Unit 2 • Congruence, Triangles, and Quadrilaterals 125
My Notes
ACTIVITY
2.5Congruent Triangle MethodsTruss Your JudgmentSUGGESTED LEARNING STRATEGIES: Close Reading, Marking the Text, Summarize/Paraphrase/Retell
T he Greene Construction Company is building a new recreation hall. In his excitement to help the company, Greg Carpenter f inds some steel beams and begins building triangular trusses that will support the roof of the hall. Greg’s boss John says that the trusses Greg uses must be identical in size and shape. According to the def inition of congruent triangles, if the three sides and the three angles of one triangle are congruent to the corresponding three sides and angles of another, then the two triangles are congruent. “Does that mean I have to measure and compare all six parts of both triangles? T here has to be a shortcut,” said Greg. John agreed and told Greg to decide which measurements are necessary to match congruent trusses.
In order to decide on the minimum number of measurements needed, Greg decides to use a scale drawing for one of the trusses he built to investigate. He will start with one measurement, then use two measures, three measures, and so on, until he f inds the minimum number of measures needed to ensure congruence.
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My Notes
ACTIVITY 2.5continued
SUGGESTED LEARNING STRATEGIES: Marking the Text, Create Representations, Quickwrite
In search of the method that would require the least amount of work, Greg begins his investigation by measuring only one of the six parts of his scale drawing. Greg wants to prove or disprove the following statement: “If one part of a triangle is congruent to a corresponding part of another triangle, then the triangles must be congruent.” Greg knows that it only takes one counterexample to disprove a statement.
1. In an ef fort to prove the statement, Greg draws two dif ferent triangles each having a side 2 inches in length.
a. Repeat Greg’s experiment below. Draw two dif ferent triangles of your own, each having a side 2 inches in length.
b. Do your two triangles allow you to prove or disprove the following statement?
“If one part of a triangle is congruent to a corresponding part of another triangle, then the triangles must be congruent.” Explain your answer.
Congruent Triangle Methods Truss Your JudgmentTruss Your Judgment
Corresponding parts result from a one-to-one matching of side lengths and angles from one fi gure to another.
Unit 2 • Congruence, Triangles, and Quadrilaterals 127
My Notes
ACTIVITY 2.5continued
Congruent Triangle Methods Truss Your JudgmentTruss Your Judgment
SUGGESTED LEARNING STRATEGIES: Marking the Text, Create Representations, Identify a Subtask
Greg continues his investigation with two pairs of congruent parts. He looks for counterexamples to the statement: “If two parts of one triangle are congruent to the two corresponding parts in a second triangle then the triangles must be congruent.”
2. Suppose Greg measures the lengths of two sides of the triangle shown below.
A B
C
Is it possible to draw a triangle that has one side congruent to ___
AB and another side congruent to
___ AC so that the new triangle is not
congruent to !ABC? If so, draw such a triangle and label the vertices D, E, and F.
Name the side of the new triangle that corresponds to ___
AB and the side of the new triangle that corresponds to
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My Notes
Congruent Triangle Methods ACTIVITY 2.5continued Truss Your JudgmentTruss Your Judgment
SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask
5. Suppose that Greg measures two angles of the triangle shown below.
A B
C
a. Is it possible to draw a triangle that has an angle congruent to ∠A and an angle congruent to ∠B so that the new triangle is not congruent to "ABC? If so, draw such a triangle and label the vertices D, E, and F.
b. Name the parts of the new triangle that correspond to ∠A and to ∠B and mark the pairs of congruent angles on the triangles above.
6. Greg wanted to prove or disprove the following statement: “If two parts of one triangle are congruent to the two corresponding parts in a second triangle, then the triangles must be congruent.” Does your work in Items 2–5 prove or disprove this statement? Explain.
Now that Greg knows that he must have at least three congruent parts to show that the trusses (triangles) are identical in size and shape, he decides to make a list of all the combinations of three congruent parts to work more ef ! ciently.
7. Greg uses A as an abbreviation to represent angles and S to represent the sides. For example, if Greg writes SAS, it represents two sides and the included angle, as shown in the ! rst triangle below. Here are the combinations in Greg’s list: SAS, SSA, ASA, AAS, SSS, and AAA.
a. Mark each triangle below to illustrate the combinations in Greg’s list.
b. Are there any other combinations of three parts of a triangle? If so, is it necessary for Greg to add these to his list? Explain.
Unit 2 • Congruence, Triangles, and Quadrilaterals 133
My Notes
ACTIVITY 2.5continued
Congruent Triangle Methods Truss Your JudgmentTruss Your Judgment
SUGGESTED LEARNING STRATEGIES: Discussion Group, Create Representations, Use Manipulatives
8b. Identify which of the ! gures in part (a) is congruent to each of the parts of !ABC.
∠A: ∠B:
∠C : ___
AB :
___
CB : ___
AC :
9. Using Greg’s list from Item 7, choose any three of the triangle parts in Item 8. Try to create a triangle that is not congruent to !ABC, but that has three corresponding congruent parts. Use the table below to organize your results.
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My Notes
Congruent Triangle Methods ACTIVITY 2.5continued Truss Your JudgmentTruss Your Judgment
SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Think/Pair/Share
10. T here are four combinations of three congruent parts that suggest that two triangles are identical. Compare your results from Item 9 with those of your classmates. Below, list the four dif ferent combina-tions that seem to guarantee a triangle congruent to !ABC. ! ese combinations are called congruent triangle methods.
11. For each of the pairs of triangles below, write the congruent triangle method that can be used to show that the triangles are congruent.
Unit 2 • Congruence, Triangles, and Quadrilaterals 135
My Notes
ACTIVITY 2.5continued
Congruent Triangle Methods Truss Your JudgmentTruss Your Judgment
SUGGESTED LEARNING STRATEGIES: Marking the Text, Self/Peer Revision
12. T hree of the triangle congruence methods are postulates. T he fourth is a theorem. Using what you know about parallel lines and the prop-erties of triangles, f ill in the reasons for the proof of this theorem.
AAS T heorem: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle, then the triangles are congruent.
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My Notes
Congruent Triangle Methods ACTIVITY 2.5continued Truss Your JudgmentTruss Your Judgment
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
13. Below are pairs of triangles in which congruent parts are marked. For each pair of triangles, name the angle and side combination that is marked and tell whether the triangles appear to be congruent.
a.
b.
c.
d.
14. We know that in general SSA does not always determine congruence of triangles. However, for two of the cases in Item 13, the triangles appear to be congruent. What do the congruent pairs of triangles have in common?
Unit 2 • Congruence, Triangles, and Quadrilaterals 137
My Notes
ACTIVITY 2.5continued
Congruent Triangle Methods Truss Your JudgmentTruss Your Judgment
SUGGESTED LEARNING STRATEGIES: Look for a Pattern, Group Presentation, Think/Pair/Share
15. In a right triangle, we refer to the correspondence SSA shown in Item 13(a) and 13(c) as hypotenuse-leg (HL). Write a convincing argument in the space below to prove that HL will ensure that right triangles are congruent.
Another way to determine if a triangle is congruent to another triangle is to use transformations, such as translations, re! ections, and rotations, to see if it can be placed over the other triangle so that they match exactly, or coincide. You can do this by tracing one of the triangles and then translating, re! ecting, and/or rotating what you traced to see if it will " t exactly over the other triangle.
16. Determine if triangle ABC is congruent to triangle DEF. Describe any transformations of triangle ABC you used.
A
E F
D
C B
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138 SpringBoard® Mathematics with Meaning™ Geometry
Congruent Triangle Methods ACTIVITY 2.5continued Truss Your JudgmentTruss Your Judgment
CHECK YOUR UNDERSTANDING
1. If !EGT, " !MXS then which of the following statements is true?
a. ∠S " ∠T b. ___
ET " ____
XM
c. !GET " !SXM d. ∠G " ∠S
2. !WIN " !LUV with m∠W = 38°, m∠V = 102° and m∠I = (7x + 5)°. Find the value of x and the measure of ∠U.
3. To prove the two triangles congruent by ASA, what other piece of information is needed?
X I
B
O P
G
4. In each of the following determine which postulate or theorem can be used to prove the triangles congruent. If it is not possible to prove them congruent, write not possible.
a.
b.
c.
d.
e.
5. MATHEMATICAL R E F L E C T I O N
Greg and his boss, John, want to discuss the report,
but John is out of town. John asked Greg to email the report to him, explaining in detail how he arrived at his conclusions. Greg’s report must contain the following.
• A brief description of what Greg did to arrive at the congruent triangle methods.
• Which congruent triangle method would be most e! ectively used for the recreation hall roof situation? Be certain to explain to John why Greg chose this particular method.
