Student Learning Targets
Unit 5A.1 – Ratios
Student Learning Targets:
· I can solve proportions.
Notes:
Solve each proportion.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
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:
Prove why the triangles shown are similar. Write a similarity statement.
1. 2. 3.
4. 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?
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?
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:
List all pairs of congruent angles and write a proportion that relates the corresponding sides.
1. 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.
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
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) Image: (-2, -1), (2, -1), (-2, 2)
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:
Use line AB and a ruler to construct a line with the given dilation factor x with endpoint C.
1. 2. 3. 4.
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’ 6. from B to B’
7. from W to W’ 8. from W to W’
9. from W to W’ 10. from W to W’
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. A(3, 2), B(6, 8); 2:1 ratio
Graph endpoint A and the point B that is on the line. Find endpoint C such that AB and BC form the indicated ratio.
14. Endpoint:A(2, -1),
point on line:B(4, 2); 1:3 ratio
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:
Find the measure of each numbered angle, and name the theorems that justify your work.
1. m∠2=26 2. m∠4=3x-1, m∠5=x+7
In the figure, m∠1=94. 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
Copy and complete the proof using the list of properties to the right.
10.
m∠3=m∠3
Given
Substitution
Definition of complementary angles
Reflexive property
Definition of congruent angles
Transitive
Symmetric
11.
Given
Definition of linear pair
Corresponding Angles
Vertical Angles
Alternate interior angle theorem
Substitution
12. Prove ∠1 ≅∠8
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:
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.
8. Write a two-column proof for the following.
9. Write a two column proof.
Given: ∠J≅∠P, JK≅ PM, JL≅ PL, and L bisects KM
Prove: ∆JLK≅∆PLM
10. Prove that the base angles of an isosceles triangle are congruent.
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:
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.
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.
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:
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.
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.
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.
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:
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.
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.
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.
17. Write a paragraph proof showing that a rectangle is a parallelogram with congruent diagonals.
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:
Find the reduced ratio of the following trig functions.
1. SIn J 2. Sin J
Cos J Cos J
Tan J Tan J
Sin L Sin L
Cos L Cos L
Tan L Tan L
3. 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.
7. 8.
9. 10.
11.
12.
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:
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?