Download Geometry Quarter 2

Geometry
Quarter2
Ø
ExploringTriangles
Ø
CongruentTriangles
Ø
PerpendicularLinesand
Triangles
Ø
Parallelogramsand
Quadrilaterals
Ø
DilationsandSimilarity
Ø
SimilarTriangles
Ø
Similarityand
Proportionality
MotherNaturelovesmath,too!
Constructaseriesofsquareswithlengths
thatfollowthenumbersintheFibonacci
sequence(1,1,2,3,5,8,13,21,…)and
thentraceacurvethroughopposite
verticesofeachsquare.
Thisformsa“Fibonaccispiral”
(#fibonaccispiral).Manyexamplesofthe
Fibonaccispiralcanbeseeninnature,
includinginthechambersofanautilusshell.
Noticehowthepetalsofarosespirals?
Howcoolisthat?
Table of Contents
L8-8.1
Introduction
1
L8-8.2
Angle and Side Relationship
3
L8-8.3
Exterior Angle Theorem
11
L8-8.4
Triangle Inequality Theorem
15
L9-9.1
Introduction
19
L9-9.2
Determining Congruence
21
L9-9.3
Identifying Congruent Triangles
29
L9-9.4
Homework 1
31
L9-9.5
Proving Congruent Triangles
35
L9-9.6
Homework 2
37
Perpendicular Lines & Triangles
L10-10.1
Construction Warmup
39
L10-10.2
Grandma’s Roof
40
L10-10.3
Homework
47
L10-10.4
Attic Access Revisited
51
L10-10.5
Finding the Center of Revolution
53
L11-11.1
Revisiting Congruence
57
L11-11.2
Properties of Parallelograms
59
L11-11.3
Homework
65
L11-11.4
Parallelogram Angle Proofs
67
L11-11.5
Homework
71
L11-11.6
Diagonals of Parallelograms
73
L11-11.7
Congruent Quadrilaterals
77
L11-11.8
Special Quadrilaterals and their Properties
81
L11-11.9
Theorems of Special Quadrilaterals
87
Lesson 9:
Congruent Triangles
Lesson 8:
Title
Exploring Triangles
Lesson
Lesson 10:
Quarter 2
Lesson 11:
Parallelograms and Quadrilaterals
Geometry
Page
i
Table of Contents
Lesson 14:
Similarity and Proportionality
Lesson 13:
Similar Triangles
Lesson 12:
Dilations and Similarity
Geometry
Quarter 1
Lesson
Title
Page
L12-12.1
Refresher on Ratios
93
L12-12.2
Grandma’s Gazebo
95
L12-12.3
Dilations
103
L12-12.4
Introduction to Similarity
109
L12-12.5
Homework
113
L13-13.1
Looking Ahead: Using Similar Triangles to Solve Problems
115
L13-13.2
Characteristics of Similar Triangles
117
L13-13.3
Similarity and Proportion
119
L13-13.4
Homework
123
L13-13.5
Theorems About Similar Triangles
125
L13-13.6
Homework
131
L13-13.7
The Geometric Mean
133
L14-14.1
Construction Warm-up
141
L14-14.2
The Altitude and Mean
143
L14-14.3
The Mid-segment Theorem
145
L14-14.4
Homework
149
L14-14.5
Using Proportionality to Find Coordinates
151
L14-14.6
Parallel Lines in Triangles (Warm-up)
157
L14-14.7
Parallel Lines in Triangles
159
L14-14.8
Medians and Altitudes in Similar Triangles
163
L14-14.9
Homework
169
ii
L8 – Exploring Triangles
8.1 – Introduction
Name___________________________
Per________ Date_________________
Inequality Review
1. Write inequalities to model the sentences below.
a. Keanu is older than Justin. Justin is older than David.
b. Mary has less money than Keani. Keani has less money than Nani.
c. In the golf match, the Vulcans shot fewer strokes than the Warriors who shot
fewer strokes than the Trojans.
d. Walmart has more employees than McDonald’s who has more employees than
Burger King.
e. 𝐴𝐵is longer than 𝐶𝐷, which is longer than 𝐸𝐹.
2. Using the above inequalities, express the following relationships with another
inequality.
a. The relationship between Keanu and David.
b. The relationship between Mary’s and Nani’s money.
c. The relationship between the Vulcans’ and Trojans’ golf scores.
d. The relationship between the number of employees in Walmart and Burger King.
e. The relationship between 𝐴𝐵 and 𝐸𝐹.
3. If Keanu is older than Justin, but Justin is older than Nathan, what can we conclude
about the relationship between Keanu and Nathan? Explain your reasoning.
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.2 – Angle and Side Relationship
Name___________________________
Per________ Date_________________
Investigating Roof Designs
Below are two possible designs for the roof on Grandma’s house. Uncle Bobby wants to
know more about the designs and realizes the designs lack measurements. He needs to
know if the roofs are too steep to build or too flat (could leak). Using your ruler and
protractor, measure each angle and each side. Record the measurements on the picture.
Roof Design #1
Roof Design #2
Based on your prior knowledge, what is the sum of the interior angles of a triangle?
Do your measurements agree with that sum?
Compare the length of a side to the measure of the angle opposite that side. Do you notice
any patterns? (hint: look at the sizes of the sides and angles compared with the other
sides and angles in the triangle).
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.2 – Angle and Side Relationship
Name___________________________
Per________ Date_________________
Investigating Your Patterns
Measure the sides and the angles of the four triangles below. Label each on the sketch.
Classify each triangle two ways:
A. By its Angle Name as Acute, Obtuse, or Right.
B. By its Sides Name as Scalene,Isosceles,orEquilateral.
1.
Angle Name ___________________
Side Name ____________________
2.
Angle Name ___________________
Side Name ____________________
3.
Angle Name ___________________
Side Name ____________________
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.2 – Angle and Side Relationship
4.
Name___________________________
Per________ Date_________________
Angle Name ___________________
Side Name ____________________
Based on your measurements, did the pattern you discovered in Grandma’s roof
designs hold with these triangles?
Trace the triangles onto patty paper. Rip off each vertex (keep each vertex with the
other two vertices of the same triangle), and arrange the vertices so that they are
adjacent (i.e. so that one ray of the first coincides with one ray of the second, and the
remaining ray of the second coincides with one ray of the third). What do you notice?
Make a conjecture about the sum of the angles.
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.2 – Angle and Side Relationship
Name___________________________
Per________ Date_________________
Sum of the Measures of the Interior Angles of a Triangle
Based on your measurements, complete the statement:
The sum of the measures of the interior angles of a triangle is __________________.
(#THM)
Let’s prove it. Use the diagram below.
Given: ∆𝐴𝐵𝐶 and line m ∥ 𝐶𝐴
Prove: 𝑚∠1 + 𝑚∠2 + 𝑚∠3 = 180°
Statement
HIDOE Geometry SY 2016-2017
Reason
6
L8 – Exploring Triangles
8.2 – Angle and Side Relationship
Name___________________________
Per________ Date_________________
Using Sum of the Interior Angles Theorem
Fill in any missing interior angle measures.
1.
2.
58.3°
54°
47°
89.5°
3.
130.8°
37.32°
4. Solve for x in each of the triangles. (Hint: the sum of the measures of the interior
angles equals 180. Set up an equation to show that sum and solve for x.)
(𝑥)°
(𝑥 + 12)°
(3𝑥 + 32)°
(𝑥 − 4)°
50°
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.2 – Angle and Side Relationship
Name___________________________
Per________ Date_________________
Proof of the Longer Side Inequality Theorem
State your conjecture from the previous work regarding the lengths of the sides and the
measures of the angles here.
In more formal language, (#THM) we can say,
One side of a triangle is longer than another side of a triangle if and only if the measure
of the angle opposite the longer side is greater than the angle opposite the shorter side.
Now let’s prove it.
Given:∆𝐴𝐵𝐶 ∶ 𝐴𝐶 > 𝐴𝐵
???? ∶ ????
Construct:𝐵𝐷
𝐴𝐷 ≅ ????
𝐴𝐵
Prove:𝑚∠𝐴𝐵𝐶 > 𝑚∠𝐶
A
D
B
C
Statement
1. DABC; AC > AB
2. BD : AD @ AB
HIDOE Geometry SY 2016-2017
Reason
1. Given
2. Congruent segment construction
8
L8 – Exploring Triangles
8.2 – Angle and Side Relationship
Name___________________________
Per________ Date_________________
Using the Longer Side Inequality
Based on the longer side inequality theorem, list the sides of the triangles in order of
longest to shortest.
1.
2.
A
E
126°
36°
70°
D
F
67°
C
B
Based on the longer side inequality, list the angles in order of size from largest to
smallest.
A
3.
4.
7.49
5.93
5.65
D
5.79
C
6.04
B
F
6.19
E
For all the above problems, explain your reasoning for listing the sides or angles in the
order that you did.
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.3 – Exterior Angle Theorem
Name_____________________________
Per________ Date_________________
Exterior Angle Theorem
Grandma has a side job for you. She wants her chicken coop repaired. We need to start
with a repair to the area where the roof meets the front wall. We need to build a support
that is the exact same angle as that. It’s too dangerous to measure that angle directly, but
luckily, we know the angle between the roof and ceiling and the angle between the
ceiling and front wall. Below is a sketch of the chicken coop and the angles that we
know.
Below is a sketch of a side view Grandma’s old chicken coop.
Angleweneedtomeasure
Roof
23° Ceiling
90°
Frontwall
Floor
How can we deduce the measure of the angle we are looking for without actually
measuring it? (Complete the next activity to find out.)
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.3 – Exterior Angle Theorem
Name_____________________________
Per________ Date_________________
The diagram below shows exterior angles, adjacent interior angles, and remote
interior angles of a triangle. (#VOC)
Remote Interior
Angle
Exterior Angle
Adjacent Interior
Angle
Measure the exterior angles and remote interior angles on the triangles below.
Compare your measurements with your classmates.
Based on your measurements, describe any patterns you see between the exterior angle
and the remote interior angles.
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.3 – Exterior Angle Theorem
Name_____________________________
Per________ Date_________________
Exterior Angle Theorem (#THM)
The exterior angle theorem states that given a triangle and an exterior angle of that
triangle, the measure of the exterior angle is equal to the sum of the measures of the
two remote interior angles.
Remote Interior
Angle
Exterior Angle
Adjacent Interior
Angle
Use the figure below to prove the exterior angle theorem.
A
Given: ∆𝐴𝐵𝐶 and exterior angle ∠𝐷𝐶𝐴
Prove: 𝑚∠𝐷𝐶𝐴 = 𝑚∠𝐴 + 𝑚∠𝐵
B
Statement
HIDOE Geometry SY 2016-2017
C
D
Reason
13
HIDOE Geometry SY 2016-2017
14
L8 – Exploring Triangles
8.4 – Triangle Inequality Theorem
Name_____________________________
Per________ Date_________________
Creating Triangles
In this exercise, you will create a triangle with straws (or pieces of spaghetti). You should
have 6 different length straws of 5, 6, 8, 12, 13, 14 cm. Create triangles with the indicated
lengths. Record whether your triangles are acute, obtuse, or right. If it is not possible to
create a triangle with the straws, write “not possible”. All the measurements are in cm
and the sides are labeled a, b, and c.
1. a = 5, b = 12, c = 13
2. a = 5, b = 6, c = 8
3. a = 5, b = 6, c = 12
4. a = 5, b = 8, c = 12
5. a = 5, b = 8, c = 13
6. a = 8, b = 12, c = 14
7. a = 6, b = 8, c = 13
8. a = 6, b = 8, c = 14
List the side lengths of the triangles below in the possible or not possible categories.
Possible
Not Possible
Do you notice any patterns? In other words, why were some combinations not
possible?
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.4 – Triangle Inequality Theorem
Name_____________________________
Per________ Date_________________
Possible or Not?
1. Determine whether you could construct a triangle given the following combinations of
side lengths. If “not possible”, explain your reasoning.
a) 4, 5, 9 cm
b) 4, 6, 9 cm
c) 7, 13, 8 cm
d)
9, 3, 5 cm
2. Dezmond decides to build a triangular garden. He has three extra 2"x4" boards lying
in his shed. He measures the board lengths and finds them to be 4, 10, and 3 ft.
a. Will he be able to build a triangular garden without adjusting his boards? Explain
your reasoning.
b. If Dezmond is only comfortable in working in whole number measurements, what
does he have to do to ensure that he is able to build a triangular garden with those
boards?
3. Gabby saw Dezmond’s triangular boxes and wants to build her own. She has 6
boards, but no way to cut the boards. Her boards have lengths of 4 ft, 7 ft, 2 ft, 3.5 ft,
6 ft, and 5 ft. What combinations of boards will not work to build a triangle? Explain
your reasoning.
The Triangle Inequality Theorem (#THM): If a, b, and c represent the side lengths of a
triangle, then each must be less than the sum of the other two.
HIDOE Geometry SY 2016-2017
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L8 – Exploring Triangles
8.4 – Triangle Inequality Theorem
Name_____________________________
Per________ Date_________________
Shortest Way – As the Nene Flies
Below is a diagram of two villages on a map, with a lake between the two villages.
Young’sVillage
Takahashi’sVillage
The dotted line represents the flight of a nene between the two villages, and the solid
line represents the walking trail. Using the triangle inequality theorem, explain why
the flight of the nene is shorter than the walking trail.
Extension: Justify the statement “the shortest distance between two points is a straight
line.”
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.1 – Introduction
Name ________________________________
Per _______ Date _______________________
How Do We Compare?
Using patty paper, compare the lengths and angles of the following triangle pairs. Record what
is the same for each pair and what is different.
1. What is common?
What is different?
Is there a rigid motion that shows they are congruent?
2. What is common?
What is different?
Is there a rigid motion that shows they are congruent?
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.1 – Introduction
Name ________________________________
Per _______ Date _______________________
3. What is common?
What is different?
Is there a rigid motion that shows they are congruent?
4. Based upon your results, what are some conjectures about when triangles are congruent?
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
Grandma’s Garden Boxes
Grandma has been watching the garden channel again. She fell in love with some triangular
shaped raised garden boxes. She asked Uncle Bobby to build them, but Uncle Bobby did not
want to measure all three sides and all three angles of all the triangles. While he was complaining
to you, you mentioned that you heard a rumor that in order to be sure that two triangles are
congruent, you only need to measure three pieces of information. You just couldn’t remember
what three pieces. Let’s investigate in the next activity. When you are finished with the activity,
write a note to Uncle Bobby explaining what three measurements he would need to make sure
the triangles are congruent.
Note to Uncle Bobby:
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
WHAT DOES IT TAKE TO BE THE SAME?
This exercise investigates critical theorems for proving when two triangles must be congruent.
You will be able to use these theorems throughout the remainder of the course.
Scenario 1: Side-Side-Side (SSS)
In this scenario you will explore if having three sides of one triangle congruent to three sides of
another triangle guarantees that the two triangles are congruent.
1. Draw a scalene triangle on a sheet of tissue paper.
2. Using three other pieces of tissue paper, trace each of the sides of the triangle onto a separate
piece of paper. Mark the ends of each segment to make them easier to see.
3. Slide the three pieces together to make a triangle and copy the new triangle onto another
piece of tissue paper.
4. Is your new triangle congruent to the original? Explain why or why not.
5. Can you rearrange the pieces to create a new triangle that is not congruent to the original?
Explain why the two triangles must be congruent, or why not.
6. State your Claim:
If three sides of one triangle are congruent to three sides of another triangle, then
______________________________________________________________________.(#THM)
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
Scenario 2: Angle-Angle-Angle (AAA)
In this scenario you will explore if having three angles of one triangle congruent to three angles
of another triangle guarantees that the two triangles are congruent.
1. Draw a scalene triangle on a sheet of tissue paper.
2.
Using three other pieces of tissue paper, trace each of the angles of the triangle onto a
separate piece of paper. Extend the rays of the angles.
3. Slide the three pieces together to make a new triangle and copy the new triangle onto another
piece of tissue paper. Recall that a ray has no end, hence you will only be using a portion of
each ray as a side.
4. Is your new triangle congruent to the original? Explain why or why not.
5. Can you rearrange the pieces to create a new triangle that is not congruent to the original?
Explain why the two triangles must be congruent, or why not.
6. State your Claim:
If three angles of one triangle are congruent to three angles of another triangle, then
__________________________________________________________________
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
Scenario 3: Side-Side-Angle (SSA)
In this scenario you will explore if having two concurrent sides and the angle adjacent to the
second side of one triangle congruent to two concurrent sides and the angle adjacent to the
second side of another triangle guarantees that the two triangles are congruent.
1. Draw a scalene triangle on a sheet of tissue paper.
2.
Using three other pieces of tissue paper, trace two concurrent sides and the angle adjacent to
the second side (i.e. opposite the first side) of one triangle onto a separate piece of paper.
Mark the ends of each segment to make them easier to see, making sure you keep track of
which side was the first side, and extend the rays of the angles.
3. Slide the three pieces together to make a new triangle, making sure the angle is still opposite
the first side, and copy the new triangle onto another piece of tissue paper. Recall that a ray
has no end, hence you will only be using a portion of each ray as a side.
4. Is your new triangle congruent to the original? Explain why or why not.
5. Can you rearrange the pieces to create a new triangle that is not congruent to the original,
where the angle is still opposite the first side? Explain why the two triangles must be
congruent, or why not.
6. State your Claim:
If two concurrent sides and the angle adjacent to the second side of one triangle are
congruent to two concurrent sides and the angle adjacent to the second side of another
triangle, then
_____________________________________________________________________
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
Scenario 4: Side-Angle-Side (SAS)
In this scenario you will explore if having two sides and the angle between them of one triangle
congruent to two sides and the angle between them of another triangle guarantees that the two
triangles are congruent.
1. Draw a scalene triangle on a sheet of tissue paper.
2. Using three other pieces of tissue paper, trace two sides and the angle between them of the
triangle onto a separate piece of paper. Mark the ends of each segment to make them easier
to see and extend the rays of the angles. Recall that a ray has no end, hence you will only be
using a portion of each ray as a side.
3. Slide the three pieces together to make a new triangle, making sure the angle is still between
the two sides, and copy the new triangle onto another piece of tissue paper. Recall that a ray
has no end, hence you will only be using a portion of each ray as a side.
4. Is your new triangle congruent to the original? Explain why or why not.
5. Can you rearrange the pieces to create a new triangle that is not congruent to the original,
where the angle is still between the two sides? Explain why the two triangles must be
congruent, or why not.
6. State your Claim:
If two sides and the angle between them of one triangle are congruent to two sides and the
angle between them of another triangle, then
___________________________________________________________________. (#THM)
HIDOE Geometry SY 2016-2017
25
L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
Scenario 5: Side-Angle-Angle (SAA)
In this scenario you will explore if having two angles and the side not between them of one
triangle congruent to two angles and one of the sides not between them, of another triangle
guarantees that the two triangles are congruent.
1. Draw a scalene triangle on a sheet of tissue paper.
2.
Using three other pieces of tissue paper, trace two angles and one of the sides not between
them of the triangle onto a separate piece of paper. Mark the ends of each segment to make
them easier to see and extend the rays of the angles.
3. Slide the three pieces together to make a new triangle, making sure the side is still not
between the two angles, and copy the new triangle onto another piece of tissue paper. Recall
that a ray has no end, hence you will only be using a portion of each ray as a side.
4. Is your new triangle congruent to the original? Explain why or why not.
5. Can you rearrange the pieces to create a new triangle that is not congruent to the original,
making sure the side is still not between the two angles? Explain why the two triangles must
be congruent, or why not.
6. State your Claim:
If two angles and the side not between them of one triangle are congruent to two angles and
the side not between them of another triangle, then
___________________________________________________________________.(#THM)
HIDOE Geometry SY 2016-2017
26
L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
Scenario 6: Angle-Side-Angle (ASA)
In this scenario you will explore if having two angles and the side between them of one triangle
congruent to two angles and the side between them of another triangle guarantees that the two
triangles are congruent.
1. Draw a scalene triangle on a sheet of tissue paper.
2.
Using three other pieces of tissue paper, trace two angles and the side between them of the
triangle onto a separate piece of paper. Mark the ends of each segment to make them easier to
see and extend the rays of the angles.
3. Slide the three pieces together to make a new triangle, making sure the side is still between
the two angles, and copy the new triangle onto another piece of tissue paper. Recall that a ray
has no end, hence you will only be using a portion of each ray as a side.
4. Is your new triangle congruent to the original? Explain why or why not.
5. Can you rearrange the pieces to create a new triangle that is not congruent to the original,
making sure the side is still between the two angles? Explain why the two triangles must be
congruent, or why not.
6. State your Claim:
If two angles and the side between them of one triangle are congruent to two angles and the
side between them of another triangle, then
___________________________________________________________________.(#THM)
HIDOE Geometry SY 2016-2017
27
L9 – Congruent Triangles
9.2 – Determining Congruence
Name __________________________________
Per _______ Date _______________________
Summary
Complete the following. Use this sheet as a summary for your class and homework.
1. List the four Congruence Theorems here. Write the acronym and then describe what that
acronym means. Be specific and clear when describing an angle or side.
2. By definition, congruent triangles have ______________________________ and
_____________________________________.
3. Thus, we can say “Corresponding parts of ____________________ triangles are
_______________________.” (#THM) We use this statement very often in geometry. When we
use it, we use an acronym, CPCTC.
Congruence Statement
If ∆𝐴𝐵𝐶 is congruent to ∆𝐷𝐸𝐹, then we write ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹.
So, if ∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹 then complete the following:
∠𝐸 ≅ ______
𝐴𝐵 ≅______
∠𝐶 ≅______
𝐸𝐹 ≅______
∠𝐷 ≅______
𝐴𝐶 ≅______
Now go back to grandma’s garden boxes and write your note to Uncle Bobby.
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.3 – Identifying Congruent Triangles
Name __________________________________
Per _______ Date ________________________
Are We Identical Twins?
Which of the following pairs of triangles are congruent? Explain which criteria for triangle
congruence you used to determine your answer.
1. Are the triangles congruent? Explain why or why not.
2. Are the triangles congruent? Explain why or why not.
3. Are the triangles congruent? Explain why or why not.
4. Are the triangles congruent? Explain why or why not.
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.3 – Identifying Congruent Triangles
Name __________________________________
Per _______ Date ________________________
5. Are the triangles congruent? Explain why or why not.
6. Are the triangles congruent? Explain why or why not.
7. Are the triangles congruent? Explain why or why not.
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.4 – Homework 1
Name __________________________________
Per _______ Date _______________________
Congruent Triangles Homework
There are four pairs of congruent triangles. State how you know they are congruent using the
measurements of the sides and/or angles. You have four methods to use to prove they are
congruent: SSS, SAS, ASA, and AAS. Once you use one method, you may not use it again.
Thus, you must use a different method for each pair. Label each triangle and write a congruence
statement for each.
1. Congruence Statement ______________________
Reason _______________
2. Congruence Statement ______________________
Reason _______________
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.4 – Homework 1
Name __________________________________
Per _______ Date _______________________
3. Congruence Statement ______________________
Reason _______________
4. Congruence Statement ______________________
Reason _______________
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.4 – Homework 1
Name __________________________________
Per _______ Date _______________________
CPCTC means _________________________________________________________
______________________________________________________________________
Using the congruent statement, solve for x. Then, give the length of the sides for each triangle.
∆𝐵𝐶𝐷 ≅ ∆𝐹𝐻𝐺
G
B
𝑥 + 5
𝑥 + 3
11.5
F
2𝑥 − 3
C
D
H
𝐵𝐶 =_________
𝐹𝐻 =_________
𝐵𝐷 =_________
𝐹𝐺 =_________
𝐷𝐶 =_________
𝐺𝐻 =_________
Using the above triangles, we are given that 𝑚∠𝐷 = (4𝑦 + 12)° and𝑚∠𝐶 = (5𝑦 − 8)° and
𝑚∠𝐹 = (6𝑦 − 34)°. Solve for y and use y to find all the angle measures.
𝑦 = ___________
𝑚∠𝐷 = ___________°
𝑚∠𝐹 = ___________°
𝑚∠𝐶 = ___________°
𝑚∠𝐻 = ___________°
𝑚∠𝐵 = ___________°
𝑚∠𝐺 = ___________°
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
34
L9 – Congruent Triangles
9.5 Proving Congruent Triangles
Name __________________________________
Per _______ Date _______________________
Now we will use SSS, SAS, ASA, AAS and CPCTC to prove statements involving congruent
triangles.
1. Given: Aisthemidpointof CE Aisthemidpointof BD Prove: ΔBCA ≅ ΔDEA Whattransformationcouldtake ΔBCA onto ΔDEA ?__________________________
Statement
Reason
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.5 Proving Congruent Triangles
2. Given:
Name __________________________________
Per _______ Date _______________________
GH ! JI ∠G ≅ ∠I GJ ≅ IH Prove:
Whattransformationcouldtake ΔJGH onto ΔHIJ ?__________________________
Statement
Reason
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.6 – Homework 2
Name __________________________________
Per _______ Date _______________________
Prove the following statements.
OR and SP bisect each other 1. Given:
ΔONP ≅ ΔRNS Prove:
Statement
Reason
2. Given: ∠JKM ≅ ∠LKM ∠JMK ≅ ∠LMK Prove:
JM ≅ LM Statement
Reason
HIDOE Geometry SY 2016-2017
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L9 – Congruent Triangles
9.6 – Homework 2
3. Given:
Name __________________________________
Per _______ Date _______________________
R is the midpoint of QS QT ≅ ST Prove:
ΔQRT ≅ ΔSRT Statement
Reason
4. Given: ∠U and ∠X are right angles. ∠W ≅ ∠Y UW ≅ YX ∠V ≅ ∠Z Prove:
Statement
Reason
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.1 – Construction Warmup
Name _____________________________
Per _____ Date _____________________
1. Use a straightedge and compass to construct the perpendicular bisector for the line below.
A
B
2. If P is a point on the perpendicular bisector that does not lie on the line segment AB,
what can you say about the lengths AP and BP?
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.2 – Grandma’s Roof
Name _____________________________
Per _____ Date _____________________
Grandma’s Roof
It’s time to design the roof to Grandma’s House. Many roofs are shaped like isosceles triangles.
1. List below all the things you know about isosceles triangles.
2. Can any of these help us build an isosceles shaped roof?
3. Sketch several examples of an isosceles triangle. What are some of the ways you can test to
see if it’s really isosceles?
4. Think about the perpendicular bisector, as depicted below. This can help us with building an
isosceles triangle. Draw an isosceles triangle on the diagram below. How do you KNOW
what you’ve drawn is isosceles?
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.2 – Grandma’s Roof
Name _____________________________
Per _____ Date _____________________
5. This construction can help us in building our roof. Imagine that this is the front view of
Grandma’s house. What are the four steps you would follow to sketch the roof in such a
manner that you could be assured your roof would be isosceles?
6. Choose how high you want Grandma’s roof, and make a sketch of the house roof below.
7. Grandma sees your plans and decides she wants a roof that is half as tall in the middle. Draw
this shorter roof on the same sketch above. In both examples, which sides of the roof triangle
appear to be congruent?
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.2 – Grandma’s Roof
Name _____________________________
Per _____ Date _____________________
8. Let’s take a break from our building project to do a quick proof.
Given: Point P is located on the perpendicular bisector of line segment AB.
Prove: 𝑃𝐴 ≅ 𝑃𝐵
C
P
A
Statement
Reason
1.
1. Given
2. ?????
𝐴𝑀 ≅ ______
2.
????? ≅ ______
3. 𝑃𝑀
3.
4.
4.
5. _________ ≅ ________
5.
???? ≅ 𝑃𝐵
????
6. 𝑃𝐴
6.
B
M
______
You just proved the Perpendicular Bisector Theorem:
Any point on the perpendicular bisector of a line segment will be __________________ from the
two endpoints of that line segment. (#THM).
You should take particular note of the Perpendicular Bisector Theorem, and its Converse, which
follows. They are very powerful theorems and appear throughout the study of geometry.
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.2 – Grandma’s Roof
Name _____________________________
Per _____ Date _____________________
9. Let’s do another isosceles triangle proof.
• First, use patty paper to reproduce the triangle STU.
• Fold the patty paper in such a way that the fold contains the midpoint of 𝑆𝑈and the
point T.
• Which angles appear to be equal? Let’s prove it!
T
Given: In triangle STU, 𝑆𝑇 ≅ 𝑇𝑈
Prove: ÐS @ ÐU
Hint: First draw the angle bisector of ∠T
Statement
S
U
Reason
The Isosceles Triangle Theorem: If two sides of a triangle are ______________________ ,
then the angles opposite those sides are also ______________________ . (#THM)
10. Make a conjecture for the converse of this theorem:
If two angles of a triangle are ____________________, then the _________ opposite
____________ are also ____________________.
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.2 – Grandma’s Roof
Name _____________________________
Per _____ Date _____________________
Note: The Isosceles Triangle Theorem is equivalent to the Converse of the Perpendicular
Bisector Theorem: If 𝑃𝐴 ≅ 𝑃𝐵 then P must lie on the perpendicular bisector of line segment
AB. (#THM) Thus, when you drop down a perpendicular line from the peak to the base, it must
bisect the base (as long as you already know the two sides are equal, which is given).
11.
Use what you just learned about the Isosceles Triangle Theorem to construct an isosceles
triangle roof for grandma’s house, using only a compass and straightedge.
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.2 – Grandma’s Roof
Name _____________________________
Per _____ Date _____________________
Distance from a point to line
12.
Given: 𝐴𝐸 bisects ÐHAT and point P lies on 𝐴𝐸
H
Prove: 𝐾𝑃 ≅ 𝑃𝐿
E
K
P
A
L
*Hint: use the triangles
Statement
T
Reason
You just proved Theorem 10.1 (#THM):
Given an angle bisector, all points on that bisector are ______________________________
from the sides of the angle.
13.
Work with a partner to come up with a converse for Theorem 10.1. Write the converse
statement below and then add it to your Theorem Booklet (#THM).
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.2 – Grandma’s Roof
14.
Name _____________________________
Per _____ Date _____________________
X
Given: Triangle XYZ is equilateral.
Prove: ÐX @ ÐY @ ÐZ
Y
Statement
Z
Reason _______________
1.
1. Given
???? ≅ ______ ≅ ______
2. 𝑋𝑌
2.
3.
3. Isosceles Triangle Theorem
4.
4. Isosceles Triangle Theorem
5.
5.
Theorem 10.2: An equilateral triangle is also _______________________________, and each
angle has a measure of ______. (#THM)
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.3 – Homework
Name _____________________________
Per _____ Date _____________________
Part I: Find the measure of the indicated side or angle.
1. 𝑊𝑌
2. 𝐽𝐷
W
3. 𝐴𝑇
F
K
12 in
7 mi
7 cm
J
9 mi
D
A
Y
U
4. mÐV
T
5.
T
mÐX
Y
68°
R
S
41°
V
X
Part II:For exercises 6 – 10, refer to the diagram on the right. Set-up and solve an equation
using the information provided.
6. LM = 5, LO = x
7. LM = 2x + 4, LO = 18
8.
9.
LM = 3x - 6, LO = 2x + 21
𝑚∠𝑀 = 20 + 𝑥, 𝑚∠0 = 90 − 𝑥
L
M
HIDOE Geometry SY 2016-2017
O
47
L10 – Perpendicular Lines and Triangles
10.3 – Homework
Name _____________________________
Per _____ Date _____________________
Part III:For exercises 11 – 14, refer to the diagram on the right. Set-up and solve an equation
using the information provided. In the diagram, 𝐸𝐾 bisects ÐYEH
10.
AP = 𝑥 − 5 and PB = −2𝑥 + 25
Y
K
A
P
11.
AP = 4𝑥 R − 12 and PB = 2𝑥 R + 6
E
12. 𝑚∠𝑌𝐸𝐻 = 82∘ 𝑎𝑛𝑑𝑚∠𝐵𝐸𝑃 = 4𝑥 + 9
13.
B
H
∠𝑌𝐸𝐻 = 82∘ 𝑎𝑛𝑑𝑚∠𝐵𝐸𝑃 = 7𝑥
C
14.
Prove the Converse of the Isosceles Triangle Theorem.
Given: ∠𝐴 ≅ ∠𝐵 in triangle ABC
Prove: 𝐴𝐶 ≅ 𝐵𝐶
Statement
Reason
A
HIDOE Geometry SY 2016-2017
B
48
L10 – Perpendicular Lines and Triangles
10.3 – Homework
15.
Name _____________________________
Per _____ Date _____________________
Given: 𝐵𝐶 ≅ 𝐶𝐷 ≅ 𝐷𝐸, ∠𝐹𝐷𝐸 ≅ ∠𝐹
C
Prove: ΔBCD ≅ ΔFED
B
D
F
E
Statement
HIDOE Geometry SY 2016-2017
Reason
49
HIDOE Geometry SY 2016-2017
50
L10 – Perpendicular Lines and Triangles
10.4 – Attic Access Revisited
Name _____________________________
Per _____ Date _____________________
In the previous lesson we proved Theorem 10.1:
Given an angle bisector, all points on that bisector are equidistant from the sides of the angle.
This theorem can be helpful when planning to design Grandma’s attic.
To ensure that opening to the attic opening is the closest distance to both sides of the roof,
Theorem 10.1 tells us that the opening should be located on the angle bisector of the peak angle
(as depicted in the sketch below).
Angle Bisector Theorem (#THM): If a ray bisects an angle of a triangle, then it divides the
opposite side into segments proportional to the other two sides of the triangle.
For example, in the diagram below, since DB is half of CA, the angle bisector theorem tell us that
Y is half of X.
C
20’
A
10’
D
X
Y
B
1. If the length of AB is 24’, what must be the values for X and Y?
2. If the length of AB is 25’, what must be the values for X and Y?
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
52
L10 – Perpendicular Lines and Triangles
10.5 – Finding the Center of Revolution
Name _____________________________
Per _____ Date _____________________
1. Δ𝐴𝐵𝐶 below was rotated 180° about point D. Find point D. Hint: when a single point P is
rotated about point D, the distance from its image P’ and D remains the same as from P to D.
a. Verify that AD = A’D, BD = B’D and CD = C’D.
b. Verify that the distance from the midpoint of BC to D is the same as the distance from the
image of the midpoint of BC to D.
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.5 – Finding the Center of Revolution
Name _____________________________
Per _____ Date _____________________
Recall the Converse of the Perpendicular Bisector Theorem: If point P is equidistant from points
A and B, then P must lie on the perpendicular bisector of 𝐴𝐵.
2. When a figure is rotated 180° about a point, it is relatively easy to find the central point of
rotation. When the figure is rotated X°, where X is NOT 180°, the problem becomes more
difficult. We can use what we’ve learned in this lesson to locate the center.
Let’s begin by rotating a point P, CW X° about a given point D, as illustrated in the diagram
below. (D has purposefully been hidden, and the value for X is irrelevant to this discussion.)
Since P is rotated about D, its image P’ must lie on the circle centered at D with radius PD.
How might you be able to find D? Assume the dotted line circle is hidden from your view.
a. Draw the line segment 𝑃𝑃′.
b. The values PD and P’D must be _________________.
c. Using your answer to part b, along with what you’ve learned in this lesson, it must be the
case that D lies somewhere on the ___________________________ of𝑃𝑃′.
d. Draw a line that must contain the point D.
Note: since in a rotation problem you are not given the dotted-line circle, it is not possible to locate
D with such limited information; you can only limit its location to being some point on a line.
However, if your figure being rotated contains more than a single point, such as with a triangle for
example, then you can use this technique on multiple points (e.g. the vertices) to find D. We do
that in the next problem.
HIDOE Geometry SY 2016-2017
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L10 – Perpendicular Lines and Triangles
10.5 – Finding the Center of Revolution
Name _____________________________
Per _____ Date _____________________
3. Δ𝐸𝐹𝐺 below was rotated clockwise X° about point D. Find point D. Hint: Follow the
technique used in the previous problem for the three vertices, along with the Converse of the
Perpendicular Bisector Theorem.
Reflections:
a. Explain how to find the center of rotation when given a triangle and its rotated image.
b. Do you think this technique would work with other figures? Why?
c. How many pre-image/image pairs of points do you need to locate the center of rotation?
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
56
L11 – Parallelograms and Quadrilaterals
11.1 – Revisiting Congruence
Name _____________________________
Per _____ Date _____________________
Congruent Triangles Review: Decide whether or not each pair of triangles is congruent. If they
are congruent, name the rule/theorem that proves that they are.
Figures
Congruent?
If congruent, state the rule/theorem
that justifies congruence
A.
Yes
No
B.
Yes
No
Yes
No
C.
50°
100°
30°
50°
100°
30°
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.1 – Revisiting Congruence
Figures
Name _____________________________
Per _____ Date _____________________
Congruent?
If congruent, state the rule/theorem
that justifies congruence
D.
Yes
No
E.
Yes
No
Yes
No
F.
10m
8m
4m
10m
8m
4m
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.2 – Properties of Parallelograms
Name _____________________________
Per _____ Date _____________________
Grandma wants to install a window in the space
shown in the diagram. Thinking about all the
shapes she knows, she thinks a parallelogram
may be a good fit for that space in the wall.
To the right is a simple sketch of the wall the
window is going in.
Sketch a picture of what the rest of the
window will look like in the diagram below.
2.
What features do you think a parallelogram has?
LEFTOFFRAME
1.
(#VOC) Parallelogram: A four-sided figure where both pairs of opposite sides are parallel.
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.2 – Properties of Parallelograms
Name _____________________________
Per _____ Date _____________________
Investigation:
3.
In order to determine what the window will look like, we need to construct the rest of the
parallelogram. Do this by drawing the two remaining sides so they are parallel to their opposite
sides.
4.
Now measure all the side lengths of the parallelogram using your ruler or compass. Make a
conjecture about the lengths of the opposite sides of a parallelogram:
Myconjectureaboutthelengthsoftheoppositesidesofaparallelogram:
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.2 – Properties of Parallelograms
Name _____________________________
Per _____ Date _____________________
Grandma is not convinced. You only measured one example of a parallelogram that you sketched on a
piece of paper.
“You think all parallelograms have
congruent opposite sides? Prove it!”
If a quadrilateral is a parallelogram, then the opposite sides are congruent. (#THM)
5.
Prove this theorem.
• To do this proof, we want to draw a diagonal inside our parallelogram; this diagonal creates
two triangles. We can use these triangles to help us prove many properties of
parallelograms.
C
B
D
A
Given: Quadrilateral ABCD is a parallelogram
Hint: Use congruent triangles.
Prove: 𝐶𝐵 ≅ 𝐷𝐴 and 𝐶𝐷 ≅ 𝐵𝐴
Statement
HIDOE Geometry SY 2016-2017
Reason
61
L11 – Parallelograms and Quadrilaterals
11.2 – Properties of Parallelograms
6.
Name _____________________________
Per _____ Date _____________________
Prove the converse of the theorem we just proved.
If two opposite sides of a quadrilateral are parallel and congruent,
then the quadrilateral is a parallelogram. (#THM)
B
C
A
D
Given: 𝐶𝐵 ∥ 𝐷𝐴 and 𝐶𝐵 ≅ 𝐷𝐴
Prove: ABCD is a parallelogram.
Statement
HIDOE Geometry SY 2016-2017
Reason
62
L11 – Parallelograms and Quadrilaterals
11.2 – Properties of Parallelograms
7.
Name _____________________________
Per _____ Date _____________________
If the only information we are given about a parallelogram is that both pairs of opposite sides of
are congruent, can we prove that the quadrilateral must be a parallelogram?
Ifthetwopairsofoppositesidesofaquadrilateralarecongruent,
thenthequadrilateralisaparallelogram.(#THM)
D
Given:𝐷𝐴 ≅ 𝐶𝐵and𝐶𝐷 ≅ 𝐵𝐴.
Prove:ABCDisaparallelogram.
A
C
B
Statement
HIDOE Geometry SY 2016-2017
Reason
63
HIDOE Geometry SY 2016-2017
64
L11 – Parallelograms and Quadrilaterals
11.3 – Homework
1.
D
Name _____________________________
Per _____ Date _____________________
Inthefigurebelow,ABCDisaparallelogram.
•
Usewhatyouknowaboutpropertiesofparallelogramstoset-upandsolvean
equationtodeterminethevalueofx.
•
Then,useyoursolutiontodeterminetheperimeteroftheparallelogram.
𝟑𝒙 + 𝟔
A
C
2.
𝒙
B
𝟓𝒙 − 𝟓
Circlethequadrilateralsbelowthatyouknowforsureareparallelograms.Diagramsarenot
drawntoscale.
A.
B.
C.
D.
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
66
L11 – Parallelograms and Quadrilaterals
11.4 – Parallelogram Angle Proofs
Name _____________________________
Per _____ Date _____________________
Tocuttheglassforthewindowintoaparallelogram,theglasscompanyneedstoknowthe
measureofeachangleoftheparallelogram.Wehavealreadyprovedthattheoppositesidesmust
bethesamelength.Butisitalsotruethatoppositeanglesmustbecongruent?
Investigation:
1. Constructthreeuniqueparallelogram-shapedwindowsusingthesegmentprovidedforone
sideoftheparallelogram.Note:constructeachparallelogramsothatadjacentsidesarenotthe
samelength.
A.ConstructparallelogramPQRSusing𝑃𝑄foronesideoftheparallelogram.Appropriatelylabel
allfourverticesofyourparallelogram.
P
Q
B.ConstructparallelogramEFGHusing𝐸𝐹foronesideoftheparallelogram.Appropriatelylabel
allfourverticesofyourparallelogram.
E
HIDOE Geometry SY 2016-2017
F
67
L11 – Parallelograms and Quadrilaterals
11.4 – Parallelogram Angle Proofs
Name _____________________________
Per _____ Date _____________________
C.ConstructparallelogramJKLMusing𝐽𝐾foronesideoftheparallelogram.Appropriatelylabel
allfourverticesofyourparallelogram.
J
K
2. Forallthreeparallelogramsthatyoujustconstructed,useaprotractortomeasureeachangle
(tothenearestdegree).Writethemeasureineachangle.
3. Analyzetheanglemeasureswithineachparallelogram.Maketwoconjecturesaboutthe
measuresoftheangleswithinaparallelogram.
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.4 – Parallelogram Angle Proofs
Name _____________________________
Per _____ Date _____________________
4.
Aretheconsecutiveanglesofparallelogramsalwayssupplementary?Let’strytoproveit.
Consecutiveanglesinaparallelogramaresupplementary.(#THM)
Given:ABCDisaparallelogram.
Prove:∠𝐶𝐷𝐴and∠𝐵𝐴𝐷aresupplementary.
B
C
A
D
***Inthediagramabove,eachsideoftheparallelogramhavebeenextended(withdashedlines)
sothatitisalittleeasiertoseetheparallellines.Whathavewelearnedpreviouslyaboutthe
relationshipsbetweenanglesformedwhenparallellinesarecutbyatransversal?
Statement
Reason
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.4 – Parallelogram Angle Proofs
5.
Name _____________________________
Per _____ Date _____________________
Aretheoppositeanglesofparallelogramsalwayscongruent?Let’strytoproveit.
Oppositeanglesofaparallelogramarecongruent.(#THM)
Given:ABCDisaparallelogram.
Prove:∠𝐶𝐷𝐴 ≅ ∠𝐶𝐵𝐴and∠𝐵𝐴𝐷 ≅ ∠𝐵𝐶𝐷
B
C
A
D
***Inthediagramabove,adiagonalhasbeendrawninsideoftheparallelogram.Whatdoesthis
diagonalcreateinsideoftheparallelogram?Howmightithelpusprove∠𝐶𝐷𝐴 ≅ ∠𝐶𝐵𝐴and
∠𝐵𝐴𝐷 ≅ ∠𝐵𝐶𝐷?
Statement
HIDOE Geometry SY 2016-2017
Reason
70
L11 – Parallelograms and Quadrilaterals
11.5 – Homework
Name _____________________________
Per _____ Date _____________________
Forexercises1and2,thequadrilateralsshownareparallelograms.Eachvariablerepresentsthe
measureoftheangle.Determinethevalueofeachvariable.
1.
x
120°
z
y
2.
34°
b
c
a
3.KPLRisaparallelogram.𝑚∠𝑃 = 7𝑥 − 75and𝑚∠𝑅 = 3𝑥 + 45.Set-upandsolveanequation,
thenuseyoursolutiontodeterminethevalueofallfouranglesofparallelogramVWXY.
K
P
R
L
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.6 – Diagonals of Parallelograms
Name _____________________________
Per _____ Date _____________________
Often,interestingdesignscanbecreatedbydrawinginthediagonalsofafigure.
Diagonalsofaparallelogramcanbedrawnbyconnectingbothpairsofoppositevertices.
(#VOC)Diagonal:Alineconnectingnon-consecutiveverticesofapolygon.
1. Intheparallelogrambelow,usearuler(orastraight-edgetool)todrawthetwodiagonals.
2. Whentwodiagonalsintersect,theycuttheothertoformfoursmallerlinesegments?
Measurethelengthsofthefourlinesegmentscreatedbythediagonalsinyourfigureabove.
Writeeachmeasurenexttoeachsegment.
3. Analyzethelengthsofthefoursegmentsyoujustmeasured.Makeaconjectureaboutthe
lengthsofthesegmentscreatedwhenyoudrawinbothdiagonalsofaparallelogram.
4.
Repeatsteps1and2abovewiththebelowparallelogram.Doesyourconjecturestillhold?
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.6 – Diagonals of Parallelograms
Name _____________________________
Per _____ Date _____________________
5.
Dothediagonalsofaparallelogramsalwayscuteachotherintotwocongruentsegments?
Let’strytoproveit.
Thediagonalsofaparallelogrambisecteachother.(#THM)
Given:ABCDisaParallelogram.
𝐶𝐴and𝐷𝐵arediagonalsofABCD.
Prove:𝐷𝐸 ≅ 𝐸𝐵and𝐶𝐸 ≅ 𝐸𝐴
C
B
E
A
D
Statement
Reason
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.6 – Diagonals of Parallelograms
Name _____________________________
Per _____ Date _____________________
Grandmalikestheideaofhavingawindowintheshapeofaparallelogram.Butshedoesn’tlike
thedesigncreatedbydrawinginthediagonalsoftheparallelogram
Instead,shewantstohavethewindowintheshapeofaparallelogram,butwithonly2sections.
Youthinkoftheideatoconnectthemidpointsofthetopandbottomsidesoftheparallelogram.
Youdrawthediagrambelowandaskherwhatshethinks.
6. Let’sproveatheoremaboutthesegmentthatconnectsthemidpointsofoppositesidesofa
parallelogram.
Inaparallelogram,thelinesegmentcontainingthemidpointsofoppositesides
isparalleltotheothertwosides.(#THM)
Given:ABCDisaParallelogram.
MandNarethemidpointsofthe
oppositesides𝐴𝐷and𝐵𝐶,respectively.
D
Prove:𝑀𝑁isparallelto𝐷𝐶and𝐴𝐵
Statement
M
C
A
N
B
Reason
HIDOE Geometry SY 2016-2017
75
L11 – Parallelograms and Quadrilaterals
11.6 – Diagonals of Parallelograms
7.
Name _____________________________
Per _____ Date _____________________
Arectangleisaspecialparallelogrambecauseithasadditionalproperties.Several
interestingthingshappenwhenwedrawinthediagonalsofarectangle.
Let’sapplywhatwe’velearnedpreviouslyaboutdiagonalsofaparallelogramandcongruent
trianglestocompletetheproofbelow.
C
X
D
Given:𝐶𝐴and𝐷𝐵arediagonalsofrectangleABCD
Prove:DDAXisisosceles
Statement
HIDOE Geometry SY 2016-2017
B
A
Reason
76
L11 – Parallelograms and Quadrilaterals
11.7 – Congruent Quadrilaterals
Name _____________________________
Per _____ Date _____________________
Recallthatininformalwaytoverifyiftwopolygonsarecongruentistoshowthatthefigures
havethesamesizeandsameshape.
1.Foreachpairoffiguresbelow,answerthefollowingquestions.
• Arethequadrilateralscongruent?Usepattyormeasuringtoolstoconfirmyouranswer.
• Ifthequadrilateralsarecongruent,writecongruencestatementsthatindicateALLofthe
correspondingsidesandanglesthatarecongruent(e.g.,𝐷𝐸 ≅ 𝐸𝐵).
A.
R
J
S
K
L
T
M
U
B.
O
V
W
P
Q
X
N
Y
HIDOE Geometry SY 2016-2017
77
L11 – Parallelograms and Quadrilaterals
11.7 – Congruent Quadrilaterals
Name _____________________________
Per _____ Date _____________________
Followingyourteacher’sinstructions,usethenextfewquestionstotakeguidednotesabout
congruentquadrilaterals.
2.CongruentQuadrilaterals
• Theyhaveexactlythesame_______________________________and_______________________________.
• Youusearigidtransformationtoverifyiftheyarecongruent.Forexample,youcould
_______________________________,_______________________________,or_______________________________
oneofthefiguressothatitfitsexactlyontheotherone.
•
•
Corresponding_______________________________andcorresponding_______________________________
arecongruent.
Whenwritingacongruencestatement,eachquadrilateralshouldbenamedsuchthatthe
correspondingverticesappearinthe________________________________________________________.
N
3. Thetwoquadrilateralsshownarecongruent.
A
A.Fillintheblankstoindicateallpairs
ofcorrespondingvertices,sidesandangles. R
Rcorrespondsto______
correspondsto____________
ÐRcorrespondsto___________
Ncorrespondsto______
correspondsto____________
ÐNcorrespondsto___________
Acorrespondsto______
correspondsto____________
ÐAcorrespondsto___________
Fcorrespondsto______
𝐹𝑅correspondsto____________
ÐFcorrespondsto___________
Y
G
F
L
P
B.Writethecongruencestatementforthetwoquadrilateralsshown:_______________________________
HIDOE Geometry SY 2016-2017
78
7.1 – Congruent Quadrilaterals
Per_____ Date_______________________
Example 2: COAT ! KIND. Find:
N
Quadrilaterals
– Congruence & Characteristics
Name_______________________________
8 cm
L11 – Parallelograms
and2Quadrilaterals
Name _____________________________
50°
7.1
–
Congruent
Quadrilaterals
Per_____
Date_______________________
11.7 – Congruent Quadrilaterals
Per _____ Date _____________________
a) Use the congruence statement to
O
50°
list corresponding
Example 2: vertices.
COAT ! KIND. Find:
A
D
N
8 cm
4.Forthefiguresshowntotheright,COAT@KIND.
a) Use the congruence statement to
O
list corresponding vertices.
A.Usethecongruencestatementtolist
130°50°
correspondingvertices.
C
T
b) m"A
____________correspondsto____________
130°
C
____________correspondsto____________
b) m"A
50°
A
D
K
8 cm
I
T
K
c) m"D
____________correspondsto____________
8 cm
I
____________correspondsto____________
c) m"D
C
!
T
d) AT
B.mÐA=
C.mÐD=____________
D.
=____________
C
T
AT
d)____________
Example 3: Find the value of the variables, x and y, if CHLD ! PNTS.
T
S
C
Example 3: Find the value of the variables, x and y, if CHLD ! PNTS.
x
T
S
5. !Forthefiguresshowntotheright,
•
CHLD@PNTS
•
xrepresentsthemeasureofÐS
•
yrepresentsthelengthof𝐶𝐷
C
40°
4xyd
y
40°
4 yd
y
D
DN
Determinethevaluesofxandy.
5 yd
N
H
3 yd
H
3 yd
•
4x+2representsthemeasureofÐR
17 mm
H
H
(8z–2)° (8z–2)°
•
5y–7representsthelengthof𝐴𝑇
•
8z–2representsthemeasureofÐH
21 mm
P
L
6. Forthefiguresshowntotheright,
MATH@PRTY
5 yd
L
R
Example 4: Find the value of the variables, x, y, and z, if MATH ! PRTY.
Example 4: Find the value of the variables, x, y, and z, if MATH ! PRTY.
•
P
T
8 mm
(4x+2)°
R
P8 mm
(4x+2)°
P
17 mm
T
38°
38°
Y
21 mm
Determinethevaluesofx,yandz.
Geometry
5y–7
M
M
90°
A
Geometry
HIDOE Geometry SY 2016-2017
5y–7
90°
A
Q2 Quadrilaterals Handouts
Q2 Quadrilaterals Handouts Page 4
Page 4 79
Y
HIDOE Geometry SY 2016-2017
80
L11 – Parallelograms and Quadrilaterals
11.8 – Special Quadrilaterals and their Properties
Name _____________________________
Per _____ Date _____________________
Followingyourteacher’sinstructions,usethenextfewquestionstotakeguidednotesabout
specialquadrilaterals.
1. Reviewthedefinitionsofthefollowingspecialquadrilaterals.
A. Quadrilateral:_____________________________________________________________________________________.
B. Parallelogram:aquadrilateralwith____________________________________________________________.
C. Rhombus:aPARALLELOGRAMwith_____________________________________________________.#VOC
D. Rectangle:aPARALLELOGRAMwith_____________________________________________________.#VOC
E. Square:aPARALLELOGRAMwith________________________________________________________.#VOC
F. Trapezoid:aquadrilateralwith____________________________________________________________.VOC
G. IsoscelesTrapezoid:aTRAPEZOID______________________________________________________.#VOC
2.
3.
H. Kite:aquadrilateralwith__________________________________________________________________.#VOC
Fromthefiguretotheright,whatcanyouconcludeabout𝐴𝐵and𝐶𝐷?Brieflyexplainhow
youknowyourconclusionistrue.
A
B
C
D
Fillintheblanktocreateatruestatement:
Basedonthedefinitionsabove,therhombus,rectangleandsquareareallspecialtypesof
_____________________________________________________.
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.8 – Special Quadrilaterals and their Properties
4.
Name _____________________________
Per _____ Date _____________________
Thediagrambelowshowsonewaytoorganizeourthinkingaboutspecialquadrilaterals.
•
Wecanclassifyquadrilateralsinto3categoriesbasedonthenumberofparallelsides
thatthefigurehas:
o nopairsofparallelsides
o exactly1pairofparallelsides
o 2pairsofparallelsides
Completethediagrambywritingthenameofthespecialquadrilateralineachfigure.
Quadrilateral
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.8 – Special Quadrilaterals and their Properties
5.
Name _____________________________
Per _____ Date _____________________
Workingwithapartner,usethedefinitionsandthediagramonthepreviouspagesto
determineifthefollowingstatementsaretrueorfalse.Ifthestatementisfalse,eitherexplain
whyordrawafigurethatshowsanexampleofwhythestatementisfalse.
Statement
True of False
A. All trapezoids are quadrilaterals.
True
False
B. All rectangles are squares.
True
False
C. An isosceles trapezoid can be a
kite.
True
False
D. All squares are rhombuses.
True
False
E. All parallelograms are rhombuses. True
False
If false, either explain why or draw
afigurethatisacounterexample.
HIDOE Geometry SY 2016-2017
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L11 – Parallelograms and Quadrilaterals
11.8 – Special Quadrilaterals and their Properties
6.
A.
B.
Name _____________________________
Per _____ Date _____________________
Findthevaluesofthevariablesandthelengthsofthesidesofeachquadrilateral.
LMNO is a parallelogram. Determine the value of m and s.
O
2m + 8 N
s+1
L
5m - 2
3m - 1 M
FGHI is a square. Determine the value of f and g.
F
3f + 2
G
2g - 5
g+6
I
H
5f - 8
HIDOE Geometry SY 2016-2017
84
L11 – Parallelograms and Quadrilaterals
11.8 – Special Quadrilaterals and their Properties
7.
A.
B.
C.
Name _____________________________
Per _____ Date _____________________
Usethequadrilateraltotherighttoanswerthefollowingquestions.
39°
Based on the information provided for the figure, what is
the most appropriate name for the quadrilateral?
Provide a brief explanation why you selected that name.
3
Determine the sum of all 4 angles of the quadrilateral. Show or explain how you determined your
answer.
U
Complete the following proof regarding the sum of the angles
of any quadrilateral.
Given:QuadrilateralQUADwithdiagonal𝑄𝐴
Prove:mÐQ+mÐU+mÐA+mÐD=360
1
Statement
A
Q
D
Reason
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
86
L11 – Parallelograms and Quadrilaterals
11.9 – Theorems of Special Quadrilaterals
Name _____________________________
Per _____ Date _____________________
PROPERTIESOFSPECIALQUADRILATERALS
1. Foreachquadrilateralbelow,dothefollowing:
-drawinandmeasurethelengthsofthediagonals.
-measureallanglesinthefigure.
-choosethebestnameforthequadrilateral
a)
b)
c)
d)
e)
f)
2. Usingwhatyoulearnedabove,
a. Whatcharacteristic(s)doallrectangleshave?
Rectangleshave___________________________________________________
b. Whatcharacteristic(s)doallrhombihave?
Rhombihave______________________________________________________
c. Whatcharacteristic(s)doallisoscelestrapezoidshave?
Isoscelestrapezoidshave_____________________________________________
d. Whatcharacteristic(s)doallkiteshave?
Kiteshave________________________________________________________
HIDOE Geometry SY 2016-2017
87
L11 – Parallelograms and Quadrilaterals
11.9 – Theorems of Special Quadrilaterals
Name _____________________________
Per _____ Date _____________________
SPECIALQUADRILATERALTHEOREMS
Theorem:Eachdiagonalofarhombus__________
twoanglesoftherhombus.#THM
A
D
B
C
A
B
D
Theorem:Thediagonalsofarhombusare__________.
#THM
C
S
T
M
Theorem:Thediagonalofarectangleare__________.
#THM
I
E
W
V
Y
X
Theorem:Baseanglesofanisoscelestrapezoidare_______________.
#THM
Theorem:Thediagonalsofanisoscelestrapezoidare___________.
#THM
P
N
Theorem:Thediagonalsofakiteare_______________.
E
HIDOE Geometry SY 2016-2017
H
P
U
N
H
U
K
I
T
88
L11 – Parallelograms and Quadrilaterals
11.9 – Theorems of Special Quadrilaterals
Name _____________________________
Per _____ Date _____________________
3.Foreachquadrilateral,
•
statethemostappropriatenameforthefigure
• findthemeasureofeachnumberedangle(e.g.,1,2,3,4).
18°
A.
B.
54°
4
1
4
23
C.
D.
121°
1
2
1
3
59° 4
E.
F.
2
1
1
79°
65°
HIDOE Geometry SY 2016-2017
2
3
1
2
2
89
L11 – Parallelograms and Quadrilaterals
11.9 – Theorems of Special Quadrilaterals
Name _____________________________
Per _____ Date _____________________
4. Theperimeterofakiteis80ft.Thelengthofoneofitssidesis5lessthan4timesthelength
ofanother.Findthelengthofeachsideofthekite.Hint:Drawapicture.
5. Findthevalueofthevariables.
A.
B.
(6x + 20)°
2x°
3x°
y°
4x°
(10x - 6)°
C.
3x - 3
x-1
x+5
HIDOE Geometry SY 2016-2017
90
L11 – Parallelograms and Quadrilaterals
11.9 – Theorems of Special Quadrilaterals
Name _____________________________
Per _____ Date _____________________
6.ProvethefollowingTheorem:Diagonalsofarhombusbisectoppositeanglesofarhombus.
S
B
Given:RhombusRMBSwithdiagonal𝑆𝑀
Prove:ÐRSM≅ÐBSMandÐRMS≅ÐBMS
R
M
7.ProvethefollowingTheorem:Thediagonalsofarectanglearecongruent. Given:RectangleRECT
M
A
Prove:𝑀𝑇 ≅ 𝐻𝐴
HIDOE Geometry SY 2016-2017
H
T
91
HIDOE Geometry SY 2016-2017
92
L12 – Dilations and Similarity
12.1 – Refresher on Ratios
Name _____________________________
Per _____ Date _____________________
1. Completethetablebelow.Thefirstrowiscompletedforyouasanexample.
Equation 1
Equation 2 (equation 1 re-written
so that the variables are on
opposite sides of the equation)
Interpretation of
Equation 2
x
=3
y
x = 3y
x is 3 times as big as y
x to y is 3
x to y is ½
x
=2
y
x=
1
y
4
x is 5 times as big as y
x to y is
0.3
2. Statetheratioofthelengthsofthefollowingpairsofsidesofthetriangle.
`a
ab
=
`b
ab
=
2 m.
1.5 m.
`a
1 m.
HIDOE Geometry SY 2016-2017
`b
=
`b
`a
=
93
L12 – Dilations and Similarity
12.1 – Refresher on Ratios
Name _____________________________
Per _____ Date _____________________
3. Ineachpicture,determineifthereisarigidmotiontransformationwhichtakesonefigureto
theother.Ifnoneexists,explainwhy.
B
A
C
D
E
HIDOE Geometry SY 2016-2017
F
94
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
Name _____________________________
Per _____ Date _____________________
1. Below is a bird’s eye view of Grandma’s Gazebo, as depicted on the architect’s blueprints.
Because of her poor eyesight, Grandma is having difficultly seeing the details of the Gazebo
drawing. Help her out by drawing a new version that is twice as big, by following the
instructionsbelow.
F
E
iii.
iv.
B
G
D
i.
ii.
A
C
Fromthecenter,extendthedottedlinesegment GA tobetheray𝐺𝐴.Repeatfortheray𝐺𝐵.
Measurethelengthofthelinesegments GA and GB .(Theyshouldbeequal,andinfactalso
equaltoAB,since∆𝐴𝐵𝐺isequilateral.)
Markthepoints A ' and B ' ontherays𝐺𝐴and𝐺𝐵,respectively,sothatthelinesegments GA '
and GB ' aretwiceaslongaslinesegments GA and GB .(Usearulerorcompass.)
Nowmeasurethelengthofthelinesegment𝐴′𝐵′.
a. Whatistherelationshipbetweenthelengthof𝐴′𝐵′andthelengthof𝐴𝐵?
!##"
!###"
b. Howarethelines AB and A' B ' related?(Note:notthelinesegments,butthelines.)
HIDOE Geometry SY 2016-2017
95
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
Name _____________________________
Per _____ Date _____________________
Imagine repeating Steps 3 and 4 at each vertex. What would be the shape of the resulting figure?
c. Tryitout.RepeatSteps3and4ateachvertex.Whatdoyounotice?
d. UsepattypaperoraprotractortocomparetheanglesABCandA’B’C’,andtheangles
ABGandA’B’G.Whatdoyounotice?
e. MeasurethedistancefromthecenterGtothemidpointof AB .Whatdoyouthink
thedistanceisfromthecenterGtothemidpointof A ' B ' ?Tryitout.
f. Repeatthiscomparisonforthelinesegmentsconnectingthemidpointsof AB and
CD ,andthoseconnectingthemidpointsof A ' B ' and C ' D ' .Whatdoyounotice?
g. Whatistheratioofthecorrespondingsidesbetweenthedouble-scaleddrawingand
theoriginal?
The ratio is:
This means that when we extend the rays from a specific point of the gazebo by a
multiple of _________, then the side lengths multiply by a factor of ______________.
h. Whatistheratioofthecorrespondinganglesbetweenthedouble-scaleddrawing
andtheoriginal?
The ratio is:
This means that when we extend the rays from a specific point of the gazebo by a
multiple of _________, then the angles ____________________________.
Whenweincreasethesizeofthegazebo,thelengthmeasurements_______________,
buttheanglemeasurements____________________________.
HIDOE Geometry SY 2016-2017
96
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
i.
j.
Name _____________________________
Per _____ Date _____________________
Whatdoyouthinktheratioofcorrespondingsideswouldbeifwehadcreatedatriplescalediagram?
Whatdoyouthinktheratioofcorrespondingangleswouldbeifwehadcreatedatriplescalediagram?
Summary: This dilation transformation seems to have preserved the _________________, but
increased the ________________.
HIDOE Geometry SY 2016-2017
97
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
Name _____________________________
Per _____ Date _____________________
2. Grandma’seyesareprettyweak.Usetheprocesswejustcompletedearliertodrawacopyof
thehexagongazebothatisthreetimeslargerthantheoriginalpicture.
F
E
A
B
G
D
C
A. What is the ratio of the diagonals of your new diagram to the original one?
B. Do you think we would get a different result if we increased the size by an integer multiple (2, 3, 5,
etc.)? For example, suppose we wanted to make the gazebo drawing 5.3 times as large; would we
end up with the same result as above or would we get a different result?
HIDOE Geometry SY 2016-2017
98
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
Name _____________________________
Per _____ Date _____________________
3. Supposewetookahexagonwithsidelengthsof3inches(i.e.yourresultsfromtheprevious
problem)andplaceditonthegroundsothatitscenterwasattheprecisecenterofwhere
Grandmawilllocatehergazebo.Grandma’sGazebohassidelengthsof8feet.Calculatehow
manytimeslargertheactualgazeboisthanthediagramplacedontheground,byfillinginthe
followingblanks.
Gazebo drawing side length: _________ inches
Gazebo actual side length: _________ feet = ____________ inches
How many times larger: _________________
a. Inordertobuildahexagonalgazebowith8ftsides,wewouldneedtoincreasethesizeof
ourdiagrambyafactorof_______,whichiswhatwecallthesizeratiobetweenthetwo
hexagons.
b. Measuretheradiusofyour3-inchsidedGazebofromProblem2,itshouldalsobe3”.
Therefore,theradiusoftheactualGazeboshouldbe_________________.
c. Ifwewantedtoincreaseourcurrentdrawing(ofsidelength3inches)toalargerdrawing
with1footsides,wewouldneedtoincreasethesizeofourdiagrambyafactorof______,the
sizeratiobetweenthetwohexagons.
d. This ratio is referred to as the scale factor (#VOC), which is a non-negative number that we use to
increase or decrease the size of our image, while maintaining the same shape.
4. AssumeyouarethecontractorandyouneedtolayouttheGazebowith8ft.sides.Youknow
wherethecenterislocated,butyouneedtolocateeachofthevertices.Youcanassumeyou
haveaverylargecompass,sinceacompasscanbeemulatedwithastringattachedtoastakein
theground(e.g.locatedatthecenteroftheGazebo),andthatyouhavea25ft.tapemeasure,
whichisessentiallyalongruler.WhatstepswouldyoufollowtolayouttheGazebo?Note:you
maychooseyourfirstvertexarbitrarily,aslongasitsdistancefromthecenteriscorrect.
HIDOE Geometry SY 2016-2017
99
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
5.
Name _____________________________
Per _____ Date _____________________
Follow your teacher’s instructions to apply the concept of “scale factor” to compare, enlarge or
compress various objects.
Original object’s side lengths: ___________, ____________, ______________, ___________
Projected object’s side lengths: __________, ____________, ______________, ___________
Ratio between lengths: ___________, ____________, ______________, ___________
Therefore, scale factor is: ______________
HIDOE Geometry SY 2016-2017
100
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
6.
Name _____________________________
Per _____ Date _____________________
For each pair of figures below, thefigureonthelefthasbeendilatedtocreatethefigureon
theright.Measurethelengthsofanysidesthatareneeded(inmillimeters)anduseyour
measurementstostatethescalefactorbetweenthetwofigures.
A.
Scalefactor:
B.
Scalefactor:
C. Scalefactor:
D. Scalefactor:
HIDOE Geometry SY 2016-2017
101
L12 – Dilations and Similarity
12.2 – Grandma’s Gazebo
Name _____________________________
Per _____ Date _____________________
7. Pairupthefollowingtrianglesthatappeartoberelatedbyadilation.
•
•
Then,measurethelengthofeachsidetohelpyoudetermineifthefiguresrepresentan
actualdilation.
Ifso,statethescalefactorthatwasusedtodilatethefirstfiguretocreatethesecond
figure.
B
A
C
D
E
Pair 1:
F & _______________
Scale Factor: _________________
Pair 2:
C & _______________
Scale Factor: _________________
Pair 3:
D & _______________
Scale Factor: _________________
HIDOE Geometry SY 2016-2017
F
102
L12 – Dilations and Similarity
12.3 – Dilations
Name _____________________________
Per _____ Date _____________________
The type of transformation that we used to enlarge Grandma’s Gazebo, where we expanded our figure
from its center, is an example of a special type of non-rigid transformation that we refer to as dilation,
which we will formally define below.
Let’s try to dilate figures from points other than a center.
1. Belowis DABC .Dilatethistrianglebyascalefactorof2fromvertexAandlabelthenew
triangle DAB ' C ' .HerepointAisplayingthesameroleaspointGintheGazeboexercise.
Followthesamestepsusedthere(i.e.drawaraywithendpointAthroughpointB,etc.)
B
A
C
2. Check that your work is accurate. Measure each side length of DABC and DAB ' C ' to determine if
the ratio of their lengths confirms our scale factor of 2. Show your work below. Remember to label
your measurements with the side lengths they represent.
3. Are the angles of the new triangle the same as the corresponding angles of the original triangle?
4. When dilating a triangle using one of its vertices as the center of dilation, how many individual
image points completely determine the entire image?
HIDOE Geometry SY 2016-2017
103
L12 – Dilations and Similarity
12.3 – Dilations
5.
Name _____________________________
Per _____ Date _____________________
WorkwithapartnertodiscussandanswerquestionsA–Dbelow.
A. Howdoestheimageofadiagramcomparetotheoriginaldiagramwhenitisdilatedusinga
scalefactorof2?
B. Howdoestheimageofadiagramcomparetotheoriginaldiagramwhenitisdilatedusinga
scalefactorof1?
C. Howdoestheimageofadiagramcomparetotheoriginaldiagramwhenitisdilatedusinga
scalefactorof½?
D. Howdoestheimageofadiagramcomparetotheoriginaldiagramwhenitisdilatedusinga
scalefactorof1.5?
6.
Nowthatyouhavepracticedtheskillofdilationfromavertex,practicethisnon-rigid
transformationonthetrianglesbelowwithnon-integerscalefactors(usingpattypaperand
astraightedge).
A. DEFG dilatedto DEF ' G ' withscalefactorof1.5
E
G
F
HIDOE Geometry SY 2016-2017
104
L12 – Dilations and Similarity
12.3 – Dilations
Name _____________________________
Per _____ Date _____________________
B. DKLM dilatedto DK ' L ' M withascalefactorof0.5
K
L
M
C. DQRS dilatedto DQ ' RS ' withscalefactor2.5
R
S
Q
HIDOE Geometry SY 2016-2017
105
L12 – Dilations and Similarity
12.3 – Dilations
7.
Name _____________________________
Per _____ Date _____________________
We now move to the most general type of dilation, where the point of dilation is outside the figure.
Follow the directions below to dilate DABC from point D with a scale factor of 2.
D
B
C
A
i.
Drawrays𝐷𝐴and𝐷𝐵.
ii.
MeasurethelengthsDAandDB.
iii.
MarkthepointA’andB’ontherays𝐷𝐴and𝐷𝐵,respectively,sothatthelinesegments DA '
and DB ' aretwiceaslongaslinesegments DA and DB .(Inotherwords,pointsA’andB’
aretwiceasfarfromDasAandB,respectively.)
Howarethelengths AB and A ' B ' related?
!##"
!###"
!##"
!####" !##"
!####"
Howarethelines AB and A' B ' related?Whatabout AC and A'C ' , BC and B 'C ' .
iv.
RepeatthisprocessforpointCtoformtriangleA’B’C’.
v.
Usepattypaperoraprotractortocomparecorrespondinganglesinthetwotriangles.What
doyounoticeaboutthecorrespondinganglesinthetwotriangles?
HIDOE Geometry SY 2016-2017
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L12 – Dilations and Similarity
12.3 – Dilations
Name _____________________________
Per _____ Date _____________________
You have now seen 3 types of dilation:
1) from the center of the figure (Grandma’s Gazebo)
2) from a vertex of a figure, and
3) from a point outside a figure.
A dilation with center D and scale factor r (#VOC) is a transformation that takes each point P (P not
equal to D) to a point P’ along the ray 𝐷𝑃 such that the ratio of the distances
(Note: Unless the scale factor is 1, a dilation is not a rigid transformation).
8.
cd e
cd
= 𝑟.
Follow your teacher’s instructions and use the space below to take guided notes.
A. A scale factor is a ___________________ number.
B. A dilation from point D with scale factor r takes a line segment of length L to a line segment of
length r L . In other words, if the image under this dilation of line segment AB is the line
segment A ' B ' , then the _______________of A ' B ' is r-times the length of _____________.
!##"
!###"
C. Furthermore, the lines AB and A' B ' are _______________. As a result, the image of a dilation
of a polygon will result in a polygon that has the same ____________________________but the
length of the _______________will be ____________________________by a factor of r.
D. A dilation with center D and scale factor r takes a line not containing D to a
_________________________________________.
E. A dilation with center D and scale factor r takes a line containing D to
_________________________________________.
F. A dilation with center D and a scale factor r takes an angle formed by rays AB and AC to a
_____________________________. In order to dilate a triangle, using one of the vertices as the
center of dilation, you only need to locate the image for __________________________ in order
to completely determine the dilated image.
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
108
L12 – Dilations and Similarity
12.4 – Introduction to Similarity
Name _____________________________
Per _____ Date _____________________
A fundamental question in Geometry is “Does Figure A have the same shape as Figure B?” What we have
seen is that the image of a dilation has the same shape as its pre-image. Two such objects, the image and
the pre-image, are said to be similar. Before formalizing the meaning of similarity, let’s intuitively
investigate the connection between this important concept and congruence.
1. Thefollowingpairsofobjectsintuitivelyappeartobesimilarinthesensethattheyappearto
havethesameshape.Usepattypaperoraprotractortocheckthecongruencyoftheir
correspondingangles.
HIDOE Geometry SY 2016-2017
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L12 – Dilations and Similarity
12.4 – Introduction to Similarity
Name _____________________________
Per _____ Date _____________________
2. Onewaytodetermineifapairoffiguresissimilaristoattempttomatchuptheangles.However,
sincethesizeofthefiguresdiffer,youcanonlymatchuponeangleatatime.
Gobackandtraceontopattypaperthefigureontherightineachofthethreepreviouspairs.
Laythepattypaperwiththefigureyoutracedapproximatelywhereitwaspreviouslylocated.
Slideitaround,asifyouwereapplyingarigidmotiontransformationuntiloneofthevertices
anditsanglematches.Note:youmayalsoneedtoflipyourpattypaperoverifareflectionis
required.Nowthatanangleismatchedup,doesthereappeartobeadilationthatwould
transformthesmallerfigureintothelargerfigure?
Whatistheapproximatescalefactorforeachpair?
3. Let’sinvestigatetheconnectionbetweensameshapeandcongruencewithafinalexample.
a. TranslateΔ𝐴𝐵𝐶belowuntilvertexAcoincideswithvertexD.Usepattypaperorcutout
Δ𝐴𝐵𝐶.
b. RotatethisimageofΔ𝐴𝐵𝐶anappropriateamountclockwiseuntil𝐴𝐶ishorizontal(asis
linesegment𝐷𝐹).
c. ReflectthesmallertriangleimageovertheverticallinepassingthroughpointD.
E
B
A
D
F
C
It should now be clear that angles A and D are congruent, and further, if you were to dilate the
smaller triangle by an appropriate scale factor it would coincide precisely with Δ𝐷𝐸𝐹, which
shows the two triangles have congruent angles.
In conclusion, a similarity transformation (#VOC) is a rigid motion transformation (e.g. the translation,
rotation, and reflection used above) followed by a dilation. Similarity transformations demonstrate the
connection between objects that have the same shape, but are not congruent (i.e. they can be made
congruent via an appropriate dilation).
Figure 1 and Figure 2 are said to be similar (#VOC) if Figure 2 is the image of Figure 1 under a
similarity transformation (e.g. Δ𝐷𝐸𝐹 coincided with the image of the similarity transformation of
Δ𝐴𝐵𝐶, so the two triangles are similar, which simply means they have congruent corresponding angles).
The notation we use is Δ𝐴𝐵𝐶~Δ𝐷𝐸𝐹.
HIDOE Geometry SY 2016-2017
110
L12 – Dilations and Similarity
12.4 – Introduction to Similarity
Name _____________________________
Per _____ Date _____________________
4. Twoboatsleaveadockatthesametime.BoatAtravels10milesperhourdirectlysouthand
BoatBtravels30milesperhourdirectlyEast(Seediagrambelow.)Ateachmomentintimethe
threepointscorrespondingtothedockandthepositionsofthetwopointsformatriangle.Is
thetriangleformedafter10minutesoftravelsimilartothetriangleformedafter30minutesof
travel?Explainthereasoningyouusedtoanswerthisquestiontoaclassmate.
Boat
Boat
B10
B30
Dock
min
min
Boat
A10
min
Boat
A30
min
5. Aboyisrunningfromsecondbasetothirdbase.Ateachmomentintimeduringhisrun,the
threepointscorrespondingtohisposition,thirdbase,andhomeplate,formatriangle.Locate
thepointscorrespondingtohispositionaquarterofthewayand halfwaytothirdbase,and
drawthetriangles.Arethetwotrianglessimilar?Explainyouranswertoaclassmate.
2ndBase
3rdBase
HomePlate
HIDOE Geometry SY 2016-2017
111
L12 – Dilations and Similarity
12.4 – Introduction to Similarity
Name _____________________________
Per _____ Date _____________________
6.
Refer to the diagram shown below.
A.
Rotate DBCD 90° CW about point A. Dilate the image by a scale factor of 0.5 with center A.
Label it DB ' C ' D '
C
B
D
A
B.
Explain how you knew which vertices to label B’, C’, and D’.
C.
The image and pre-image are related, in that DBCD and DB ' C ' D ' are _____________________ .
HIDOE Geometry SY 2016-2017
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L12 – Dilations and Similarity
12.5 – Homework
Name _____________________________
Per _____ Date _____________________
Complete the following rigid motion transformation and dilations to create new images of the given
figures.
1. Rotate DEFG 90°CCWaboutpointAanddilatetheimagebyascalefactorof1.5,andlabelthe
resultingimage DE ' F ' G ' .
E
F
G
A
Check your work by measuring the corresponding angles:
mÐE = _________
mÐE’ = _________
mÐF = _________
mÐF’ = _________
mÐG = _________
mÐG’ = _________
The image and pre-image are related, in that DEFG and DE ' F ' G ' are ____________________ .
HIDOE Geometry SY 2016-2017
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L12 – Dilations and Similarity
12.5 – Homework
Name _____________________________
Per _____ Date _____________________
2. Reflect DHIJ aboutlinem,translatetheresultingimageusingthetranslationdefinedbyT(x,
y)=(x,y–5),anddilatetheresultingimagebyascalefactorof2abouttheleft-mostpointof
thetriangle.Labeltheresultingtriangle DH ' I ' J ' .
m
H
I
J
The image and pre-image are related, in that DHIJ and DH ' I ' J ' are ____________________ .
HIDOE Geometry SY 2016-2017
114
L13 – Similar Triangles
Name _____________________________
13.1 – Looking Ahead: Using Similar Triangles to Solve Problems
Per _____ Date ____________
Motivation:Attheendofthisunit,youwillbeabletosolvethefollowingproblems.
Real-WorldProblem:Grandma needs to measure the palm tree in her front yard in order to have it
removed by professionals. She is perplexed by this task since measuring the height of the tree with a tape
measure is difficult, if not impossible. As the kids are playing in the yard, grandma notices the shadow
her granddaughter casts on the ground looks to be in the same proportion to her height as the tree’s shadow
on the ground is to the tree’s height. If her 5-foot tall granddaughter has a shadow 8-feet long, sketch a
diagram to find the tree’s height if its shadow is 24 feet.
Real-WorldProblem:Grandmaistakingoutarowofbushesalongthesideofthehouseinorder
toextendhertriangularvegetablegarden,shownbelow.Shewantsthevegetablegardentobeas
largeaspossible,butretainitstriangularshape.Sincethenewgardenwillrunalongthesideofthe
house,shecanonlyextendthe8-footsidetobe18feet,andtheedgealongthehousewillbeparallel
withthe9-footside.Sketchadiagramofgrandma’snewgarden.
18 feet
8 feet
9 feet
Grandma’s
House
14 feet
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
116
L13 – Similar Triangles
13.2 – Characteristics of Similar Triangles
Name _____________________________
Per _____ Date _____________________
SIMILARTRIANGLES:Investigatethecharacteristicsofsimilartriangles
1. UsingthediagramofDABCbelow,measurethesides,andlabeleachsidewithitslengthin
centimeters.
B
C
A
2. Inthespaceabove,usearulertoextend AB to AB' todoubleitsoriginallength.Dothe
samefor AC to AC' .ConnectB’andC’,andmeasureeachsideofthelargertriangleand
labelitslength.
€
€
€
€
HIDOE Geometry SY 2016-2017
117
L13 – Similar Triangles
13.2 – Characteristics of Similar Triangles
Name _____________________________
Per _____ Date _____________________
3. Comparethelengthsof BC and B'C' .Whatdoyounotice?Explain.
€
€
4. Repeatthisprocess,triplingtheoriginalsidelengthsusingtheextendedpointsB”andC”,
andhalvingtheoriginalsidelengthsusingthepointsB’’’andC’’’.Doeswhatyounoticed
aboutthethirdsidestillholdtrue?Explain.
5. Usewhatyoudiscoveredtomakeaconjectureaboutcorrespondingsidesofsimilar
triangles.
Insimilartriangles,correspondingsides___________________________________________
_________________________________________________________________________________________
6. Now,measurealltheanglesineachtriangle.Whatdoyounotice?
7. Usewhatyoudiscoveredtomakeaconjectureaboutthecorrespondinganglesofsimilar
triangles.
Insimilartriangles,correspondinganglesare_______________________________________.
HIDOE Geometry SY 2016-2017
118
L13 – Similar Triangles
13.3 – Similarity and Proportion
Name _____________________________
Per _____ Date _____________________
Part I: Follow your teacher’s instructions and use the space below to take guided notes.
1. Similar ( ~ ) Triangles (#VOC): corresponding angles are ___________, and corresponding
sides are _________________________ (meaning the ratios of their lengths are __________).
2. Proportion (#VOC): a statement that two ratios are ________________.
•
For example,
a
b
=
c
d
(read as “________________________________________________”)
a
3. Similarity Ratio (written as b ) (#VOC): the ratio of lengths of ____________________ sides;
€
always measured in the same ______________.
4.
R
Are the triangles€to the right similar?
Y
If so, write
the similarity statement and state the similarity ratio.
2
4
6
X
3
12
Z
Q
9
S
5. Extended Proportion: a statement that __________ or more ratios are ___________. (#VOC)
6. Write an “extended proportion” to show the equivalence between the ratios of all pairs of
corresponding sides of DXYZ and DQRS.
7. If
h
i
j
Properties of Proportions
k
= , which of the following must be true?
A.8x=5y
x y
=
B. 5 8 y 8
=
C. x 5 isequivalentto
1)ad=bc
(Cross-ProductProperty)
€
HIDOE Geometry SY 2016-2017
€
2)
3)
119
L13 – Similar Triangles
13.3 – Similarity and Proportion
Name _____________________________
Per _____ Date _____________________
8.
Determine the value of x that makes each statement true.
A.
13 =
9.
In the diagram, ΔART ~ ΔBUS .
𝟔
𝒙
B.
j
R
=
h
C.
l
8
𝑥
=
𝑥
2
T
R
mno
=
p
kmno
B
A.
Determine the similarity ration between DART and DBUS.
B.
What is the measure of ÐU?
C.
What is the length of 𝑆𝐵?
D.
What is the measure of ÐA?
E.
F.
D.
€
S
10 cm
18 cm
U
56°
A
12 cm
R
What is the measure of ÐS?
If the given similarity statement was written as ΔBUS ~ ΔART , how would that affect the
similarity ratio?
HIDOE Geometry SY 2016-2017
€
120
L13 – Similar Triangles
13.3 – Similarity and Proportion
Name _____________________________
Per _____ Date _____________________
Part II: Practice exercises
10.If xy = 47 ,whichofthefollowingmustbetrue?Justifyyouranswer.
A.
x y
= 4 7
B.
x y
= 7 4
C.
x 12
= y 21
C.
3
5
=
x x +1
€
€
11.Solveforx.
7
5
x
3
A. = €
€
€
2
x
B. =
x
32
€
12.DVOC~DTMH.Findthevalueofthevariablesx&y.
H
V
10 in
T
24in
8 in
x
y
M
C
O
40 in
HIDOE Geometry SY 2016-2017
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HIDOE Geometry SY 2016-2017
122
L13 – Similar Triangles
13.4 – Homework
Name _____________________________
Per _____ Date _____________________
Solve for x.
1.
9 12
=
24 x
2.
€
x 9
=
4 x
3.
€
3.
x +1 6x − 2
=
9
36
€
ΔWXZ ~ ΔDFG
Z
G
€
3 in
W
37°
4 in
10 in
X
D
6 in
F
Determine each of the following:
A. the similarity ratio
B. m∠Z
€
C. m∠G
€
D. GF
€
E. m∠D
€
F. WZ
€
HIDOE Geometry SY 2016-2017
123
L13 – Similar Triangles
13.4 – Homework
Name _____________________________
Per _____ Date _____________________
5. Find the value of the variables if Δ𝑀𝑄𝐴~ΔEOD.
O
Q
108°
4m
z
3m
(2x+4)°
A
9m
M
6m
y°
(3x–14)°
E
D
6.
The triangles below are similar. Determine the value of the variables x and y.
3 m.
y
7 m.
x
HIDOE Geometry SY 2016-2017
5 m.
17 m.
124
L13 – Similar Triangles
13.5 – Theorems About Similar Triangles
Name _____________________________
Per _____ Date _____________________
Part I: Investigate what information you need to prove two triangles are similar.
Materials:
- protractor
- ruler
1. In DABC below, use a protractor to label ÐA with its measure (to the nearest degree).
B
C
A
2. Using the line drawn below and a protractor or patty paper, create an angle congruent to ÐA
from Question 1 at point R. Given this angle, attempt to draw a triangle that is not similar to
DABC.
S
R
3. Were you able to draw a triangle not similar to DABC? Explain.
HIDOE Geometry SY 2016-2017
125
L13 – Similar Triangles
13.5 – Theorems About Similar Triangles
Name _____________________________
Per _____ Date _____________________
4. Measure and label ÐB from Question 1. Again, create an angle congruent to ÐA at point K and
an angle congruent ÐB at point L. Attempt to draw a triangle that is not similar to DABC.
L
K
5. Use what you discovered to make a conjecture about the minimum amount of angle measures
needed to prove triangles similar.
The minimum number of angle measures needed to prove triangles are similar
is _________ sets of corresponding congruent angles.
6. If you have two sets of corresponding congruent angles in a triangle, what does that say about the
last pair of corresponding angles? Explain.
HIDOE Geometry SY 2016-2017
126
L13 – Similar Triangles
13.5 – Theorems About Similar Triangles
Name _____________________________
Per _____ Date _____________________
Part II: Follow your teacher’s instructions and use the space below to take guided notes.
àSimilarTriangles:
1)aretwotrianglesthathaveexactlythesame____________butNOTnecessarilythe
same_____________.
2)occurwhencorresponding___________arecongruentANDcorresponding
______________areproportional.
3)showtheirrelationshipusinga_____________________________________,the
ratioofthelengthsofcorresponding______________.
Three Ways to Prove Two Triangles are Similar Without Relying on the Definition
Angle-AngleSimilarityTheorem(AA~Postulate)(#THM)
C
A
D
O
If______anglesofone______are_________to______angles
G
ofanother_________,thenthetwo__________are__________.
Side-Angle-SideSimilarityTheorem(SAS~Theorem)(#THM)
A
Ifanangleofone______is______toanangleofanother______,
andthesides____________________thetwoanglesare
________________________,thenthetwo________are______.
T
H
3
E 40°
2
40°
T
M
G
Side-Side-SideSimilarityTheorem(SSS~Theorem)(#THM)
Ifthe___________________________sidesoftwo________are
K
N
24
6
16
_________________________,thenthetwo________are______.
HIDOE Geometry SY 2016-2017
6
4
4
L
4
Q
1
P
M
127
L13 – Similar Triangles
13.5 – Theorems About Similar Triangles
Name _____________________________
Per _____ Date _____________________
PartIII:Practiceexercises
7. Is enough information provided to prove that the following triangles are similar?
•
If so, write the similarity statement and name the postulate or theorem you used.
•
If not, explain why not.
A.
B.
U
Y 8 Z
5
4
30°
C
V
T
R
6
X
10
8
30°
A
12
B
Q
S
13
C.
A
2
4
B
Q
4
M
8
P
8.
Each diagram contains 2 triangles that are similar. Determinethevalueofx.
42
A.
9
B.
C.
12
15
17
10
x
8
x
5 x
6
12
14
HIDOE Geometry SY 2016-2017
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L13 – Similar Triangles
13.5 – Theorems About Similar Triangles
9.
10.
€
€
Name _____________________________
Per _____ Date _____________________
Lookingbackatgrandma’sgarden,tothenearestfoot,howmuchedgingwillgrandmaneed
topurchasetomakeanewborderaroundherentirevegetablegarden?
18 feet
8 feet
9 feet
14 feet
Justify the following statements using the similarity postulates or theorems.
A. ΔABC ~ ΔADB B
B. ΔABC ~ ΔBDC A
D
C
C. ΔADB ~ ΔBDC €
HIDOE Geometry SY 2016-2017
129
HIDOE Geometry SY 2016-2017
130
L13 – Similar Triangles
13.6 – Homework
1.
Name _____________________________
Per _____ Date _____________________
Is enough information provided to prove that the following triangles are similar?
A.
•
If so, write the similarity statement and name the postulate or theorem you used.
•
If not, explain why not.
K
N
3
J
70°
M
C.
B.
20°
L
6
S
110°
B
T
R
P 20°
A
50°
C
Q
A
40°
40°
X
B
2.
A.
70°
2
4
R
C
Each diagram contains 2 triangles that are similar. Determinethevalueofx.
B.
C.
15
4
12
x
3
12
10
x
x
7
11
8
4
8
HIDOE Geometry SY 2016-2017
131
L13 – Similar Triangles
13.6 – Homework
3.
Name _____________________________
Per _____ Date _____________________
The figure below contains three similar triangles: DABC ~ DADB ~ DBDC
B
y
6
A
A.
x
D
9
C
In the given figure above, DADB and DBDC are embedded in DABC, making them a little
difficult to see.
The diagram below separates the 3 triangles to make them a little easier to distinguish from each
other.
Label the vertices of each of the 3 triangles below. To help you, use the given similarity statement
and use any rigid motion transformations to help you figure out the corresponding vertices.
B.
Determine the values of x and y in the given figure above.
HIDOE Geometry SY 2016-2017
132
L13 – Similar Triangles
13.7 – The Geometric Mean
Name _____________________________
Per _____ Date _____________________
Grandma wants to put in a pool and Jacuzzi in the Northwest (back left side) corner of her back yard.
She wants to be different and make both of them triangular shaped. She also wants to build them so that
they share a common wall so people can easily climb from one to the other. See figure below. The
common wall is an altitude for the larger triangle.
Pool
4X
Jacuzzi
X
Grandma knows she wants the hypotenuse of the pool and Jacuzzi together (the largest triangle) to be 40
ft. and that the Pool portion of the hypotenuse should be four times as long as the Jacuzzi portion of the
hypotenuse. Help her find the rest of the measurements so she can buy the right amount of materials.
1. What kind of triangle is formed by the pool and Jacuzzi together?
2. What kind of triangles are the pool and Jacuzzi separately?
3. Determine the value of X and 4X.
HIDOE Geometry SY 2016-2017
133
L13 – Similar Triangles
13.7 – The Geometric Mean
Name _____________________________
Per _____ Date _____________________
In order for the contractors to lay out the pool and Jacuzzi, what they really need are the measurements
along the property boundaries (i.e., the other two legs of the largest triangle). They will also need the
length of the dividing wall. Before we find the remaining measurements for Grandma, let’s explore and
make conjectures to help us.
4. Each larger triangle below contains two smaller inner triangles. For each set of three triangles thus
formed, determine which of the three triangles are similar. Write similarity statements and explain
why you think they are similar.
E
A
58°
G
7.5
32°
B
C
D
4.5
10
6
H
8
F
5. Make a conjecture about similarity of the three triangles determined by the altitude to the hypotenuse
in a right triangle.
HIDOE Geometry SY 2016-2017
134
L13 – Similar Triangles
13.7 – The Geometric Mean
Name _____________________________
Per _____ Date _____________________
6. Now let’s prove it!
Given:
Right ΔABC
𝐵𝐷 is an altitude for ΔABC
𝑚∠𝐴𝐵𝐶 = 90∘
Prove: ΔABC~ΔADB~ΔBDC
Statement
Reason
7. Theorem 19.4: The altitude to the hypotenuse of a right triangle forms _________ triangles that are
similar to _______________________ and ________________________. (#THM)
8. Use Theorem 19.4 to find the missing labeled lengths in the triangles below.
HIDOE Geometry SY 2016-2017
135
L13 – Similar Triangles
13.7 – The Geometric Mean
Name _____________________________
Per _____ Date _____________________
R
Y
4
C
S
x
5
16
T
x
12
Z
X
Q
9. What do you notice about the proportions you made within the last two triangles?
Geometric mean: x is the geometric mean between positive numbers p and q if and only iff
p x
= .
x q
This is equivalent to pq = x 2 . (#VOC).
Why do they call this the Geometric Mean?
•
The Arithmetic mean of two numbers is their midpoint (add them together and divide by 2).
•
To find the Geometric mean you multiply them together and find their positive square root.
HIDOE Geometry SY 2016-2017
136
L13 – Similar Triangles
13.7 – The Geometric Mean
Name _____________________________
Per _____ Date _____________________
Example:Determinethegeometricmeanof3and12
t
h
h
oR
=
𝑥 R = 36
𝑥 = 6
10. Practicefindingthegeometricmeanofthefollowingnumbers.
A.3,27
B.20,5
11. Given:RightΔABCandaltitude𝐵𝐷
Statement
`c
ac
Prove:
= ac
C.6,8
Reasoning
cb
A
D
B
C
HIDOE Geometry SY 2016-2017
137
L13 – Similar Triangles
13.7 – The Geometric Mean
Name _____________________________
Per _____ Date _____________________
12. Theorem13.7:Thealtitudetothehypotenuseofarighttriangleisthe_________________________
________________________ofthetwosegmentsintowhichitdividesthehypotenuse.(#THM).
Let’s try to make sense of this theorem. Referring to the diagram in the proof for the previous problem
(question #11):
•
Imagine if you copy 𝐴𝐶 from the diagram and use it form a rectangle such that 𝐷𝐶 is perpendicular
to 𝐴𝐷.
o The area of this rectangle would be the product of lengths of those 2 sides of the rectangle:
Area = (𝐴𝐷)(𝐷𝐶).
A
A
D
D
C
B
•
C
Now, look at the altitude of the triangle (i.e., 𝐵𝐷 ) and picture a square with each side having a
length of 𝐵𝐷 .
o The area of this square would be the length of 𝐵𝐷 squared: Area = (𝐵𝐷)(𝐵𝐷).
B
D
•
Therefore, one way to interpret what Theorem 13.7 means is, “The area of the rectangle formed
by the two segments in the hypotenuse of the triangle is equal to the area of square formed
by the altitude of the triangle.”
•
This relationship is a reason the altitude of a triangle is can be referred to as the Geometric Mean.
HIDOE Geometry SY 2016-2017
138
L13 – Similar Triangles
13.7 – The Geometric Mean
Name _____________________________
Per _____ Date _____________________
13. Now let’s return to Grandma’s pool/Jacuzzi. Using the theorems we just proved, together with other
facts you know about right triangles, determine all the missing lengths for her pool and Jacuzzi.
Hint: determine the value for h first.
X
Pool
Y
h = _____________
X = _____________
Y = ____________
HIDOE Geometry SY 2016-2017
139
HIDOE Geometry SY 2016-2017
140
L14 – Similarity and Proportionality
14.1 – Construction Warm-up
Name _____________________________
Per _____ Date _____________________
The Great Gazebo Project: Grandma is thinking ahead to her
days of leisure, sitting in her Gazebo and watching her
grandkids play, sipping on iced tea.
As a helpful grandson you volunteer to help her with some of
the planning, while your sister has offered to do most of the
actual work.
Before you start though, she wants to be sure you remember
your geometry. After all, how can you be trusted to modify
blueprints and calculate distances if you can’t even bisect a line
or angle!
Grandma’s First Day Challenge!
Completethesimpletasks
belowtoprovethatyou
knowyourstuff.
Using a compass and straightedge, or patty paper, construct the following on the next page:
1. Draw a Line Segment and draw its Perpendicular Bisector.
2. Draw a line and a point not on the line, and then draw another line that is Parallel to the first line
passing through the point.
3. Draw an Acute Angle and Bisect it.
4. Construct an Equilateral Triangle
HIDOE Geometry SY 2016-2017
141
L14 – Similarity and Proportionality
14.1 – Construction Warm-up
Name _____________________________
Per _____ Date _____________________
1.
2.
3.
4.
HIDOE Geometry SY 2016-2017
142
L14 – Similarity and Proportionality
14.2 – The Altitude and Mean
Name _____________________________
Per _____ Date _____________________
Project 1: Your first project is to help Grandma with two windows she has chosen to add into her house.
Both options she has chosen are triangular and will be located using the orientations shown below. She
wants to know which is taller, but the manufacturer only lists the area of the glass and the length of each
side. She needs the Altitude of each window.
(#VOC) Altitude of a Triangle: The Line Segment containing a designated vertex that is perpendicular
to the line containing the opposite side. Note: Since each triangle has three vertices, each triangle has
exactly three altitudes.
Two possible window designs are shown below:
Task A: Construct the altitude of the glass window pane shown below, using the highest point as the
vertex from which the altitude is drawn.
Task B: Construct the altitude of the glass window pane shown below, using the highest point as the
vertex from which the altitude is drawn. This may appear to be more difficult, since the altitude is drawn
outside the triangle. But remember, the altitude is drawn to the line containing the opposite side, so simply
extend the base of the window prior to drawing the altitude.
HIDOE Geometry SY 2016-2017
143
L14 – Similarity and Proportionality
14.2 – The Altitude and Mean
Name _____________________________
Per _____ Date _____________________
Project 2: We also need to cut each window in half (for some reason, perhaps the glass is too large and
needs to be supported in the middle). For symmetry, each half must have the same area. If we find the
median of the base of the triangle and connect that point to the opposite vertex, it will create two new
triangles that have the same height (as determined by the common vertex) and the same base (half the
original), and hence the same area. This line is called the Median of a Triangle.
(#VOC) Median of a triangle: The Line segment that extends from a designated vertex to the midpoint
of the opposite side. Note: Each triangle has exactly three medians.
Task A: Construct the Median containing the highest point, for the glass pane shown below.
Task B: Construct, using dotted lines, the Altitudes for the two smaller triangles you just created with
the median. Include the highest point on each smaller triangle in each altitude. How many dotted lines
did you draw? What does this imply about the areas of the two smaller triangles?
HIDOE Geometry SY 2016-2017
144
L14 – Similarity and Proportionality
14.3 – The Mid-segment Theorem
Name _____________________________
Per _____ Date _____________________
Objective: To find proportional lengths formed by the mid-segment in triangles and trapezoids.
Grandma is finally working on finishing touches for her house, and realizes she needs
more storage. She wants to include a shelf in the attic that is located halfway up the
triangular wall at the end of the attic. How can she determine where to put the shelf?
How long should she make the shelf?
MID-SEGMENT (A __________ that joins the __________ of two segments) #VOC
1. Draw and cut out a large scalene triangle. Each member should
have a different kind of triangle (right, acute, and obtuse triangles).
Label your vertices A, B, and C.
2. Find and mark the midpoint of 𝐴𝐶 by folding A onto C. Find and
mark the midpoint of 𝐵𝐶 in the same way. Label the midpoints
D and E correspondingly, and draw 𝐷𝐸.
C
A
B
C
D
A
E
B
3. Fold your triangle on𝐷𝐸.
A
B
4. Fold A to C. Do the same for B.
B
5. a) What type of quadrilateral does the folded triangle appear to form?
b) What does this tell you about 𝐷𝐸 and 𝐴𝐵? Explain.
6. What conjecture can you make about how the mid-segment of a triangle is related to the third
side of a triangle?
HIDOE Geometry SY 2016-2017
145
L14 – Similarity and Proportionality
14.3 – The Mid-segment Theorem
Name _____________________________
Per _____ Date _____________________
Theorem14.1:TriangleMid-segmentTheorem
Ifasegmentjoinsthe________________of______sidesofa
____________,thenthe________________is________tothe
A
S
K
________________sideand____________itslength.#THM
C
B
Example1:𝐷𝐸and𝐸𝐹aremidsegments.DE=7andAB=10.
a)FindAC.
B
E
D
b)FindEF.
C
c)WhatistheperimeterofADEF?
A
F
Example2:Simoneisdesigningakiteforacompetition.Sheplanstouseadecorativeribbonto
connectthemidpointsofthesidesofthekite.Thediagonalsofthekitemeasure
64cmand90cm.Findtheamountofribbonshewillneedtobuy.
HIDOE Geometry SY 2016-2017
146
L14 – Similarity and Proportionality
14.3 – The Mid-segment Theorem
Name _____________________________
Per _____ Date _____________________
àMid-segmentofatrapezoid:a__________________thatjoinsthe
the______________ofthe___________________sidesofa________________.#VOC
Theorem14.2:TrapezoidMid-segmentTheorem
Themid-segmentofatrapezoidis_______________tothe
basesandhalfaslongasthe_____________ofthelengths
ofthe____________________.#THM
Example3:𝐷𝐸isthemid-segmentofthetrapezoidQRST.
a) 𝐷𝐸 is parallel to _____ and ______
T
R
M
S
P
A
Q
R
D
E
T
S
b) If QR = 14 and ST = 7, then DE = _______
c) If DE = 7 and ST = 11, then QR = _______
d) If DE = 2x+3, QR = 7x–2, and ST = 3x–10, then QR = _______
Example4:LayerCake.Abakerismakingacakeliketheoneatthe
right.Thetoplayerhasadiameterof8inchesandthe
bottomlayerhasadiameterof20inches.What
shouldthediameterofthemiddlelayerbe?
HIDOE Geometry SY 2016-2017
147
HIDOE Geometry SY 2016-2017
148
L14 – Similarity and Proportionality
14.4 – Homework
Name _____________________________
Per _____ Date _____________________
For1–5,intriangleABC,thepointsD,E,andFaremidpoints.
1. 𝐷𝐸 is parallel to ____________
F
D
2. 𝐹𝐸 is parallel to ____________
3. If AB = 14, then EF = ____________
A
B
C
E
4. If DE = 6, then AC = ____________
M
For6–9,intriangleMNO,thepointsX,Y,andZaremidpoints.
5. If YZ = 3x+1 and MN = 10x–6, then YZ = _____________
X
Z
O
6. If YX = x–1 and MO = 3x–7, then MO = _____________
Y
N
7. If mÐMON = 48°, then mÐMZX = _____________
8. If mÐMXZ = 37°, then mÐMNO = _____________
HIDOE Geometry SY 2016-2017
149
L14 – Similarity and Proportionality
14.4 – Homework
Name _____________________________
Per _____ Date _____________________
Quadrilateral ABCD is a trapezoid with midsegment 𝐸𝐹.
A
B
E
F
D
C
9. If mÐB = 73°, then mÐC = ________________
10. If AB = 28 and DC = 13, then EF = ________________
11. If EF = 13 and DC = 6, then AB = ______________
12. If DC = x+5 and DC + AB = 4x+6, then EF = ______________
HIDOE Geometry SY 2016-2017
150
8
L14 –CB
Similarity
= 2.48 cmand Proportionality
14.5 –ACUsing
to Find Coordinates
= 4.96Proportionality
cm
AC
Name _____________________________
Per _____ Date _____________________
7
Question:= How
2.00 do we find the coordinates of point C in the diagram below, if we know that C lies on
CB
line segment 𝐴𝐵 and is located 2/3 of the way from A to B? In general, how do we find the coordinates
6
A: (3.00, 1.03)
of C if we know that C lies on 𝐴𝐵 and the ratio
`b
`a
= 𝑝, where 0 < p < 1?
5
B
4
3
C
2
1
2
A
2
4
6
8
10
12
14
1
Let's investigate, by starting off with some easier examples. We will return to solve the problem above
later in the lesson. 2
1. Find the coordinates of the point C that lies on the line segment AB below such that:
3
a.
b.
c.
d.
`b
`a
`b
`a
`b
`a
`b
`a
=
o
=
o
=
t
=
v
R
t
v
4
Coordinates for C = ________________
5
Coordinates for C = ________________
6
Coordinates for C = ________________
7
Coordinates for C = ________________
j
8
9
HIDOE Geometry SY 2016-2017
151
L14 – Similarity and Proportionality
14.5 – Using Proportionality to Find Coordinates
Name _____________________________
Per _____ Date _____________________
2. FindthecoordinatesofthepointCthatlieson𝐴𝐵belowsuchthat:
`b
t
a.
= CoordinatesforC=________________
`a
b.
`b
`a
c.
`b
`a
l
o
= R
o
= j
CoordinatesforC=________________
CoordinatesforC=________________
3. FindthecoordinatesofthepointCthatlieson𝐴𝐵belowsuchthat:
a.
b.
`b
`a
`b
`a
c.
`b
`a
d.
`b
`a
o
= CoordinatesforC=________________
R
= CoordinatesforC=________________
CoordinatesforC=________________
CoordinatesforC=________________
R
t
t
= v
t
= j
HIDOE Geometry SY 2016-2017
152
L14 – Similarity and Proportionality
14.5 – Using Proportionality to Find Coordinates
Name _____________________________
Per _____ Date _____________________
4. FindthecoordinatesofthepointCthatlieson𝐴𝐵belowsuchthat:
`b
o
a.
= CoordinatesforC=________________
`a
b.
`b
`a
c.
`b
`a
R
R
= .
j
l
= k
CoordinatesforC=________________
CoordinatesforC=________________
5. Let'strytogeneralizethislastsituation.Supposenowthatpisanumberstrictlybetween0
`b
and1.FindthecoordinatesofthepointCthatlieson𝐴𝐵abovesuchthat = 𝑝.
HIDOE Geometry SY 2016-2017
`a
153
L14 – Similarity and Proportionality
14.5 – Using Proportionality to Find Coordinates
Name _____________________________
Per _____ Date _____________________
Thelastsetofproblemswasrelativelyeasybecausethelinesegmentswereeitherhorizontalor
vertical,whichmeantonecoordinatewasalreadydetermined,leavingyoutoonlyneedtofindthe
othercoordinate.Wenowmovetonon-verticallinesegmentswithanon-zeroslope.Let'sbegin
withthestartingpointAattheorigin.
Example1:FindthecoordinatesofthepointC,locatedon𝐴𝐵belowsuchthat
`b
`a
o
= .
R
o
Solution:ClearlyΔ𝐴𝐶𝐸issimilartoΔ𝐴𝐵𝐷.Itthenfollowsthatif𝐴𝐶 = 𝐴𝐵(whichwasgiven)it
o
o
R
R
R
mustfollowthat𝐴𝐸 = 𝐴𝐷and𝐶𝐸 = 𝐵𝐷.Thisisexactlythetechniqueweusedforourprevious
horizontalandverticallinesegments,respectively.SinceAD=6andBD=3itfollowsthatAE=3
andBD=3/2,whichimpliesC=(3,3/2),whichwecouldhaveguessedfromthediagram.
6. FindthecoordinatesofthepointCthatlieson𝐴𝐵abovesuchthat:
`b
o
a.
= CoordinatesforC=________________
`a
b.
c.
`b
`a
`b
`a
d.
e.
f.
`b
`a
`b
`a
`b
`a
t
R
= CoordinatesforC=________________
o
= CoordinatesforC=________________
= .
CoordinatesforC=________________
o
= CoordinatesforC=________________
t
= .
CoordinatesforC=________________
t
v
t
v
j
j
HIDOE Geometry SY 2016-2017
154
L14 – Similarity and Proportionality
14.5 – Using Proportionality to Find Coordinates
Name _____________________________
Per _____ Date _____________________
Example2:FindthecoordinatesofthepointC,locatedon𝐴𝐵below,suchthat
`b
`a
o
= .
t
o
Solution:Justasbefore,clearlyΔ𝐴𝐶𝐸issimilartoΔ𝐴𝐵𝐷.Itthenfollowsthatif𝐴𝐶 = 𝐴𝐵(which
o
o
t
t
t
wasgiven)itmustfollowthat𝐴𝐸 = 𝐴𝐷and𝐶𝐸 = 𝐵𝐷.SinceAD=7andBD=4,itfollowsthat
AE=7/3andCE=4/3.What'sdifferenthere,incomparisontoExample1,isthatthesearenotthe
coordinatesofC,thesearethesegmentlengths.
InordertofindthecoordinatesofCwemustaddtheselengthstothecoordinatesofA.So,thexl
v
v
o
coordinateforCis−1 + = ,andthey-coordinateforCis−1 + = ,whichisconsistentwith
t
t
thecoordinatesforConthegraph.
t
t
7. FindtheexactcoordinatesofthepointCthatlieson𝐴𝐵abovesuchthat:
`b
o
a.
= CoordinatesforC=________________
`a
b.
c.
`b
`a
`b
`a
d.
e.
f.
`b
`a
`b
`a
`b
`a
l
v
= CoordinatesforC=________________
o
= CoordinatesforC=________________
= CoordinatesforC=________________
o
= CoordinatesforC=________________
R
= CoordinatesforC=________________
l
R
t
v
j
j
HIDOE Geometry SY 2016-2017
155
L14 – Similarity and Proportionality
14.5 – Using Proportionality to Find Coordinates
Name _____________________________
Per _____ Date _____________________
8. Returntotheproblemattheverybeginningofthelessonandfindtheexactcoordinatesofthe
`b
R
`b
pointCthatlieson𝐴𝐵suchthat = ,andmoregenerallythecasewhere = 𝑝for
`a
0<p<1.
t
`a
Hintforthefollowingproblem:EventhoughΔ𝐴𝐵𝐷isnotdrawnin,asinthepreviousexamples,
youmaywishtoincludeitinordertovisualizethesimilarityproperty.
9. FindtheexactcoordinatesofthepointCthatlieson𝐴𝐵abovesuchthat:
`b
o
a.
= CoordinatesforC=________________
`a
b.
c.
`b
`a
`b
`a
R
R
= CoordinatesforC=________________
l
= CoordinatesforC=________________
= CoordinatesforC=________________
=
CoordinatesforC=________________
CoordinatesforC=________________
t
k
d.
e.
f.
`b
`a
`b
`a
`b
`a
t
=
v
o
ow
p
ow
HIDOE Geometry SY 2016-2017
156
L14 – Similarity and Proportionality
14.6 – Parallel Lines in Triangles (Warm-up)
Name _____________________________
Per _____ Date _____________________
Attic Shelves
Grandma is thinking of building a set of 4 shelves along one end of the attic. (See dotted line shelves in
the sketch below.)
Y=20ft.
X=30ft.
She knows the attic floor is level and she wants to make sure her shelves are level. She already knows the
lengths X & Y of her roof. What she wants to know is if she can just divide each of these by 5, mark those
lengths on the side walls and connect the two points on each side with a line (the dotted lines). She can
then follow the line when she puts up the shelf supports. Label each section of the roof in terms of X or
Y. One section has been done for you. We ask some preliminary questions below, and will return to
answer this question, in full, later in the lesson.
1. Do you think this would work? Why or why not? Discuss your thoughts with a classmate.
2. She can also check the heights along the line from each shelf below to make sure it is the same along
the entire run of the shelf. How do you think Grandma could most easily determine the desired
height from the base to the first shelf? Discuss your ideas with a classmate.
HIDOE Geometry SY 2016-2017
X
Y
157
HIDOE Geometry SY 2016-2017
158
L14 – Similarity and Proportionality
14.7 – Parallel Lines in Triangles
Name _____________________________
Per _____ Date _____________________
Let’s experiment with several triangles to make conjectures that may help us with Grandma’s attic
shelves.
1. For each scalene triangle below, determine the missing lengths, then write ratios to compare.
L
LN=18
Q
15
LM=____
10
6
R
T
M
K
P
QP=14
6
5
10
S
N
J
KJ
=
LJ
MN
=
LN
KJ
=
LK
MN
=
LM
LM
=
LN
LK
=
LJ
SQ
=
QT
RQ
=
QP
ST
=
QT
PR
=
QP
SQ
=
TS
RQ
=
PR
2. What do you notice about the ratios?
3. Corresponding lengths are ____________________.
4. How could we have determined the value for LM even if we were NOT given the value for LN, and
similarly for PR if we were NOT given QP? Hint: The next question may help you answer this
question.
5. How do you know DNJL
6.
DMKL without using the fact that LN = 18?
KM
= ________
JN
HIDOE Geometry SY 2016-2017
159
L14 – Similarity and Proportionality
14.7 – Parallel Lines in Triangles
Name _____________________________
Per _____ Date _____________________
In the triangle below, using a straightedge and compass draw line l parallel to line segment AC passing
through the midpoint D of side AB and intersecting side BC at E. Notice that by design, l divides side
AB into two line segments with equal length AD and DB .
7. Discuss with your partner how you know that DDBE is similar to DABC .
8. Use a compass or ruler to check if l also divides BC evenly; that is, BE = EC. Discuss with your
partner why this must be true.
9. Using a straightedge and compass, draw the altitude from B to AC . Discuss with your partner why l
must also divide this perpendicular line segment into two line segments of equal length.
10. Repeat the process above, locating point D ¼ of the way from A to B. Explain why the line parallel
to AC passing through D must also divide BC into two line segments of the same proportion.
HIDOE Geometry SY 2016-2017
160
L14 – Similarity and Proportionality
14.7 – Parallel Lines in Triangles
Name _____________________________
Per _____ Date _____________________
Reflection:
If a line segment within DABC connects the midpoint on side AB with a point on side BC in such a
manner that it is parallel to the base AC , then the endpoint on side BC must be its
________________________________________.
Using the same reasoning as for the midpoint, if a line segment within DABC connects sides AB and
BC in such a manner that it is parallel to the base AC , then it cuts the two sides
________________________________________.
11.
Given: ΔABC and 𝐷𝐸 || 𝐴𝐶
Prove:
`c
ca
=
bx
xa
(*Hint: first prove similar triangles)
Statement
Reasoning
1)
1) Given
2)
2) Corresponding Angles
3)
3) Corresponding Angles
4)
4)
5) ΔABC ~ ΔDEB
5)
6)
7)
8)
`a
ca
=
ba
ca
`c
ca
=
=
bx
xa
D
A
E
C
6)
xa
`ayca
B
bayxa
xa
7)
8)
From this proof we can state the following theorem:
Theorem 14.3: If a line (or line segment) parallel to one side of a triangle intersects the other two sides,
then it divides the other two sides _____________________. (#THM)
Further, the length of this parallel line segment is __________________ to the length of the parallel
third side.
HIDOE Geometry SY 2016-2017
161
L14 – Similarity and Proportionality
14.7 – Parallel Lines in Triangles
Name _____________________________
Per _____ Date _____________________
12.UseTheorem14.3(onthepreviouspage)tohelpyousolvethefollowingproblem.
Given𝐽𝐾||𝑌𝑍,solveforx.
X
4
J
4
K6
x
Y
Z
4
13. WritetheconverseofTheorem14.3(#THM)
Ifalinedividestwosidesofatriangle________________________________________,thenitis
________________________________________tothethirdsideofthetriangle.
14. UsingtheconverseofTheorem14.3(thatyoujuststatedabove),
•
identifyifthesegmentinsideofeachtriangleisparalleltothebase;
•
ifitisparallel,solveforx.
A.
B.
Parallel?____________ Parallel?____________
x=__________
x=__________ HIDOE Geometry SY 2016-2017
162
L14 – Similarity and Proportionality
14.8 – Medians and Altitudes in Similar Triangles
Name _____________________________
Per _____ Date _____________________
MediansinSimilarTriangles
1. Recallthatthemedianconnectsa_______________tothe__________________ontheoppositesideofa
triangle.Drawtheremainingtwomediansinthetrianglebelow.
Z
Y
M
X
2. Giventhateachpairoftrianglesbelowissimilar,withamediandrawnin,fillinthemissing
lengthsandwritetheratiooftheirsidesandoftheirmedians.
𝑌𝑍 Z
=
𝐴𝑇
𝑍𝑀 =
𝑇𝑃
T
12
𝑍𝑀 =
𝑌𝑍
𝑇𝑃 =
𝐴𝑇
Y
M
3
YM=_____
𝐸𝐿 =
𝑇𝐴
𝐸𝑀 =
𝑇𝑃
8
X
3
A
6
2
6
C
P
T
E
10
𝑇𝑃 =
𝑇𝐶
𝐸𝑀 =
𝐸𝐹
9
9
6
5
4
3
HIDOE Geometry SY 2016-2017
8
L
1
M
F
A
P
2
C
163
L14 – Similarity and Proportionality
14.8 – Medians and Altitudes in Similar Triangles
Name _____________________________
Per _____ Date _____________________
3. Makeaconjectureabouttheratiosofmedianstocorrespondingsidesinsimilartriangles.
4. Nowlet’sproveit.
B
Given:ΔABC~ΔXYZ
𝐵𝑀and𝑌𝑁aremedians
C
A
M
`a
a•
Y
Prove: =
•€
€‚
*Hint:Usesimilartriangletheorems.
X
N
Z
Statement
Reason
5. Theorem14.4:Correspondingmediansofsimilartrianglesare___________________tothe
corresponding___________________.(#THM)
HIDOE Geometry SY 2016-2017
164
L14 – Similarity and Proportionality
14.8 – Medians and Altitudes in Similar Triangles
Name _____________________________
Per _____ Date _____________________
Useknowntheoremstohelpyousolvethefollowingproblem.
6. ΔOLA~ΔFET,solveforx.
F
O
20
12
15
9
M
L
16
x
A
E
C
T
AltitudesinSimilarTriangles
7. Eachpairoftrianglesbelowissimilar.Usearulertofindtheratiosofcorrespondingsidesand
thealtitudeswithineachtrianglepair.
A
C
D
C
B
X
O
G
A
T
Z
Y
8. Whatdoyounoticeabouttheratiosbetweencorrespondingsidesandtheratiobetween
altitudesforeachsimilarpair?
HIDOE Geometry SY 2016-2017
165
L14 – Similarity and Proportionality
14.8 – Medians and Altitudes in Similar Triangles
Name _____________________________
Per _____ Date _____________________
9. WhymustthisbetrueforALLpairsofsimilartrianglesandtheiraltitudes?Whatprevious
theoremsorpostulatesgiveevidenceofthis?
10. Useyourexplanationinthepreviousproblemtohelpyouwiththefollowingproof.
A
Given: ΔABC~ΔXYZ
𝐴𝑃and𝑋𝑆arealtitudes
Prove:
`a
•€
=
ab
€ƒ
=
`b
•ƒ
=
`d
•„
B
C
P
X
Y
Statement
S
Z
Reason
11. Theorem14.5:Correspondingaltitudesofsimilartrianglesare_________________________________
tothecorresponding_________________________________.(#THM)
HIDOE Geometry SY 2016-2017
166
L14 – Similarity and Proportionality
14.8 – Medians and Altitudes in Similar Triangles
Name _____________________________
Per _____ Date _____________________
12. Restatingthe4resultsfromthelasttwolessons:
• Ifalineparalleltoonesideofatriangleintersects________________________,thenitdivides
________________________________________________________________.
•
Ifalinedividestwosidesofatriangle__________________________,thenitis
___________________________________________________________ofthetriangle.
•
Corresponding__________________ofsimilartrianglesare____________________tothe
correspondingsides.
•
Corresponding__________________ofsimilartrianglesare____________________tothe
correspondingsides.
13. Usetheaboveresultstofindthemissingmeasuresinthefiguresbelow.Assumepairsof
trianglesaresimilar.
a)
B
b)
10
8
15
6
E
D
20
?
?
4
A
C
c) ?
6
3
4
3
4
HIDOE Geometry SY 2016-2017
167
L14 – Similarity and Proportionality
14.8 – Medians and Altitudes in Similar Triangles
Name _____________________________
Per _____ Date _____________________
14. Grandmadecidesthattheeasiestwaytodeterminethedistancebetweenheratticshelvesisto
measurethealtitude,fromthepeakofherrooftothefloor,anddividethatdistanceby5(why
5andnot4,sincethereare4shelves?).Shecanthenmarkthisdistanceatvariouslocations
acrosstheatticendwallwheretheshelveswillbelocated,measuringfromthefloor,then
connectthedotsandformaguideforthebottomshelf.Shecanthencontinuethisforthe
secondshelf,measuringfromthefirstshelf,etc.,untilshearrivesatthelastshelf.Discusswith
yourpartnerwhythisworks,sightingappropriateresultsfromthelasttwolessons,andwrite
yourexplanationbelow.
15. Supposetheceiling(i.e.thebottomofthemaintriangle)is25ft.HowlongwillGrandmaneed
tocuteachshelf?Supposeinsteadthattheceilinglengthis24ft.HowlongwillGrandmaneed
tocuteachshelfinthisinstance?
X
Y
HIDOE Geometry SY 2016-2017
168
L14 – Similarity and Proportionality
14.9 – Homework
Name _____________________________
Per _____ Date _____________________
1. Findthemissingvaluesbelow.
X=____________
Y=____________
2. Findthemissingvaluesbelow.
X=____________
Y=____________
HIDOE Geometry SY 2016-2017
169
L14 – Similarity and Proportionality
14.9 – Homework
Name _____________________________
Per _____ Date _____________________
3. Findthemissingvaluesbelow,giventhat∆𝐴𝐵𝐶~∆𝐴… 𝐵… 𝐶 … .
W=____________
X=____________
Y=____________
4. Findthemissingvaluesbelow,giventhat∆𝐴𝐵𝐶~∆𝐴… 𝐵… 𝐶 … .
X=____________
Y=____________
W=______________
Z=_______________
HIDOE Geometry SY 2016-2017
170