Download Congruent Triangle Methods Truss Your Judgment

Congruent Triangle Methods
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Close Reading,
Marking the Text, Summarize/Paraphrase/Retell
ACTIVITY
2.5
My Notes
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.
MATH TERMS
congruent triangles
C
B
© 2010 College Board. All rights reserved.
A
Unit 2 • Congruence, Triangles, and Quadrilaterals
125
ACTIVITY 2.5
continued
Congruent Triangle Methods
Truss Your Judgment
My Notes
Corresponding parts
result from a one-to-one
matching of side lengths
and angles from one
figure to another.
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.
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.
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a. Repeat Greg’s experiment below. Draw two dif ferent triangles of
your own, each having a side 2 inches in length.
Congruent Triangle Methods
ACTIVITY 2.5
continued
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Create Representations, Identify a Subtask
My Notes
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.
C
A
B
___
© 2010 College Board. All rights reserved.
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 AC and mark the pairs of
congruent sides on the triangles above.
Unit 2 • Congruence, Triangles, and Quadrilaterals
127
ACTIVITY 2.5
continued
Congruent Triangle Methods
Truss Your Judgment
My Notes
SUGGESTED LEARNING STRATEGIES:
Create Representations, Identify a Subtask
3. Suppose that Greg measures an angle and an adjacent side of the
triangle shown below.
C
A
B
___
b. Name the parts of the new triangle that correspond to ∠A and AB
and mark the pairs of congruent parts on the triangles above.
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a. Is it possible to draw a triangle that has___
an angle congruent to
∠A and an adjacent side congruent to AB so that the new triangle
is not congruent to "ABC? If so, draw such a triangle and label
the vertices D, E, and F.
Congruent Triangle Methods
ACTIVITY 2.5
continued
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Create Representations,
Identify a Subtask
My Notes
4. Suppose that Greg measures an angle and the opposite side of the
triangle shown below.
C
A
B
© 2010 College Board. All rights reserved.
a. Is it possible to draw a triangle that has___
an angle congruent to
∠A and an opposite side congruent to CB 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 CB
and mark the pairs of congruent parts on the triangles above.
Unit 2 • Congruence, Triangles, and Quadrilaterals
129
ACTIVITY 2.5
continued
Congruent Triangle Methods
Truss Your Judgment
My Notes
SUGGESTED LEARNING STRATEGIES: Create Representations,
Identify a Subtask
5. Suppose that Greg measures two angles of the triangle shown below.
C
A
B
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.
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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.
Congruent Triangle Methods
ACTIVITY 2.5
continued
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Quickwrite,
Self/Peer Revision
My Notes
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 ficiently.
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 first triangle below. Here are the
combinations in Greg’s list: SAS, SSA, ASA, AAS, SSS, and AAA.
© 2010 College Board. All rights reserved.
a. Mark each triangle below to illustrate the combinations in
Greg’s list.
SAS
SSA
ASA
AAS
SSS
AAA
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
131
ACTIVITY 2.5
continued
Congruent Triangle Methods
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Use Manipulatives
My Notes
8. Three segments congruent to the sides of !ABC and three angles
congruent to the angles in !ABC are given in Figures 1–6, shown
below.
C
A
B
a. Use the manipulative(s) supplied to recreate the six figures given
below.
Figure 1
Figure 3
Figure 2
Figure 5
Figure 6
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Figure 4
Congruent Triangle Methods
ACTIVITY 2.5
continued
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Discussion Group,
Create Representations, Use Manipulatives
My Notes
8b. Identify which of the figures in part (a) is congruent to each of the
parts of !ABC.
∠A:
∠C :
___
CB:
∠B:
___
AB:
___
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.
Could You Create
a Triangle Not
Congruent to !ABC
© 2010 College Board. All rights reserved.
Combination
Name the Three Figures
Used by Listing the
Figure Numbers
Unit 2 • Congruence, Triangles, and Quadrilaterals
133
ACTIVITY 2.5
continued
Congruent Triangle Methods
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Interactive Word
Wall, Think/Pair/Share
My Notes
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 combinations that seem to guarantee a triangle congruent to !ABC. These
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.
b.
c.
d.
© 2010 College Board. All rights reserved.
a.
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Congruent Triangle Methods
ACTIVITY 2.5
continued
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Marking the Text,
Self/Peer Revision
My Notes
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 properties 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.
Remember: Postulates are
statements accepted without
proof, while theorems need
proof before they are used.
Given: !MNO and !PQR
____
___
∠N # ∠Q, ∠O # ∠R, and MO # PR
Prove: !MNO # !PQR
M
N
P
O
© 2010 College Board. All rights reserved.
Q
Statements
1. !MNO and !PQR
2. m∠M + m∠N + m∠O = 180°;
m∠P + m∠Q + m∠R = 180°
3. m∠M + m∠N + m∠O =
m∠P + m∠Q + m∠R
4. ∠N # ∠Q; ∠O # ∠R
5. m∠N = m∠Q; m∠O = m∠R
6. m∠M + m∠N + m∠O =
m∠P + m∠N + m∠O
7. m∠M = m∠P
R
Reasons
1.
2.
3.
4.
5.
6.
7.
8. ∠M # ∠P
8.
9. MO # PR
9.
____
___
10. !MNO # !PQR
10.
Unit 2 • Congruence, Triangles, and Quadrilaterals
135
ACTIVITY 2.5
continued
Congruent Triangle Methods
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Think/Pair/Share
My Notes
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.
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?
136
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d.
Congruent Triangle Methods
ACTIVITY 2.5
continued
Truss Your Judgment
SUGGESTED LEARNING STRATEGIES: Look for a Pattern,
Group Presentation, Think/Pair/Share
My Notes
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.
© 2010 College Board. All rights reserved.
Another way to determine if a triangle is congruent to another triangle is
to use transformations, such as translations, reflections, 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, reflecting, and/or rotating what you traced to see if it will fit
exactly over the other triangle.
16. Determine if triangle ABC is congruent to triangle DEF. Describe any
transformations of triangle ABC you used.
A
C
B
E
F
D
Unit 2 • Congruence, Triangles, and Quadrilaterals
137
Congruent Triangle Methods
ACTIVITY 2.5
continued
Truss Your Judgment
CHECK YOUR UNDERSTANDING
1. If !EGT, " !MXS then which of the
following statements is true?
___
c.
____
a. ∠S " ∠T
b. ET " XM
c. !GET " !SXM
d. ∠G " ∠S
d.
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
e.
I
O
P
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.
5. MATHEMATICAL Greg and his boss, John,
R E F L E C T I O N 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 effectively used for the recreation hall
roof situation? Be certain to explain to John
why Greg chose this particular method.
Write Greg’s report on your own paper.
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G
B