Draw the following angles in standard position, draw the
reference triangle, label the sides, and give the trig ratio. Rationalize answers.
a. sin 60º
b. cos (-120º)
c. tan 240º
d. sin 330º
e. cos 120º
f. tan (- 150º)
g. sin 330º
h. cos 210º
i. sin 45º
j. cos (-45º)
k. sin 225º
l. tan 300º
m. tan 90º
n. cos 270º
o. tan (- 360º)
p. sin 180º
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Secondary II
7. If θ is in standard position and includes the point (-3, -4), find sin, cos, and tan. NO calculator.
8. A 25 foot ladder leans against a building. The ladder’s base is 13.5 feet from the building. Find the
angle which the ladder makes with the ground. Calculator allowed.
9. An airplane climbs at an angle of 11° with the ground. Find the ground distance it has traveled when
it has attained an altitude of 400 feet. Calculator allowed.
10. Analyze triangle ABC and triangle DEF. Use
and
as the reference angles. Calculator allowed.
a.
Calculate the length of the hypotenuse of triangle ABC. Round your answer to the nearest tenth.
b.
Calculate the ratios
,
, and
for the reference angle in triangle ABC. Round
your answers to three decimal places.
c.
Describe the relationship between
d.
Calculate the length of the hypotenuse in
e.
Calculate the ratios
,
and
. Explain your reasoning.
without using the Pythagorean Theorem. Show your work.
, and
for the reference angle in
. Round your
answers to 3 decimal places.
f.
Compare the values of the three ratios for
true?
and
. What do you observe? Why do you think this is
84
Secondary II
Worksheet – After Chapter 8 Test
Name ______________________
1. Find the value of each trigonometric ratio to the nearest ten-thousandth.
a.
b.
c.
d.
2. Use a calculator to find the value of each to the nearest ten-thousandth.
a. sin 21°
b. tan 22°
c. cos 20°
d. sin 77°
3. Find the value of the trig function indicated.
a.
b.
c.
4. Find the measure of each angle indicated. Round to the nearest tenth.
a.
b.
c.
5. Find the measure of each side indicated. Round to the nearest tenth.
a.
b.
c.
d.
6. Solve each triangle, find all missing sides and angles. Round answers to the nearest tenth.
a.
b.
c.
85
Secondary II
9.1-9.3 - Worksheet
Name ___________________________
1. Identify an instance of each term in the diagram.
a. center of the circle
b. chord
c. secant of the circle
d. tangent of the circle
e. point of tangency
f. central angle
g. inscribed angle
h. arc
i. major arc
j. minor arc
k. diameter
l. semicircle
2. Identify each angle as an inscribed angle or a central angle.
a. ∠𝑍𝑂𝑀
b. ∠𝑅𝑂𝐾
3. Classify each arc as a major arc, minor arc, or a semicircle.
a.
b.
4. Draw the part of a circle that is described.
a.
b.
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Secondary II
5. Determine the measure of the minor arc.
6. Determine the measure of the central angle.
a.
b.
7. Determine the measure of each inscribed angle.
a.
b.
8. Determine the measure of each intercepted arc.
a.
b.
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Secondary II
9. The measure of ∠𝐶𝑂𝐷 is 98°.
10. The measure of ∠𝐾𝑂𝐿 is 148°.
What is the measure of ∠𝐶𝐸𝐷?
What is the measure of ∠𝐾𝑀𝐿?
̂ = 65° and 𝑚𝑋𝑍
̂ = 38°.
11. In circle C, 𝑚𝑊𝑍
What is 𝑚∠𝑊𝐶𝑋?
12. In circle C, 𝑚∠𝑊𝐶𝑌 = 83°.
What is ∠𝑋𝐶𝑍 ?
13. List the intercepted arc(s) for the given angle.
a.
b.
14. Use the diagram shown to determine the measure of each angle or arc.
a.
b.
c.
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Secondary II
15. Solve the system by elimination.
3𝑥 − 2𝑦 = 16
𝑓(𝑥) = {
5𝑥 + 𝑦 = 18
16. Solve for x: 3x  4t  7
17. Find the distance and the midpoint of A (-1, 5) and B (11, -3). Do not use decimals.
18. Determine each measure of a regular nonagon.
a) The sum of the interior angles.
b) The measure of an interior angle.
c) The measure of an exterior angle.
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Secondary II
9.4-9.5 - Worksheet
Name ___________________________
̅̅̅̅ intersects 𝐸𝐺
̅̅̅̅ at
1. If diameter 𝐹𝐻
a right angle, how does the length of
̅̅̅
𝐸𝐼 compare to the length of ̅̅̅
𝐼𝐺 ?
̅̅̅̅ ≅ 𝑍𝑂
̅̅̅̅, what is the
2. If 𝑌𝑂
̅̅̅̅ and 𝑋𝑉
̅̅̅̅
relationship between 𝑇𝑈
?
̅̅̅̅ is 13
3. If the length of 𝐴𝐵
millimeters, what is the length of
̅̅̅̅
𝐶𝐷 ?
4. If segment ̅̅̅̅
𝐴𝐶 is a diameter,
what is the measure of ∠𝐴𝐸𝐷 ?
̅̅, how does the
5. If ̅̅̅̅
𝑄𝑅 ≅ ̅̅
𝑃𝑆
̂ and 𝑃𝑅𝑆
̂
measure of 𝑄𝑃𝑅
compare?
6. If ∠𝐸𝑂𝐻 ≅ ∠𝐺𝑂𝐹 , what is the
relationship between ̅̅̅̅
𝐸𝐻 and ̅̅̅̅
𝐹𝐺 ?
7. Use each diagram and the Segment Chord Theorem to write an equation involving the segments of the chords.
a.
b.
8. Find the value of x.
̅̅̅̅ and ̅̅̅̅
9. Find 𝐴𝐵
𝐷𝐸 .
11. If ̅̅̅
𝑅𝑆̅ is a tangent segment and
̅̅̅̅
𝑂𝑆 is a radius, what is the measure
of ∠𝑅𝑂𝑆 ?
12. If ̅̅̅̅
𝑁𝑃 and ̅̅̅̅
𝑄𝑃 are tangent
segments, what is the measure of
∠𝑁𝑃𝑄 ?
̅̅̅̅ is a radius, what is the
10. If 𝑂𝐷
measure of ∠𝑂𝐷𝐶 ?
13. If ̅̅̅̅
𝐴𝐹 and ̅̅̅̅
𝑉𝐹 are tangent
segments, what is the measure of
∠𝐴𝑉𝐹 ?
90
Secondary II
14. Name two secant segments and two external secant segments for circle O. Then use the Secant Segment
Theorem to write an equation involving the secant segments.
a.
b.
15. Find the value of x.
a.
b.
16. Name a tangent segment, a secant segment, and an external secant segment for circle O. Then use the
Secant Tangent Theorem to write an equation involving the secant and tangent segments.
a.
b.
17. Solve for x.
a.
b.
18. Determine the measure of each missing angle in each figure.
a.
b.
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Secondary II
19. Write the prime factorization for each number.
a.
12
b. 30
c. 24
d. 40
e. 60
20. Identify the greatest common factor between each set of terms.
a.
12a 2b3 and 18a 3b
b. 30hp 5 and 24h 4 p 3
c. 20 y 2 m3 x 5 and 12 y 6 mx 4
21. Write the equation of a line with the given information.
a. passes through the points (4,-9) and (4,3)
b. passes through the point (2,-6) with a slope of -4
c. Parallel to y  2 x  5 that passes through ( 3 , - 2 )
d. Perpendicular to y  3x  1 that passes through point ( 3 , - 2 )
92
Secondary II
10.1-10.2 - Worksheet
Name ___________________________
1. Draw a triangle inscribed in the circle through the three points. Then determine if the triangle is a right triangle.
a.
b.
Answer: The Triangle is not a
right triangle because none of the
legs is a diameter of the circle.
2. Draw a triangle inscribed in the circle through the given points. Then determine the measure of the indicated angle.
In ABC, mA = 55°. Determine mB.
a. In ABC, mB = 380, Determine mA
Answer: mB = 1800 – 900 – 550 = 350
b. In ABC, mC = 490. Determine mA
c. In ABC, mB = 51 0. Determine mA
3. Draw a quadrilateral inscribed in the circle through the given four points. Then determine the measure
of the indicated angle.
In quadrilateral ABCD, mB = 810. Determine
mD.
a. In quadrilateral ABCD, mC = 750. Determine
mA.
Answer: m D = 1800 – 810 = 990
b. In quadrilateral ABCD, mB = 1120.
Determine mD.
c. In quadrilateral ABCD, mD = 930. Determine
mB.
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Secondary II
4. In the figure shown, ABC is inscribed in circle D and mA = 550. What is mC?
Explain your reasoning.
5. In the figure shown, RST is inscribed in circle Q, RS = 18 centimeters, and ST = 24 centimeters.
What is RT? Explain your reasoning.
6. In the figure shown, quadrilateral LMNP is inscribed in circle R, mP = 570, and
mL = mN. What are mM, mL and mN? Explain your reasoning.
7. Plot the following angles on the coordinate system.
a.
3
4
b.
5𝜋
8
c.
6𝜋
5
d.
9𝜋
4
Review:
8. Find the missing side and
following triangle.
62°
9. Draw the angle 𝜃 = 89° in standard angle for the
position then find the reference angle.
B = ________
y
𝜽′ = ________
y = ________
22.6
B
10. Use your calculator to find the following:
a. sin A  .866 __________

b. cos 25 __________
94
Secondary II
11. Simplify each expression without a calculator.
a.
3
125
b.
4
16
c.
5
32
d.
72
12. Determine each measure of a regular 15-gon.
a) The sum of the interior angles.
b) The measure of an interior angle.
c) The measure of an exterior angle.
13. Write a system of equations to represent the problem situation. Solve the system and explain
what your solution means in context of the problem.
The high school band is selling fruit baskets as a fundraiser. They sell a large basket containing 10 apples and
15 oranges for $20. They sell a small basket containing 5 apples and 6 oranges for $8.50. How much is the
band charging for each apple and each orange?
95
Secondary II
10.3 - WORksheet
Name ___________________________
1. Calculate the ratio of the length of each arc to the circle’s circumference.
̂ is 900.
a. The measure of 𝐶𝐷
̂is 1050.
b. The measure of 𝐼𝐽
̂ is 750.
c. The measure of 𝐾𝐿
̂ You do not need to simplify
2. Write an equation that you can use to calculate the length of 𝑅𝑆.
the expression.
a.
b.
c.
3. Calculate each arc length. Write your answer in terms of .
̂ is 45º and the radius is 12 meters, what is the arc length of 𝐴𝐵
̂?
a. If the measure of 𝐴𝐵
̂ is 120º and the radius is 15 centimeters, what is the arc length of 𝐶𝐷
̂?
b. If the measure of 𝐶𝐷
0
̂ is 90 , what is the arc length of 𝐸𝐹
̂?
c. If the measure of 𝐸𝐹
4. Calculate each arc length in radians. Write your answer in terms of .
̂ is 450, what is the arc length of 𝐶𝐷
̂?
a. If the measure of 𝐶𝐷
̂ is 60º and the radius is 8 inches, what is the arc length of 𝐸𝐹
̂?
b. If the measure of 𝐸𝐹
̂
c. If the length of the radius is 4 cm., what is the arc length of 𝐼𝐽?
96
Secondary II
5. Use the given information to answer each question. Where necessary use 3.14 to approximate .

a. If  = 3 and r = 3, what is the length of the intercepted arc?
b. If r = 8 and the intercepted arc length is 6 , what is the measure of the central angle?
c. If the measure of the central angle is 800. The length of the radius of 40mm. Determine the arc length
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒
using the formula
· 2𝑟.
3600
d. If the measure of the central angle is 300. The length of the radius of 10mm. Determine the arc length
𝑚𝑒𝑎𝑠𝑢𝑟𝑒 𝑜𝑓 𝑎𝑛𝑔𝑙𝑒
using the formula
· 2𝑟.
3600
6. If the radius of the circle is 9 centimeters, what is the area of sector AOB?
7. If the radius of the circle is 16 meters, what is the area of sector COD?
8. If the radius of the circle is 15 feet, what is the area of sector EOF?
9. If the radius of the circle is 10 inches, what is the area of sector GOH?
10. If the radius of the circle is 24 centimeters and 𝜃 =
2𝜋
3
, what is the area of sector MON?
HINT: Convert to degrees first!
97
Secondary II
11. If the radius of the circle is 21 meters and 𝜃 = 4, what is the area of sector POQ?
HINT: 4 is in radians, convert to degrees!
12. Calculate the area of each segment. Round your answer to the nearest tenth, if necessary use 3.14 for .
a. If the radius of the circle is 6 centimeters, what is the area of the shaded
segment?
b. If the radius of the circle is 14 inches, what is the area of the shaded segment?
c. If the radius of the circle is 25 meters, what is the area of the shaded segment?
In problems 13 & 14, calculate the area of the shaded segment AB of circle C. Express your answer in terms of  and
as a decimal rounded to the nearest hundredth.
13. In circle C shown, ABC is an equilateral triangle and ̅̅̅̅
𝐴𝐶 = 10 inches. Calculate the area
of the shaded segment AB of circle C.
Express your answer in terms of  and as a decimal rounded to the nearest hundredth.
14. In circle C, the radius is 18 centimeters and  ABC is an equilateral triangle.
C
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Secondary II
Review:
15. Determine the relationship between ∠4 𝑎𝑛𝑑 ∠8 and wriite a postulate or theorem that justifies your
answer.
_____________________________________
_____________________________________
16. Evaluate the trigonometric function. Show your triangle!!!
a. tan 210°
c. cos  60 
b. sin 225
17. A five-meter-long ladder leans against a wall, with the top of the ladder being four meters above the
ground. What is the approximate angle that the ladder makes with the ground?
18. Find the value of x.
19. If the measure of ∠𝑃𝑈𝑄 = 34° and the
radius of ⨀ 𝑈 is 8 inches, what is the arc
̂ ?
length of 𝑃𝑄
20. Solve each exponential equation for the missing variable.
a.
4 x  256
b. 63 x  216
c. 32 x 
1
729
99
Secondary II
11.3/11.6 - Worksheet
Name ___________________________
Estimate the approximate area or volume of each irregular or oblique figure. Round your answers to the nearest
tenth, if necessary.
1. The height of each recangle is .6 inches
and the base of each rectangle is 2 inches.
2.
Calculate the volume of each cone. Use 3.14 for π.
3.
4.
5.
6.
Calculate the volume of each pyramid.
7.
8.
100
Secondary II
9.
10.
11.
Calculate the volume of each cylinder. Use 3.14 for π. Round decimals to nearest tenth if necessary.
12.
13.
14.
Calculate the volume of each sphere. Use 3.14 for π. Round decimals to nearest tenth if necessary.
15.
16.
Calculate the volume of each solid. Use 3.14 for π. Round decimals to nearest tenth if necessary.
17.
18.
19.
Find each measurement.
20.
21.
22.
101
Secondary II
23.
24.
25.
26.
27. Solve for x.
29.
30.
28.
31.
32. If (3, -1) is a point on a graph, give the sin, cos and tan.
33. Find the sin 180°. Show your triangle!
102
Secondary II
Review - Chapter 9, 10 & 11
Name: ________________________
Show neat, complete work! Complete as many problems as possible without your calculator.
1. AC=6, AB = ?
2. DC & BC are tangent
segments. AB=6, AC=10,
find DC.
̂ = 180°. Find
3. 𝑚𝐵𝐷
∠BCD.
̂.
4. ∠𝐷𝐶𝐵 = 47°. Find 𝑚𝐷𝐵
5. ∠𝐷𝐶𝐵 = 125°. Find
̂.
𝑚𝐷𝐶𝐵
̂ = 117°, 𝐸𝐷
̂ = 125°,
6. 𝐶𝐹
find 𝑚∠𝐶𝐺𝐹.
̂ = 90°, 𝐻𝐽
̂ = 100°,
7. 𝐺𝐼
̂ = 120°, find
𝐼𝐽
𝑚∠𝐺𝐾𝐻.
8. ∠D = 80°, ∠C = 95°, Find
𝑚∠E.
9. ∠D=95°, ∠C=105°,
Find 𝑚∠ E.
10. EH = 4x+5, HF = 6x-9.
Solve for x.
11. CD = x+8, CB=5x-4
Solve for x
12. AB=5, BC=12, find AC.
13. ∠𝐷𝐶𝐵=?°,
Major arc DB=198°.
14. ∠𝐼𝐾𝐽=29°, arc GH=56°.
̂.
Find 𝑚𝐼𝐽
15. ∠𝐵𝐶𝐷=2x+7,
∠𝐷𝐸𝐵=5x-12.
Solve for x.
16. CG=x, GD=4, EG=8, FG=3.
Find x.
17.
In ∆𝐴𝐵𝐶, 𝑚∠𝐶 = 49°
Determine
.
18. Find the measure of
angles A, B, and C.
19. Find the length of
arc RS.
20. If the measure of
is
and the diameter is 6
millimeters, what is the arc
length of
?
103
Secondary II
21. If the measure of
is
5
and the
4
diameter is 20
millimeters, what is the
arc length of
?
25. If the radius of the
circle is 14 inches,
what is the area of the
shaded segment?
22. If θ is in standard
position and includes the
point (-3, -4), find sin, cos,
23. If the radius of the
circle is 9 centimeters,
what is the area of sector
AOB?
24. If the radius of the circle
is 21 meters, what is the area
of sector POQ?
̅̅̅̅ and 𝑉𝐹
̅̅̅̅ are
27. If 𝐴𝐹
tangent segments, what
is the measure of ∠𝐴𝑉𝐹 ?
̂ = 166°,
28. 𝑚𝐵𝐶
and tan. NO calculator.
̅̅̅̅ ≅ 𝑍𝑂
̅̅̅̅, and 𝑇𝑌
̅̅̅̅ =
26. If 𝑌𝑂
̅̅̅̅
3, what is the 𝑚𝑋𝑉 ?
̂ = 122°
𝑚𝐸𝐷
________
________
29. MC = 6, IC = 3, find
QI
30. AT = 15, B is the
midpoint of AT. Find AX.
31. WX = 6mm, XY =
8mm, YZ = 9mm, find WV.
̅̅̅̅ are tangent
32. If ̅̅̅̅̅
𝑀𝑇 and 𝑅𝑇
segments, what is the measure
of
?
33. PQ=3cm,
PR=10cm, LQ=5cm,
find LN.
34. Find the missing side
or angle.
35. CB is a diameter.
36. Calculate the measure of
̂ , 𝑚∠𝐶𝐷𝐵.
Find 𝑚𝐶𝐷𝐵
Given CB=10, CD=3, find
DB.
∠𝐶𝐴𝐵 if
104
Secondary II
37. A 25 foot ladder
leans against a
building. The ladder’s
base is 13.5 feet from
the building. Find the
38. Given: tan 𝜃 = √3,
where. Find  .
39. Find the following using
your calculator:
(Show triangle!)
a. sin 290
4
b. cos 𝜃 = 5
angle which the ladder
makes with the ground.
Calculator allowed.
c. csc(120 )
Find the volume of each figure.
40.
41.
42.
43.
44.
45.
46.
48.
47.
49.
50.
105
Secondary II
Worksheet – After Chpt 9/10/11 Test
Name ______________________
Simplify. Your answer should contain only positive exponents.
106