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Angles and Triangles
ACTIVITY 22
Triangle Trivia
Lesson 22-1 Properties of Triangles and Side Lengths
My Notes
Learning Targets:
Determine when three side lengths form a triangle.
Use the Triangle Inequality Property.
Classify triangles by side length.
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SUGGESTED LEARNING STRATEGIES: Interactive Word Wall,
Summarizing, Look for a Pattern, Graphic Organizer
Students in Mr. Mira’s math class made up some geometry games.
Here are the rules for the game Matt and Allie created.
Triangle Trivia Rules
Properties of Triangles—Perimeter Variation
Players:
Materials:
Directions:
Three to four students
Three number cubes and a “segment pieces” set
of three each of the following lengths: 1 inch,
2 inches, 3 inches, 4 inches, 5 inches, and
6 inches.
Take turns. Roll the three number cubes. Find
a segment piece to match each number rolled.
See whether a triangle can be formed from those
segment pieces. The value of the perimeter
of any triangle that can be formed is added
to that player’s score. The first player to reach
50 points wins.
© 2014 College Board. All rights reserved.
Amir wonders what the game has to do with triangles.
1. Play the game above to see how it relates to triangles. Follow the
rules. Record your results in the table.
Player 1
Numbers
Player 2
Score
Numbers
Player 3
Score
Numbers
Player 4
Score
Numbers
Score
Activity 22 • Angles and Triangles
277
Lesson 22-1
Properties of Triangles and Side Lengths
ACTIVITY 22
continued
My Notes
2. There is more to the game than just adding numbers. How does the
game relate to triangles?
Amir noticed that he could tell whether the lengths would form a triangle
even without the segment pieces.
3. Explain how Amir can determine whether a triangle can be formed
from three given lengths.
MATH TIP
When sides of a figure have the
same length, this can be shown by
drawing marks, called tick marks,
on those sides. For example, the
equal sides of the isosceles and
equilateral triangles in the table
below right have the same number
of tick marks.
Matt and Allie’s game illustrates the following property that relates the
side lengths of a triangle.
Triangle Inequality Property
For any triangle, the sum of any two sides must be greater than the
length of the third side.
Before students play another game, Mr. Mira wants to review the
vocabulary terms scalene, isosceles, and equilateral with the class. He
draws the following examples of triangles.
Isosceles Triangles
Equilateral Triangles
© 2014 College Board. All rights reserved.
Scalene Triangles
278
Unit 5 • Geometric Concepts
Lesson 22-1
Properties of Triangles and Side Lengths
Activity 22
continued
My Notes
4. Based on Mr. Mira’s examples, describe each type of triangle.
a. scalene triangle
Math Tip
b. isosceles triangle
A triangle can be identified as
scalene, isosceles, or equilateral
by the lengths of its sides.
c. equilateral triangle
Amir creates a variation of Matt and Allie’s game. Here are the rules for
Amir’s game.
Triangle Trivia Rules - Name the Triangle
Players:
Materials:
Directions:
Three to four students
Three number cubes
Take turns rolling three number cubes.
• If you can, form
a scalene triangle .............add 5 points
an isosceles triangle ........add 10 points
an equilateral triangle .....add 15 points
no triangle ............................add 0 points
• If you make a mistake, deduct 10 points from
your last correct score.
• The first player to reach 25 points wins.
© 2014 College Board. All rights reserved.
5. Make use of structure. When playing Amir’s variation of Triangle
Trivia, suppose that the cubes landed on the following numbers. Tell
how many points you would add to your score and why.
a. 5, 5, 5
b. 1, 6, 4
c. 3, 2, 4
d.6, 6, 4
e. 1, 4, 1
Share your responses with your group members. Make notes as you
listen to other members of your group. Ask and answer questions
clearly to aid comprehension and to ensure understanding of all
group members’ ideas.
Activity 22 • Angles and Triangles 279
Lesson 22-1
Properties of Triangles and Side Lengths
Activity 22
continued
My Notes
Check Your Understanding
6. Can a triangle be formed using the side lengths below? If so, is the
triangle scalene, isosceles, or equilateral? Explain.
a . 4 m, 4 m, and 8 m
b. 8 ft, 6 ft, and 4 ft 7. If three segments form a triangle, what must be true about the sum
of any two side lengths of the triangle?
Lesson 22-1 Practice
For Items 8–14, use the Triangle Inequality Property to determine
whether a triangle can be formed with the given side lengths in inches.
If a triangle can be formed, classify the triangle by the lengths of its
sides. Explain your thinking.
8. a = 5, b = 5, c = 5
9. a = 3, b = 3, c = 7
10. a = 7, b = 4, c = 4
11. a = 8, b = 4, c = 5
12. a = 1, b = 2, c = 8
13. a = 8, b = 12, c = 4
15. Which of the following are possible side lengths of a triangle?
A.12, 20, 15 B
.33, 20, 12 C . 12, 20, 11 16. Reason abstractly. Is it necessary to find the sum of all three
possible pairs of side lengths to use the Triangle Inequality Property
when deciding if the sides form a triangle? Include an example in
your explanation.
17. Construct viable arguments. Two sides of a triangle are 9 and
11 centimeters long.
a. What is the shortest possible length for the third side?
b. What is the longest possible length for the third side?
280 Unit 5 • Geometric Concepts
© 2014 College Board. All rights reserved.
14. a = 12, b = 5, c = 13
Lesson 22-2
Properties of Triangles and Angle Measures
ACTIVITY 22
continued
Learning Targets:
Classify angles by their measures.
Classify triangles by their angles.
Recognize the relationship between the lengths of sides and measures
of angles in a triangle.
Recognize the sum of angles in a triangle.
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My Notes
MATH TIP
If the rays are too short to measure
with a protractor, extend the
length of the sides of the angle.
SUGGESTED LEARNING STRATEGIES: Interactive Word Wall,
Summarizing, Visualization, Graphic Organizer
Another way to classify triangles is by their angles. A right angle has a
measure of 90°. An acute angle has a measure of less than 90°. An obtuse
angle is greater than 90° and less than 180°.
1. Use the angles shown.
C
A
B
E
© 2014 College Board. All rights reserved.
D
F
a. Estimate the measure of each angle.
∠A ≈
∠B ≈
∠C ≈
∠D ≈
∠E ≈
∠F ≈
b. Use appropriate tools strategically. Use a protractor to find
the measure of each angle to the nearest degree. Then classify
each angle as acute, obtuse, or right by its measure.
∠A =
∠B =
∠C =
∠D =
∠E =
∠F =
Activity 22 • Angles and Triangles
281
Lesson 22-2
Properties of Triangles and Angle Measures
Activity 22
continued
My Notes
Now Mr. Mira draws the following examples of triangles.
Math Tip
Acute Triangles
A box at the vertex of an
angle indicates an angle with
measure 90°.
A
Right Triangles
Obtuse Triangles
E
I
D
B
G
H
C
F
2. Based on Mr. Mira’s examples, describe each type of triangle.
a. acute triangle
b. obtuse triangle
3. A triangle can be labeled using both its angle measure and the
lengths of its sides.
a. Label the triangles that Mr. Mira drew by side length.
b. Choose one of the triangles and give the two labels that describe it.
c. Explain how the two labels together provide a better description
of the triangle than either one alone. Share your ideas with our
group and be sure to explain your thoughts using precise language
and specific details to help group members understand your ideas
and reasoning.
282 Unit 5 • Geometric Concepts
© 2014 College Board. All rights reserved.
c. right triangle
Lesson 22-2
Properties of Triangles and Angle Measures
Activity 22
continued
Mr. Mira has his class investigate the sum of the measures of a triangle.
Students measured the angles of some scalene, isosceles, and equilateral
triangles. They recorded their results as shown.
Isosceles Triangles
Scalene Triangles
30°
135°
15°
20°
60°
45°
15°
70°
55°
90°
150°
20°
70°
85°
40°
Equilateral Triangles
15°
40°
40°
120°
My Notes
60°
90°
60°
45°
30°
75°
60°
60°
60°
60°
60°
70°
60°
75°
4. a. Find the sum of the angle measures for each triangle.
The Triangle Sum Theorem states that the sum of the three angle
measures in any triangle is always equal to a certain number.
A theorem is a statement or
conjecture that has been proven
to be true.
© 2014 College Board. All rights reserved.
b. What is the sum of the angle measures in any triangle?
MATH TERMS
Activity 22 • Angles and Triangles 283
Lesson 22-2
Properties of Triangles and Angle Measures
Activity 22
continued
My Notes
The Triangle Sum Theorem allows you to find the measure of the third
angle in a triangle when you are given the other two angle measures. 5. Students played a game in which they chose two angle measures of a
triangle and then determined the third angle measure. What must be
true about the two angle measures the students choose?
6. Some of the angle measures students created for triangles are shown.
For each pair of angle measures, find the measure of the third angle
in the triangle.
a . 43°, 94°
b. 38°, 52°
c. 57°, 39°
d.140°, 12°
© 2014 College Board. All rights reserved.
e. 60°, 60°
284 Unit 5 • Geometric Concepts
Lesson 22-2
Properties of Triangles and Angle Measures
Activity 22
continued
The angle measures of a triangle can be used to determine if the triangle
is scalene, isosceles, or equilateral. Look back at the triangles Mr. Mira drew.
Isosceles Triangles
Scalene Triangles
30°
135°
15°
60°
45°
15°
70°
70°
85°
55°
90°
150°
20°
120°
40°
Equilateral Triangles
15°
40°
40°
20°
My Notes
60°
90°
60°
45°
30°
75°
60°
60°
60°
70°
60°
60°
60°
75°
7. Compare the angle measures of the triangles. Look for patterns in
Mr. Mira’s examples to help you determine if the triangles described
below are scalene, isosceles, or equilateral.
a. a triangle with three different angle measures
b. a triangle with exactly two congruent angle measures
Math TERMS
c. an equiangular triangle
A triangle with three equal angles
is called equiangular.
© 2014 College Board. All rights reserved.
8. Look back at Item 6. Classify each triangle by its side length and by
its angle measure.
Another relationship exists between the angles and the sides of a triangle.
In a triangle, the side opposite the angle with the greatest measure is the
longest side.
9. Compare the angle measure to the side opposite the angle in a
scalene triangle. What is true about the side opposite the angle with
the least measure?
Activity 22 • Angles and Triangles 285
Lesson 22-2
Properties of Triangles and Angle Measures
ACTIVITY 22
continued
My Notes
Check Your Understanding
For Items 10–12, sketch a triangle described by each pair of words
below or state that it is not possible. Use tick marks and right angle
symbols where appropriate. If it is not possible to sketch a triangle,
explain why not.
10. scalene, obtuse
11. isosceles, acute
12. equilateral, right
13. Two angles in a triangle measure 35° and 50°. Explain how to find
the measure of the third angle.
LESSON 22-2 PRACTICE
For Items 14–19, sketch a triangle described by each pair of words
below or state that it is not possible. If it is not possible to sketch a
triangle, explain why not.
14. scalene, right
15. isosceles, obtuse
16. equilateral, acute
17. isosceles, right
18. scalene, acute
19. equilateral, obtuse
21. Two angles in a triangle measure 65° each. What is the measure of
the third angle?
22. Reason quantitatively and abstractly. Find the missing angle
measure or measures in each triangle below. Then classify the
triangle by both its angle measures and its side lengths.
a. The three angles in a triangle have the same measure.
b. Two angles in a triangle measure 45° each.
c. Two angles in a triangle measure 25° and 50°.
23. Construct viable arguments. Determine whether each statement
below is always true, sometimes true, or never true. Explain your
reasoning.
a. The acute angles of an isosceles triangle add up to 90°.
b. An isosceles triangle has two equal angles.
c. An equilateral triangle has a right angle.
d. The largest angle of a scalene triangle can be opposite the
shortest side.
286
Unit 5 • Geometric Concepts
© 2014 College Board. All rights reserved.
20. Use appropriate tools strategically. Use a ruler and a protractor
to sketch a triangle that is scalene and has an angle that measures
30°. Is the triangle acute, right, or obtuse? Explain.