Download UnderstandConditions for Drawing Triangles

Focus on Math Concepts
Lesson 19
Part 1: Introduction
CCSS
7.G.A.2
Understand Conditions for Drawing Triangles
What side lengths make a triangle?
You have worked with triangles for several years. You know they have three sides and three
angles and they seem pretty simple. But, can a triangle have just any side lengths? Can a
triangle have just any angle measures? Try an experiment. Pick random numbers, or roll a
number cube and try drawing a triangle with those side lengths. You will find some
combinations of side lengths make a triangle. In this lesson you will learn which
combinations do or don’t work as sides of a triangle.
Can these side lengths form a triangle?
Think If you know its side lengths, you know everything about a triangle.
Cut a straw into lengths 3 cm, 4 cm, and 5 cm. Use the three pieces
to form a triangle, and carefully trace the triangle on paper. Then
rearrange the sides to form another triangle and trace that one too.
Compare your two triangles, and compare yours to other students’
triangles. Compare your triangles to the one shown here. Be sure to
measure the angles with a protractor and compare the angles, too.
Trace one of your
triangles next to the
one shown here.
If you work carefully, you will be able to understand why we can say that if you know the
side lengths you know “everything” about a triangle. Any two triangles with given side
lengths always have the same angle measures, too.
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L19: Understand Conditions for Drawing Triangles
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Part 1: Introduction
Lesson 19
Think If you know its angle measures, you don’t know everything about a triangle.
Use a protractor to draw a triangle with one 908 angle, one 308
angle, and one 608 angle. Try to draw another triangle with the
exact same angle measures, but with at least one side a
different length. If you can do that, try another one.
How do these triangles
compare to yours?
When you have drawn several triangles with angles measuring
308, 608, and 908, compare them. Compare your triangles to the
two shown here, and to other students’ triangles.
Reflect
1 In what way does knowing the measures of three sides of a triangle tell you more than
just knowing the measures of the three angles?
L19: Understand Conditions for Drawing Triangles
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Part 2: Guided Instruction
Lesson 19
Explore It
Explore each set of conditions described below. On a separate sheet of paper draw
several triangles to match the description. Then make a conjecture about the number
of different triangles possible, and justify your thinking.
2 Three sides are given: Use side lengths 6 cm, 6 cm, and 4 cm. How many different
triangles can you make?
3 Three angles are given: Start with angle measures 808, 408, and 608. Try other sets of
angles such as 208, 1208, and 408. Try three measures that don’t add up to 1808.
4 Two sides and the angle between them are given: Start with side lengths 5 cm and 4 cm,
meeting to form a 458 angle. Then try side lengths of 2 cm and 4 cm, meeting to form a
90° angle.
5 Two angles and the side between them are given: Draw triangles with a side 4 cm long
and a 45° angle at each endpoint. Try different angles measures and different lengths for
the side between them.
Now try this problem.
6 How many triangles can you draw with side lengths 2 cm, 3 cm, and 5 cm? Try sides 2 cm,
3 cm, and 8 cm. Explain what you find.
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L19: Understand Conditions for Drawing Triangles
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Part 2: Guided Instruction
Lesson 19
Make Sense of It
Solve the problems below as a group.
7 Use straws to form a quadrilateral with side lengths
5 cm, 5 cm, 3 cm, and 3 cm. Make a sketch and
label the sides of your quadrilateral in the space
to the right.
8 Now form different quadrilaterals using the same side lengths. Sketch and label the sides
of at least two of the quadrilaterals you find.
9 Experiment with some other dimensions, such as 3 cm, 4 cm, 5 cm, and 6 cm, or 4 cm,
4 cm, 4 cm, and 4 cm. Can you find a set of 4 side lengths that will form only one
quadrilateral? Explain why or why not.
Try It Another Way
Work with your group to understand why triangles are used in building.
10 When you have the three side lengths of a triangle, there is only one triangle that can be
formed. How does that explain why a gate like the one here is built with a diagonal board?
L19: Understand Conditions for Drawing Triangles
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Part 3: Guided Practice
Lesson 19
Connect It
Talk through these problems as a class, then write your answers below.
11 Analyze: Draw an isosceles triangle with exactly one 408 angle. Is this the only possibility
or can you draw another triangle that will also meet these conditions? How is this
different from drawing a triangle given 2 sides and the angle between them?
40°
12 Explain: How many different triangles can be drawn with two obtuse angles? Explain
how you can be sure of your answer without drawing many examples.
13 Create: Sketch several triangles with one side 6 cm long, one side 10 cm long, and the
third side x cm long. What are some possible lengths of the third side? Write an
inequality to express the least possible value of x. Write another inequality to express the
greatest possible value of x. Use a sketch to explain your thinking.
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L19: Understand Conditions for Drawing Triangles
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Part 4: Common Core Performance Task
Lesson 19
Put It Together
14 Use what you have learned in this lesson to complete this task.
A
Give conditions (measurements) for a triangle that would result in more than one
possible triangle. Include at least one side length. Explain your thinking and provide
an illustration.
B
Give conditions for a triangle that would result in only one possible triangle. Explain
your thinking and provide an illustration.
L19: Understand Conditions for Drawing Triangles
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