Congruence and Similarity
Congruence and Similarity
Objectives
This lesson connects previous knowledge of angle measurement, length of sides and proportional triangles to new concepts of proving triangle congruency and similarity. Students will:
- use the definition of congruent triangles to prove congruence.
- use triangle congruence postulates to prove congruence.
- use triangle similarity postulates to prove similarity.
- use proportions to solve similar triangles.
Essential Questions
- How can we describe relationships between shapes and use these relationships to better understand the specific properties of shapes?
- What geometric situation would you choose to model a real-world situation and how would you explain your solution to the problem?
Related Unit and Lesson Plans
Related Materials & Resources
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Formative Assessment
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View
Multiple-Choice Items:
1. The definition of congruent triangles states
A
that all pairs of corresponding sides are congruent.
B
that all pairs of corresponding angles are congruent.
C
that one pair of corresponding sides and one pair of corresponding angles are congruent.
D
that all pairs of corresponding sides and all pairs of corresponding angles are congruent.
2. Given two triangles, you know that they have two pairs of corresponding, congruent angles. Based on this, what can you conclude about the two triangles?
A
The two triangles are congruent, but not necessarily similar.
B
The two triangles are similar, but not necessarily congruent.
C
The two triangles are both congruent and similar.
D
You cannot conclude anything about the congruence or similarity of the two triangles.
3. Which statement is true about two similar triangles?
A
They always have equal areas.
B
They always have equal perimeters.
C
They are always congruent.
D
They are sometimes congruent.
Use the following information for items 4–6:
Triangle DEF (ΔDEF) is congruent to ΔNOM.
4. Which side in ΔNOM corresponds to?
5. Which angle in ΔNOM corresponds to ÐFED?
6. In ΔDEF,
is 6 cm and 
is 5 cm. What is the length of 
?A
1 cm
B
5 cm
C
6 cm
D
11 cm
In items 7–9, use the following triangles:
7. Which theorem or postulate can be used to prove that the two triangles are similar?
A
the definition of similar triangles
B
SSS
C
AA
D
SAS
8. What is the length of
?A
4 cm
B
13 cm
C
16 cm
D
40 cm
9. What is the scale factor that compares ΔABC to ΔDEF?
A
1:4
B
1:9
C
4:1
D
9:1
Multiple-Choice Answer Key:
1. D
2. B
3. D
4. C
5. A
6. B
7. C
8. C
9. A
Short-Answer Items:
In items 10 and 11 below, use the following information to answer the questions:
At 3 p.m., Chuck, who is 6 feet tall, goes outside. He measures his shadow and finds that his shadow is 2 feet long. He wants to measure the height of a nearby building that is casting a shadow that is 140 feet long.
10. Draw two triangles to illustrate the situation. Clearly label all known parts of each triangle.
11. How tall is the nearby building? Show your work and list any theorems or postulates that you used to find your answer.
12. How would you determine if two figures are congruent? Be sure to include your understanding of congruency, diagrams and what they mean, and state the appropriate theorems that support your reasoning.
A
Draw two congruent figures.
B
Definition congruency as it applies to your examples.
C
Use supporting theorems and explain how they apply to your examples.
13. List three ways to prove triangles are congruent. Explain the reasoning that supports your answers.
Short-Answer Key and Scoring Rubrics:
In items 10 and 11 below, use the following information to answer the questions:
At 3 p.m., Chuck, who is 6 feet tall, goes outside. He measures his shadow and finds that his shadow is 2 feet long. He wants to measure the height of a nearby building that is casting a shadow that is 140 feet long.
10. Draw two triangles to illustrate the situation. Clearly label all known parts of each triangle.
The student should draw two similar right triangles, one with the height labeled
6 feet, and the other leg labeled 2 feet, and a right-angle mark; the other triangle should have the shorter leg labeled 140 feet and also have a right-angle mark.Points
Description
5
The student’s answer includes:
- the height of the smaller triangle labeled as 6 feet.
- the shorter leg of the small triangle labeled as 2 feet.
- the right angle indicated within the smaller triangle.
- the shorter leg of the larger triangle labeled as 140 feet.
- the right angle indicated within the larger triangle.
4
The student’s answer includes four of the five requirements.
3
The student’s answer includes three of the five requirements.
2
The student’s answer includes two of the five requirements.
1
The student’s answer includes one of the five requirements.
0
The student’s answer does not include any of the five requirements.
11. How tall is the nearby building? Show your work and list any theorems or postulates that you used to find your answer.
Using AA Similarity and the proportion
, the height of the building is 420 feet.Points
Description
3
The student:
- indicates the usage of the AA similarity postulate.
- correctly sets up the proportion (or an equivalent proportion).
- correctly determines the height of the building.
2
The student’s answer includes two of the three requirements.
1
The student’s answer includes one of the three requirements.
0
The student’s answer does not include any of the three requirements.
12. How would you determine if two figures are congruent? Be sure to include the definition of congruency, diagrams and state theorems to support your answer.
A
Draw two congruent figures.
B
Definition of congruent
C
Supporting theorems
Congruency is determined by the definition (proving all pairs of corresponding parts are congruent) using SAS, AAS, or SSS. Congruency means the sides and angles are exactly the same. Students should draw any diagrams demonstrating the figures are equal and congruent.
13. List three ways to prove triangles are congruent.
Students can receive up to 3 points, 1 point for each valid method to prove triangle congruence.
Performance Assessment:
- Are the two triangles congruent, similar, or neither? If they are congruent or similar, write a congruence statement for the two triangles. If they are neither congruent nor similar, explain why.
- Determine the value of x and y.
- Determine the values of the two missing angles (one in each triangle).
- Are the two triangles congruent, similar, or neither? If they are congruent or similar, write a congruence statement for the two triangles. If they are neither congruent nor similar, explain why.
Performance Assessment Answer Key and Scoring Rubric:
The two triangles are similar. The similarity statement is “ΔABC is similar to ΔDEF.”
- Determine the value of x and y.
x = 12 cm
y = 6.5 cm
3. Determine the values of the two missing angles (one in each triangle).
The missing angle in each triangle is 23 degrees.
Points
Description
4
The student’s answer includes:
- a similarity statement equivalent to “ΔABC is similar to ΔDEF.”
- the correct value for x.
- the correct value for y.
- the correct values of the two missing angles.
3
- The student’s answer includes three of the four requirements.
2
- The student’s answer includes two of the four requirements.
1
- The student’s answer includes one of the four requirements.
0
- The student’s answer does not include any of the requirements.
DRAFT 10/12/2011
