Similar Triangles – Grade Seven
Ohio Standards
Connection
Geometry and Spatial
Sense
Benchmark E
Use proportions to express
relationships among
corresponding parts of
similar figures.
Indicator 1
Use proportional reasoning
to describe and express
relationships between parts
and attributes of similar
and congruent figures.
Benchmark G
Describe and use properties
of triangles to solve
problems involving angle
measures and side lengths
of right triangles.
Indicator 5
Apply properties of
congruent or similar
triangles to solve problems
involving missing lengths
and angle measures.
Benchmark J
Apply properties of
equality and
proportionality to solve
problems involving
congruent or similar
figures; e.g., create a scale
drawing. missing lengths
and angle measures.
Indicator 5
Apply properties of
congruent or similar
triangles to solve problems
involving missing lengths
and angle measures.
Lesson Summary:
Students explore pairs of triangles to develop a conceptual
understanding for similar figures. Students use this definition to
explore the proportional relationships involved in similar
triangles and to find missing lengths and angle measures.
Estimated Duration: One and one-half to two hours
Commentary:
The concept of similarity is not new to students at this grade
level. However, there are aspects that require further
investigation. One is the use of an asymmetrical triangle in the
study of similar triangles. This takes some students out of their
comfort zone but provides a deeper understanding of the students’
knowledge of similarity. The second focus is related to the
orientation of two similar objects. Identifying corresponding
angles and sides becomes more of a challenge to students when
the orientation is changed from one object to the other.
Research has also shown that computer assisted drawing tools
help develop conceptual understanding of scale factor and
proportionality among similar figures. These tools allow students
to easily make changes to a figure by a given scale factor, change
the orientation and compare lengths and angle measures of
corresponding parts quickly and accurately to determine if objects
are similar.
Pre-Assessment:
 Distribute Geometric Figures, Attachment A, to each
student.
 Direct the students to match each figure on the page to a
similar figure or group of figures on the page.
 Pair the students and direct the pairs to share their ideas and
to write explanations as to what makes the figures similar.
 Group two pairs and have each student draw an additional
similar figure that belongs in this group. Have them include
an explanation to justify why the figures are similar.
 Observe students as they work. Use the Pre-Assessment
Checklist, Attachment B, to determine the students’ prior
knowledge and to inform instructional decisions.
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Similar Triangles – Grade Seven
Ohio Standards
Connection
Mathematical Processes
Benchmark
B. Apply and adapt
problem-solving
strategies to solve a
variety of problems,
including unfamiliar
and non-routine
problem situations.
Scoring Guidelines:
Use Pre-Assessment Checklist, Attachment B, during the
observation to assess student understanding of the Geometric
Shapes activity, discussion and written responses. Evaluate
the methods used by students to determine similar shapes.
Strategies and discussion may include:
 identifying the number of sides and angles,
 finding corresponding sides and angles,
 distinguishing types of lines and angles and
 comparing sizes of figures using words such as
“shrunk,” “blown up,” “twice as large” or “three times
bigger.”
Post-Assessment:
Allow students to choose one of the following postassessment options.
 Have students design squares for a patchwork quilt or
stained-glass window on grid paper by using six to eight
types of polygons. Include two sets of congruent and two
sets of similar, non-congruent figures to create the square
using construction paper. Use proportional reasoning to
prove that the figures are similar or congruent. Have
students explain the reasoning used to prove the figures
are similar or congruent.
 Instruct students to create a blueprint of a gymnasium
floor or football field. Include the lines painted on the
field or floor. Use measurements, ratios and proportions
to show that the blueprint is similar to the real floor or
field. Have students explain the reasoning used to prove
the blueprint is similar to the real gymnasium floor or
football field.
 Use Lost Lengths, Attachment C, to determine if the
students can use their knowledge of similar figures to find
the missing lengths.
Scoring Guidelines:
Check each drawing for correct proportional reasoning to
explain how the figures are similar or congruent. Students
having difficulty in creating congruent or similar figures may
need additional instruction prior to completing this lesson.
Instructional Procedures:
Part One
1. Distribute one copy of Similar Triangles, Attachment D,
to each student. Have students work with partners.
2. Tell students that the pairs of triangles are similar.
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Similar Triangles – Grade Seven
3. Instruct the students to measure all the angles and side lengths of each triangle.
4. Direct the students to compare their findings for the pair of similar triangles. Ask questions
such as:
 What did you notice about the angle measures?
 What did you notice about the measures of the sides?
 Is there a relationship? Describe the relationship.
Instructional Tip:
Assist students who may have difficulty in seeing the relationships among the corresponding
angle measures of similar triangles and among the corresponding side lengths by asking guiding
questions.
5. Ask partners to make conjectures about the properties of similar triangles based on their
findings, such as:
Similar triangles have congruent corresponding angle measures, but the lengths of the sides
are different.
If proportional relationships are discussed, include in the conjecture.
6. Have students share conjectures and record them on the board or chart paper.
7. Have the class determine if each conjecture holds for all similar triangles by looking for
examples that support or contradict the conjecture.
8. Facilitate a class discussion to create a general definition for similar triangles. Use the correct
conjectures that the class agreed upon to develop the definition.
Instructional Tip:
The definition should include the fact that the corresponding angle measures are congruent, but
the side lengths do not have to be congruent. Develop the fact that corresponding side lengths are
to be proportional throughout this lesson.
9. Have students identify the ratio or scale factor between the corresponding side lengths for
each pair of similar triangles.
10. Have students show their understanding of similar figures by writing a lesson summary in
their mathematics journals. They should include a definition using mathematical terms to
explain the characteristics. Read through student summaries to identify level of
understanding.
Part Two
11. Draw two similar triangles on the board. Label the sides with the lengths 6 inches, 8 inches
and 12 inches. Label the side of the second triangle corresponding to 6 inches in the first
triangle with the measurement of 4 inches. If proportions are not understood, use grid paper
to draw accurate triangles and the grid squares to estimate the missing side length.
12. Ask students to determine the ratio of the two triangles (6:4). Have partners find the lengths
of the other two sides of the triangle. Observe the strategies students use to find the lengths
of the sides. Students who have proportional reasoning will use the ratio 6 to 4 and make
equivalent ratios and proportions.
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Similar Triangles – Grade Seven
13. Have the partners share their methods for finding the solution. Modify the class definition to
include the fact that the side lengths of similar triangles are proportional.
14. Provide practice to reinforce the concepts of similarity and strengthen the evidence to support
the conjecture.
a. Draw two similar triangles on the board with two side lengths missing. Ask students to
write explanations of how to find these missing side lengths. Instruct students to use
correct vocabulary words such as similar, proportion, congruent and corresponding.
b. Have students create triangles or pairs of triangles to give to partners. Have the partners
create similar triangles or prove that two triangles are similar and provide mathematical
evidence for their solutions.
15. Pose the question,
Think of the minimum requirements you need to know if two triangles are congruent, such as
angle, side and angle. Can these minimum requirements be used to determine if two angles
are similar?
16. Have small groups test the minimum requirements and make conjectures. Select students to
share the groups’ discoveries.
17. Have students play the following game as a final review.
a. Arrange the class in groups of three or four.
b. Tell the students that there is a triangle drawn on the index card in your hand.
c. Instruct the students to come up with the criteria that they would need to know in order to
draw a triangle that was similar to your triangle and one and a half times larger than your
triangle.
Differentiated Instructional Support:
Instruction is differentiated according to learner needs, to help all learners either meet the intent
of the specified indicator(s) or, if the indicator is already met, to advance beyond the specified
indicator(s).
 Have students find and draw similar shapes other than triangles. Use easier or more difficult
measurements depending on the students.
Extensions:
 Ask students to determine the minimal information required to prove that two triangles are
similar.
 Extend the content to include objects that physically cannot be measured. (e.g., use shadows
and other indirect measurements to find the height of a tall tree, a building, a flagpole.) Using
what the students learned about similar triangles, take them outside to use their shadows to
find the height of a very tall tree or building. Have students stand outside in the sun. Have
them measure their heights and the length of their shadows. (Do this activity in small
groups.) Have the students sketch this in the form of a right triangle with their heights and the
shadows as the legs of the triangles. Select a tall tree or building and sketch it with its
shadow as a right triangle. Have students use proportional reasoning to measure the length of
the shadow and use it to find the height of the tall object.
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Similar Triangles – Grade Seven
Home Connection:
Instruct students to list examples of objects that are similar or congruent; measurements should
be included to verify their conclusion. For example, they could compare two different boxes, two
different size cans, (e.g., coffee and soup) or windows in their house.
Materials and Resources:
The inclusion of a specific resource in any lesson formulated by the Ohio Department of
Education should not be interpreted as an endorsement of that particular resource, or any of its
contents, by the Ohio Department of Education. The Ohio Department of Education does not
endorse any particular resource. The Web addresses listed are for a given site’s main page,
therefore, it may be necessary to search within that site to find the specific information required
for a given lesson. Please note that information published on the Internet changes over time,
therefore the links provided may no longer contain the specific information related to a given
lesson. Teachers are advised to preview all sites before using them with students.
For the teacher: attachments, chalkboard or overhead projector
For the student:
paper, rulers, protractors
Vocabulary:
 congruent
 corresponding
 equivalent fraction
 proportion
 similar
Technology Connections:
Use geometry software to create two figures and verify that they are similar or congruent.
Research Connections:
Marzano, Robert J., Jane E. Pollock and Debra Pickering. Classroom Instruction that Works:
Research-Based Strategies for Increasing Student Achievement, Alexandria, Va.: Association for
Supervision and Curriculum Development, 2001.
Attachments:
Attachment A, Geometric Figures
Attachment B, Pre-Assessment Checklist
Attachment C, Lost Lengths
Attachment D, Similar Triangles
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Similar Triangles – Grade Seven
Attachment A
Geometric Figures
Name_______________________________
Date__________________________
Directions: Cut out each of the figures and organize them into groups of similar figures.
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Similar Triangles – Grade Seven
Attachment B
Pre-Assessment Checklist
Name_______________________________
Date__________________________
Directions: Use this checklist to pre-assess students for the content of this lesson.
_____
1. Figures are appropriately grouped.
_____
2. Figures are divided by more criteria than just shape.
_____
3. There is understanding of the meaning of similar, as seen by the grouping of
figures.
_____
4. Similar attributes have been identified for different groups of figures.
_____
5. Correct explanations are given as to why the figures in each group are similar.
_____
6. Additional similar figures have been drawn that fit into each category.
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Similar Triangles – Grade Seven
Attachment C
Lost Lengths
Name_______________________________
Date__________________________
Directions: Complete each of the following problems.
1. Emily needs to create a smaller version of her club’s symbol seen on this poster. The new
drawing needs to fit on a T-shirt. Draw two smaller, similar triangles that she could use.
Hint: 1ft = 12 inches
2 ft
Good Book
Club
2 ft
3 ft
2. The following triangles are similar triangles taken from the roofline of this home. Find the
missing side lengths labeled x and y.
10 ft
10 ft
14 ft
y
7 ft
x
3. Describe what you know about the angle measures and side lengths of two similar triangles.
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Similar Triangles – Grade Seven
Attachment D
Similar Triangles
Name_______________________________
Date ___________________
Directions: Four pairs of similar triangles are shown. Determine the measurements of the
lengths of the sides and angles. Compare the measurements of the corresponding sides and
angles.
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Similar Triangles – Grade Seven
Attachment D (continued)
Similar Triangles
Name_______________________________
Date__________________________
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Similar Triangles – Grade Seven
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