Download Unit 5A.2 – Similar Triangles

Unit 5A.1 – Ratios
Student Learning Targets:
 I can solve proportions.
Notes:
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Notes (Continued for 5A.1):
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Assignment 5A.1
Solve each proportion.
1.
6
21
=
𝑥 31.5
2.
6
9
=
18.2 𝑦
3.
𝑥 11
=
4 −6
4.
11
55
=
20 20𝑥
5.
4𝑥
56
=
24 112
6.
6𝑥
= 43
27
7.
4𝑥 − 5 −26
=
3
6
2𝑥 + 5 42
=
10
20
9.
3𝑥 − 1 2𝑥 + 4
=
4
5
10.
𝑎+2 3
=
𝑎−2 2
8.
11.
3𝑥 − 6 4𝑥 − 2
=
2
4
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12.
7
9
=
𝑧−1 𝑧+4
Unit 5A.2 – Similar Triangles
Student Learning Targets:
 I can prove that if two angles of one triangle are congruent to two angles of another
triangle, the triangles are similar (AA) using the properties of similarity transformations.

I can solve problems in context involving sides or angles of congruent or similar triangles.
Angle-Angle (AA) Similarity – If two angles of one triangle are congruent to two angles of
another triangle, then the triangles are similar.
Notes:
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Notes (Continued for 5A.2):
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Assignment 5A.2
Prove why the triangles shown are similar. Write a similarity statement.
1.
4.
2.
3.
5.
6.
Find the indicated measure (show your work). Draw a picture if one is not provided for you.
7. Josh wanted to measure the height of the Sears Tower in Chicago. He used
a 12-foot light pole and measured its shadow at 1pm. The length of the
shadow was 2 feet. Then he measured the length of the Sears Tower’s shadow
and it was 242 feet at the same time. What is the height of the Sears Tower?
8. Hallie is estimating the height of the Superman roller coaster. She is 5
feet 3 inches tall and her shadow is 3 feet long. If the length of the
shadow of the roller coaster is 40 feet, how tall is the roller coaster?
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Assignment 5A.2 (Continued)
9. A local furniture store sells two versions of the same chair: one for
adults and one for children. Find the value of x such that the chairs are
similar.
10. The two sailboats shown are participating in a regatta.
Find the value of x.
11. Adam is standing next to the Palmetto Building in Columbia, South Carolina. He is 6 feet
tall and the length of his shadow is 9 feet. If the length of the shadow of the building is 322.5
feet, how tall is the building?
12. A cell phone tower casts a 100-foot shadow. At the same time, a 4-foot 6-inch post near
the tower casts a shadow of 3 feet 4 inches. Find the height of the tower.
13. Angie is standing next to a statue in the park. If Angie is 5 feet tall, her shadow is 3 feet
long and the statue’s shadow is 10 ½ feet long, how tall is the statue?
14. When Alonzo, who is 5’11” tall, stands next to a basketball goal, his shadow is 2’ long, and
the basketball goal’s shadow is 4’4” long. About how tall is the basketball goal?
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Unit 5A.3 – Similar Polygons
Student Learning Targets:



I can understand that in similar triangles corresponding sides are proportional and
corresponding angles are congruent.
I can find lengths of measures of sides and angles of congruent and similar triangles.
I can decide whether two figures are similar using properties of transformations.
Distance Formula: d =
Notes:
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Notes (Continued for 5A.3):
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Assignment 5A.3
List all pairs of congruent angles and write a proportion that relates the corresponding sides.
∆𝑨𝑩𝑪~∆𝒁𝒀𝑿
∆𝑭𝑮𝑯~∆𝑱𝑲𝑳
1.
2.
∆𝑨𝑩𝑪~∆𝑹𝑺𝑻
3.
Explain whether each pair of figures is similar. If so, write the similarity statement.
4.
5.
6.
Each pair of polygons is similar. Find the missing values.
7.
8.
∆𝐴𝐵𝐶~∆𝐷𝐸𝐹, 𝑓𝑖𝑛𝑑 𝐷𝐹 𝑎𝑛𝑑 𝐸𝐹
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∆𝐽𝐾𝐿~∆𝑊𝑌𝑍, 𝑓𝑖𝑛𝑑 𝑥 𝑎𝑛𝑑 𝑦
Assignment 5A.3 (Continued)
9. Find x, ED and FD.
10. Find x, ST and SU.
Find each measure.
11. BE and AD
12. QP and MP
13. WR and RT
14. RQ and QT
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Assignment 5A.3 (Continued)
Identify the similar triangles. Then find each measure.
15. JK
16. ST
17. WZ and UZ
18. HJ and HK
Verify that the dilation is a similarity transformation.
19. Original: A(-6, 3), B(3, 3), C(3, -3)
20. Original: (-6, -3), (6, -3), (-6, 6)
Image: X(-4, -2), Y(2, 2), Z(2, -2)
2)
Image: (-2, -1), (2, -1), (-2,
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Unit 5A.4 – Dilations
Student Learning Targets:
 Given a line segment, a point not on the line segment, and a dilation factor, I can
construct a dilation of the original segment. (10-8)
 I can recognize that the length of the resulting image is the length of the original
segment multiplied by the scale factor and that the original and dilated image are
parallel to each other (10-8)
 I can use coordinate geometry to divide a segment into a given ratio.
Distance Formula: d =
Notes:
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Notes (Continued for 5A.4):
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Assignment 5A.4
Use line AB and a ruler to construct a line with the given dilation factor x with endpoint C.
1.
2.
𝑘=3
𝑘 = 0.5
∙𝐶
∙𝐶
3.
𝑘 = 1.5
4.
𝑘 = 2.25
∙𝐶
∙𝐶
Determine whether the dilation is an enlargement or a reduction. Then find the scale factor
of the dilation and x.
5. from Q to Q’
B’
6. from B to
7. from W to W’
W’
8. from W to
9. from W to W’
10. from W to W’
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𝑨
𝑩
Assignment 5A.4 (Continued)
11. from K to K’
12. from K to K’
DISCOVERY ACTIVITY
Graph segment AB and then find point C on
the line such that AC and CB form the indicated
ratio.
13.
Graph endpoint A and the point B that is on
the line. Find endpoint C such that AB and BC
form the indicated ratio.
A(3, 2), B(6, 8); 2:1 ratio
14. Endpoint:A(2, -1),
point on line:B(4, 2); 1:3 ratio
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Unit 5B.1- Parallel Lines
Student Learning Targets:
I can prove and use theorems about lines and angles, including but not limited to:




Vertical angles are congruent.
When parallel lines are cut by a transversal congruent angle pairs are created.
When parallel lines are cut by a transversal supplementary angle pairs are created.
Points on the perpendicular bisector of a line segment are equidistant from the
segment’s endpoints.
Notes:
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Notes (Continued for 5B.1):
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Assignment 5B.1
Find the measure of each numbered angle, and name the theorems that justify your work.
1. 𝑚∠2 = 26
2. 𝑚∠4 = 3(𝑥 − 1), 𝑚∠5 = 𝑥 + 7
In the figure, 𝒎∠𝟒 = 𝟏𝟎𝟏. Find the measure of each angle. Tell which postulate(s) or theorem(s) you
used.
3. ∠6
4. ∠7
5. ∠5
Find the value of the variable(s) in each figure. Explain your reasoning.
6.
7.
Find each measure.
8. XW
9. LP
10. AC
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Assignment 5B.1 (Continued)
Copy and complete the proof using the list of properties to the right.
Answer Bank
11.
𝑚∠3 = 𝑚∠3
Given
Substitution
Definition of
complementary
angles
Reflexive property
Definition of
congruent angles
Transitive
Symmetric
12.
GivenAnswer Bank
Definition of linear
pair
Corresponding
Angles
Vertical Angles
Alternate interior
angle theorem
Substitution
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Assignment 5B.1 (Continued)
13. Prove ∠1 ≅ ∠8
1 2
3 4
5 6
7 8
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Unit 5B.2- Triangles
Student Learning Targets:
I can prove and use theorems about triangles including, but not limited to:
 Prove that the sum of the interior angles of a triangles = 180.
 Prove that the base angles of an isosceles triangle are congruent. Prove that if two
angles of a triangle are congruent, the triangle is isosceles.
 Prove the segment joining midpoints of two sides of a triangle is parallel to the third
side and half the length.
 Prove the medians of a triangle meet at a point.
Notes:
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Notes (Continued for 5B.2):
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Assignment 5B.2
Find the measures of each numbered angle.
1.
2.
Find each measure.
3. 𝑚∠𝐵𝐴𝐶
4. 𝑚∠𝑆𝑅𝑇
5. 𝑇𝑅
6. 𝐶𝐵
7. Write a paragraph explaining why the segment joining midpoints of two sides of a triangle is
parallel to the third side.
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Assignment 5B.2 (Continued)
8. Write a two-column proof for the following.
9. Write a two column proof.
̅̅̅ ≅ 𝑃𝑀
̅̅̅̅̅, 𝐽𝐿
̅ ≅ 𝑃𝐿
̅̅̅̅, 𝑎𝑛𝑑 𝐿 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝐾𝑀
̅̅̅̅̅
Given: ∠𝐽 ≅ ∠𝑃, 𝐽𝐾
Prove: ∆𝐽𝐿𝐾 ≅ ∆𝑃𝐿𝑀
10. Prove that the base angles of an isosceles triangle are congruent.
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Unit 5B.3- Prove Theorems Involving Similarity
Student Learning Targets:

I can prove that a line constructed parallel to one side of a triangle intersecting the
other two sides of the triangle divides the intersected sides proportionally.

I can prove that a line that divides two sides of a triangle proportionally is parallel to the
third side.
I can prove that if three sides of one triangle are proportional to the corresponding sides
of another triangle, the triangles are similar.
I can prove the Pythagorean Theorem using similarity.


Notes:
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Notes (Continued for 5B.3):
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Assignment 5B.3
1. A triangle is intersected by a segment parallel to one side. Prove that the result is a proportional
division of the sides.
2. Prove the Pythagorean Theorem using similarity.
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Assignment 5B.3 (Continued)
3. Prove that if three sides of one triangle are proportional to the corresponding sides of another
triangle, the triangles are similar.
4. Prove that a line that divides two sides of a triangle proportionally is parallel to the third side.
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Unit 5B.4 –Solving Problems Using Congruency and Similarity
Student Learning Targets:



I can find lengths of measures of sides and angles of congruent and similar triangles.
I can solve problems in context involving sides or angles of congruent or similar
triangles.
I can prove conjectures about congruence or similarity in geometric figures using
congruence and similarity criteria.
Notes:
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Notes (Continued for 5B.4):
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Assignment 5B.4
Find the value of the variable that yields congruent
triangles. Explain.
1.
2.
3.
4.
5. A high school wants to hold a 1500-meter regatta on Lake Powell but is unsure if the lake is
long enough. To measure the distance across the lake, the crew members locate the vertices of
the triangles below and find the measures of the lengths of triangle HJK as shown below.
a. Explain how the crew team can use the triangles formed to estimate the distance FG across
the lake.
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Assignment 5B.4 (Continued)
b. Using the measures given, is the lake long enough for the team to use as the location for
their regatta? Explain your reasoning.
6. The length of George Washington’s face at Mt. Rushmore is 60 feet. Describe a method for
determining the length of his nose using similar triangles. Justify your reasoning.
Write a two-column proof.
7.
8.
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Assignment 5B.4 (Continued)
9. The pattern shown is created using regular polygons.
a. What two polygons are used to create the pattern?
b. Name a pair of congruent triangles.
c. Name a pair of corresponding angles.
d. If CB=2 inches, what is AE? Explain.
e. What is the measure of angle D? Explain.
10. Write a two-column proof.
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Unit 5B.5- Parallelograms
Student Learning Targets:
I can prove and use theorems about parallelograms including, but not limited to:
 Opposite sides of a parallelogram are congruent.
 Opposite angles of a parallelogram are congruent.
 The diagonals of a parallelogram bisect each other
 Rectangles are parallelograms with congruent diagonals.
Notes:
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Notes (Continued for 5B.5):
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Assignment 5B.5
Determine whether each quadrilateral is a parallelogram. Justify your answer.
1.
2.
3.
4.
Find the value of each variable in each parallelogram.
5.
6.
7.
8.
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Assignment 5B.5 (Continued)
The quadrilateral WXYZ is a rectangle. Use it to answer questions 9-14.
9. If 𝑍𝑌 = 2𝑥 + 3 and 𝑊𝑋 = 𝑥 + 4, find 𝑊𝑋.
10. If 𝑃𝑌 = 3𝑥 − 5 and 𝑊𝑃 = 2𝑥 + 11, find 𝑍𝑃.
11. If 𝑚∠𝑍𝑌𝑊 = 2𝑥 − 7 and 𝑚∠𝑊𝑌𝑋 = 2𝑥 + 5., find 𝑚∠𝑍𝑌𝑊.
12. If 𝑍𝑃 = 4𝑥 − 9 and 𝑃𝑌 = 2𝑥 + 5, find 𝑚∠𝑌𝑋𝑍.
13. If 𝑚∠𝑋𝑍𝑌 = 3𝑥 + 6 and 𝑚∠𝑋𝑍𝑊 = 5𝑥 − 12, find 𝑚∠𝑌𝑋𝑍.
14. If 𝑚∠𝑍𝑋𝑊 = 𝑥 − 11 and 𝑚∠𝑊𝑍𝑋 = 𝑥 − 9, find 𝑚∠𝑍𝑋𝑌.
15. Four jets are flying in formation. Three of the jets are shown in the graph. If the four jets
are located at the vertices of a parallelogram, what are the three possible locations of the
missing jet?
16. Write a two-column proof.
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Assignment 5B.5 (Continued)
17. Write a paragraph proof showing that a rectangle is a parallelogram with congruent
diagonals.
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Unit 5C.1 – Basic Trig Functions
Student Learning Targets:
 I can understand that the ratio of two sides in one triangle is equal to the ratio of the
corresponding two sides of all other similar triangles.
 I can define sine, cosine, and tangent as the ratio of sides in a right triangle.
 I can demonstrate the relationship between sine and cosine in the acute angles of a
right triangle.
 I can explain the relationship between the sine and cosine in complementary angles.
Pythagorean Theorem: 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
Notes:
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Notes (Continued for 5C.1):
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Assignment 5C.1
Find the reduced ratio of the following trig functions.
1.
SIn J
3.
2.
Sin J
Cos J
Cos J
Tan J
Tan J
Sin L
Sin L
Cos L
Cos L
Tan L
Tan L
SIn J
4.
Sin J
Cos J
Cos J
Tan J
Tan J
Sin L
Sin L
Cos L
Cos L
Tan L
Tan L
Fill in the sides of the special right triangles and express each trigonometric ratio as a reduced
fraction.
5.
6.
30°
60°
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Assignment 5C.1 (Continued)
7.
8.
5
12
45°
9.
4
10.
10
11.
Find the second acute angle of a
right triangle given that the first
acute angle has measure of 39°.
Complete the following statement:
1
1
If sin 30°=2 , then the cos_____ = 2
12.
Explain why the sine of x is the same regardless of which triangle is used to find it in
the figure.
x
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Unit 5C.2 – Pythagorean Theorem and Trig Ratios
Student Learning Targets:
 I can use the Pythagorean Theorem and trigonometric ratios to find missing measures in
triangles in contextual situations.
Pythagorean Theorem: 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐
Notes:
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Notes (Continued for 5C.2):
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Assignment 5C.2
1. Leah wants to see a castle in an amusement park. She sights the top of the castle at an angle of
elevation of 38°. She knows that the castle is 190 feet tall. If Leah is 5.5 feet tall, how far is she
from the castle to the nearest foot?
2. A hockey player takes a shot 20 feet away from a 5-foot goal. If the puck travels at a 15° angle
of elevation toward the center of the goal, will the player score?
3. A search and rescue team is airlifting people from the scene of a boating accident when they
observe another person in need of help. If the angle of depression to this other person in need
of help. If the angle of depression to this other person is 42° and the helicopter is 18 feet above
the water, what is the horizontal distance from the rescuers to this person to the nearest foot?
4. A lifeguard is watching a beach from a line of sight 6 feet above the ground. She sees a
swimmer at an angle of depression of 8°. How far away from the tower is the swimmer?
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Assignment 5C.2 (Continued)
5. The cross bar of a goalpost is 10 feet high. If a field goal attempt is made 25 yards from the base
of the goalpost that clears the goal by 1 foot, what is the smallest angle of elevation at which
the ball could have been kicked to the nearest degree?
6.
A digital camera with a panoramic lens is described as having a view with an angle of elevation
of 38°. If the camera is on a 3-foot tripod aimed directly at a 124-foot-tall monument, how far
from the monument should you place the tripod to see the entire monument in your
photograph?
7. A teenager whose eyes are 5’ above ground level is looking into a mirror on the ground and can
see the top of a building that is 30’ away from the teenager. The angle of elevation from the
center of the mirror to the top of the building is 75°. How far away from the teenager’s feet is
the mirror? How tall is the building?
8. To estimate the height of a tree she wants removed, Mrs. Long sights the tree’s top at a 70°
angle of elevation. She then steps back 10 meters and sights the top at a 26° angle. If Mrs.
Long’s line of sight is 1.7 meters above the ground, how all is the tree to the nearest meter?
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Assignment 5C.2 (Continued)
9. Austin is standing on the high dive at the local pool. Two of his friends are in the water as
shown. If the angle of depression to one of his friends is 40°, and 30° to his friend who is 5 feet
beyond the first, how tall is the platform?
10. While traveling across flat land, you see a mountain directly in from of you. The angle of
elevation to the peak is 3.5°. After driving 14 miles closer to the mountain, the angle of
elevation is 9° 24′ 36". Explain how you would set up the problem, and find the approximate
height of the mountain.
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Unit 5C.3 – Trig Functions and the Pythagorean Identity
Student Learning Targets:


Given sin (θ), cos (θ), or tan (θ) for 0< θ <90, I can find sin (θ), cos (θ), or tan (θ)
I can prove 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1 for right triangles in the first quadrant.
Notes:
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Notes (Continued for 5C.3):
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Assignment 5C.3
Let 𝜽 be an acute angle of a right triangle. Find the values of the other two trig functions,
(𝒔𝒊𝒏𝜽, 𝒄𝒐𝒔 𝜽, 𝒕𝒂𝒏𝜽 ).
5
1. 𝑠𝑖𝑛𝜃 = 6
7
2. 𝑡𝑎𝑛𝜃 = 3
3. 𝑐𝑜𝑠𝜃 =
5
12
𝑐𝑜𝑠𝜃 =
𝑡𝑎𝑛𝜃 =
𝑠𝑖𝑛𝜃 =
𝑐𝑜𝑠𝜃 =
𝑠𝑖𝑛𝜃 =
𝑡𝑎𝑛𝜃 =
Work through the following problems to discover a Pythagorean Trig Identity
Use Angle L
4.
4
3
5
𝑠𝑖𝑛𝜃 = ___________
𝑐𝑜𝑠𝜃 = __________
𝑠𝑖𝑛2 𝜃 =___________
𝑐𝑜𝑠 2 𝜃 = __________
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = _________
Use Angle J
5.
6
13
𝑠𝑖𝑛𝜃 = ___________
𝑐𝑜𝑠𝜃 = __________
𝑠𝑖𝑛2 𝜃 =___________
𝑐𝑜𝑠 2 𝜃 = __________
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = _________
What does 𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 ALWAYS equal? _________________
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