triangle inequalities activity the long and the...

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ACTIVITY

My Notes

Unit 2 • Congruence, Triangles, and Quadrilaterals 105

2.3Triangle Inequalities The Long and the Short of ItSUGGESTED LEARNING STRATEGIES: Marking the Text, Visualization, Think/Pair/Share, Use Manipulatives, Notetaking, Quickwrite

Triangles are rigid ! gures. Structures, such as bridges and towers are o" en reinforced with diagonals to create triangles that give the structures added strength.

In this activity, you will explore some angle and side relationships in triangles.

Exploring Angles and Sides

Follow these steps to explore a relationship between an angle of a triangle and the side opposite that angle.

• Select two non-congruent straws.• Use the straws as sides to create an angle.• Imagine the third side opposite the angle you created.

1. Increase the size of the angle to make a second triangle. How does the length of the third side of this triangle compare to the length of the third side of the ! rst triangle you made?

Your observation in Item 1 can be stated formally as the Hinge # eorem.

! e Hinge ! eorem If two sides of one triangle are congruent to two sides of another triangle, and the included angle in the ! rst triangle is larger than the included angle in the other triangle, then the third side of the ! rst triangle is longer than the third side of the other triangle.

2. If another student in your class starts with two straws that are the same lengths as the ones you used, but the included angle (between the given straws) is smaller than the included angle you created, what can you conclude about the third side of that student’s triangle com-pared to the third side of your triangle?

3. # e Converse of the Hinge # eorem is also true. State the Converse of the Hinge ! eorem.

105-110_SB_Geom_2-3_SE.indd 105105-110_SB_Geom_2-3_SE.indd 105 1/21/10 4:17:27 PM1/21/10 4:17:27 PM

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106 SpringBoard® Mathematics with Meaning™ Geometry

My Notes

Triangle Inequalities ACTIVITY 2.3continued The Long and the Short of ItThe Long and the Short of It

SUGGESTED LEARNING STRATEGIES: Visualization, Think/Pair/Share, Quickwrite, Self/Peer Revision, Use Manipulatives

4. Two students discussed their geometry assignment on the phone. Tony drew !ABC with AB = 6 cm, BC = 10 cm, AC = 8 cm.Angelica drew !ABC with AB = 6 cm, BC = 10 cm, AC = 11 cm.

Using the Converse of the Hinge ! eorem, what conclusion can you state about their triangles?

Exploring Angles and Sides within a Triangle

5. Draw a scalene triangle. Label the vertex of the largest angle A, the vertex of the smallest angle T, and the third angle R. Measure all three angle measures and all three side lengths.

6. a. What is the longest side of your triangle?

b. What is the shortest side of your triangle?

7. Compare the answers to item 6 with your other group members. What do you notice about the longest sides compared to the largest angles in each triangle drawn?


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My Notes

ACTIVITY 2.3continued

Triangle Inequalities The Long and the Short of ItThe Long and the Short of It

! e results from this investigation indicate the existence of several theorems. Complete each theorem.

8. ! e longest side of a triangle lies opposite the angle with the _______ measure, and the shortest side lies opposite the angle with the ______ measure.

9. Conversely, in a triangle, the angle with the greatest measure lies opposite the _______ side and the angle with the least measure lies opposite the _______ side.

Exploring Triangle Inequalities

10. Create a quadrilateral with straws.

• Use the four smaller straws provided by your teacher. Reserve the longest straw for later.

• Insert a thread through the straws to create a quadrilateral. Tie a knot to close the quadrilateral. In groups, create di" erent shapes.

• Trace the quadrilateral on paper. • Measure the four sides in centimeters. Notice that the quadrilateral

easily changes angle measures as you handle it. Unlike triangles, quadrilaterals are not always rigid # gures.

• Thread the longest straw and use it to create a diagonal of the quadrilateral.

• Explore possible diagonal lengths for both diagonals of the quadrilat-eral (one at a time).

11. What is the longest length a of the straw that can be used as a diagonal for your quadrilateral? Explain why.

SUGGESTED LEARNING STRATEGIES: Use Manipulatives, Quickwrite


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My Notes


SUGGESTED LEARNING STRATEGIES: Think/Pair/Share, Quickwrite

12. What is the shortest length b of the straw that can be used as a diagonal for your quadrilateral? Explain why.

13. Choose a length x for the diagonal for your quadrilateral so thatb < x < a from Items 11 and 12. Tie the diagonal into place. Notice that your quadrilateral maintains its shape with the support of the diagonal.

14. You can formally state the relationship you have just investigated as the Triangle Inequality ! eorem. Complete the theorem below.

Triangle Inequality ! eorem ! e sum of the lengths of any two sides of a triangle must be ______________________________________.

Using the Triangle Inequality ! eorem, you know that RA + AM > RM.

15. Use the Triangle Inequality ! eorem and !RAM to complete thefollowing inequalities.

a. _____ + _____ > RA b. MR < ____ + ____

c. Write an inequality comparing the length of AM to the sum of the other two sides of !RAM.

A

RM


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My NotesMy Notes

ACTIVITY 2.3continued

Triangle Inequalities The Long and the Short of ItThe Long and the Short of It

TRY THESE

Write your answers on notebook paper. Show your work.

For Items a through d, use the Triangle Inequality ! eorem to determine whether a triangle can be formed with the given side lengths.

a. 8 in., 6 in., 4 in. b. 3 cm, 4 cm, 7 cm

c. 7 yd, 4 yd, 4 yd d. 8 m, 8 m, 8 m

e. In !PAL, ______ < x < ______ f. In !IBU _____ > BU > _____

IL

B

8.5 ft

9 ft

P

x

AL

4.9 cm

6.2 cm


1. Use the given angle measures to put side measures x, y, and z in numerical order. ! e triangle is not drawn to scale.

2. Tina plans to create triangular " ower bed frames using each of the sets of board lengths listed below. Will each set of boards create a triangle? Explain why or why not.

a. 1 # , 2 # , 3 #

b. 6 # , 9 # , 11 #

c. 2 # , 5 # , 8 #

3. Given !CAT and !DOG with measures as shown (not to scale), which of the following must be true?

a. CT < DG b. CT = DG

c. CT > DG d. Not enough information

4. ! e given triangle isnot drawn to scale.State which anglemust be the largestand explain howyou know.

CHECK YOUR UNDERSTANDING

x

yz

60°

75°

A T

C

83°O G

D

49°

O L

W

b + 2

b - 1

b


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CHECK YOUR UNDERSTANDING (continued)

5. Use the diagram with the given measures to compare each pair of segments. Use <, =, or > to state the relationships.

m∠CEA = 130°

a. CR ____ ET b. EA ____ ET

c. RE ____ ET d. CR ____ EA

90°

70°85°

T

E

C

R A

6. MATHEMATICAL R E F L E C T I O N

Daniel is a student who missed class for this

activity. Write an email message telling him the mathematical ideas you’ve learned.


triangle inequalities activity the long and the...

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