correctionkey=nl-a;ca-a 5 . 4 do not edit--changes must be … · 2017-12-13 · module 5 255...

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Explore Constructing Triangles Given Three Side Lengths

Two triangles are congruent if and only if a rigid motion transformation maps one triangle onto the other triangle. Many theorems can also be used to identify congruent triangles.

Follow these steps to construct a triangle with sides of length 5 in., 4 in., and 3 in. Use a ruler, compass, and either tracing paper or a transparency.

A Use a ruler to draw a line segment of length 5 inches. Label the endpoints A and B.

B Open a compass to 4 inches. Place the point of the compass on A, and draw an arc as shown.

C Now open the compass to 3 inches. Place the point of the compass on B, and draw a second arc.

D Next, find the intersection of the two arcs. Label the intersection C. Draw

_ AC and

_ BC . Label the

side lengths on the figure.

E Repeat steps A through D to draw △DEF on a separate piece of tracing paper. The triangle should have sides with the same lengths as △CAB. Start with a segment that is 4 in. long. Label the endpoints D and E as shown.

Resource Locker

A B

4 in.

5 in.A B

4 in. 3 in.

5 in.

A B

C

4 in. 3 in.

5 in.

D E

F

4 in.

3 in.5 in.

Module 5 255 Lesson 4

5.4 SSS Triangle CongruenceEssential Question: What does the SSS Triangle Congruence Theorem tell you about triangles?

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Common Core Math StandardsThe student is expected to:

G-CO.B.8

Explain how the criteria for triangle congruence (... SSS) follow from the definition of congruence in terms of rigid motions. Also G-CO.B.7, G-CO.C.10, G-SRT.B.5

Mathematical Practices

MP.7 Using Structure

Language ObjectiveHave small groups of students complete a triangle congruence chart.

COMMONCORE

COMMONCORE

HARDCOVER PAGES 221230

Turn to these pages to find this lesson in the hardcover student edition.

SSS Triangle Congruence

ENGAGE Essential Question: What does the SSS Triangle Congruence Theorem tell you about triangles?If three sides of one triangle are congruent to three

sides of another triangle, you can conclude that the

triangles are congruent.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Point out that the “structural beams” mentioned in the Preview refer to the steel sides of the triangles. Then preview the Lesson Performance Task.

255

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore Constructing Triangles Given

Three Side Lengths

Two triangles are congruent if and only if a rigid motion transformation maps one triangle onto

the other triangle. Many theorems can also be used to identify congruent triangles.

Follow these steps to construct a triangle with sides of length 5 in., 4 in., and 3 in.

Use a ruler, compass, and either tracing paper or a transparency.

Use a ruler to draw a line segment of length 5 inches. Label the endpoints A and B.

Open a compass to 4 inches. Place the point of

the compass on A, and draw an arc as shown. Now open the compass to 3 inches. Place

the point of the compass on B, and draw

a second arc.

Next, find the intersection of the two arcs. Label

the intersection C. Draw _ AC and

_ BC . Label the

side lengths on the figure.

Repeat steps A through D to draw △DEF on

a separate piece of tracing paper. The triangle

should have sides with the same lengths as

△CAB. Start with a segment that is 4 in. long.

Label the endpoints D and E as shown.

Resource

Locker

G-CO.B.8 Explain how the criteria for triangle congruence (... SSS) follow from the definition of

congruence in terms of rigid motions. Also G-CO.B.7, G-CO.C.10, G-SRT.B.5

COMMONCORE

A

B

4 in.

5 in.A

B

4 in.

3 in.

5 in.

A

B

C

4 in.3 in.

5 in. D

E

F

4 in.

3 in.

5 in.

Module 5

255

Lesson 4

5 . 4 SSS Triangle Congruence

Essential Question: What does the SSS Triangle Congruence Theorem tell you about triangles?

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255 Lesson 5 . 4

L E S S O N 5 . 4

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InDesign Notes1. This is a list

InCopy Notes1. This is a list Bold, Italic, Strickthrough.

InCopy Notes1. This is a list

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Compare △CAB and △DEF. Are they congruent? How do you know?

Reflect

1. Discussion When you construct △CAB, how do you know that the intersection of the two arcs is a distance of 4 inches from A and 3 inches from B?

2. Compare your triangles to those made by other students. Are they all congruent? Explain.

Explain 1 Justifying SSS Triangle CongruenceYou can use rigid motions and the converse of the Perpendicular Bisector Theorem to justify this theorem.

SSS Triangle Congruence Theorem

If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

Example 1 In the triangles shown, let ̄ AB ≅ ̄ DE , ̄ AC ≅ ̄ DF , and ̄ BC ≅ ̄ EF . Use rigid motions to show that △ABC ≅ △DEF.

A

B D E

F

C

Yes. Triangle CAB can be mapped to △DEF by a translation that maps

point B to point F, and then a rotation about point F.

The arcs are sections of circles sets of points, respectively, 3 inches and 4 inches from A. The other circle

is the set of points that is 3 inches away from B. The intersection has the properties of both circles.

Yes, because there is a sequence of rigid motions that maps any one of the

triangles onto any of the others.


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Math BackgroundThe Side-Side-Side Triangle Congruence Theorem is often presented as the first of the Triangle Congruence Theorems because it is easy to demonstrate concretely. In this course, it is the third theorem presented because the justification requires students to apply the Perpendicular Bisector Theorem. Reinforce this justification throughout the lesson.

EXPLORE Constructing Triangles Given Three Side Lengths

INTEGRATE TECHNOLOGYStudents may use geometry software to explore the concept of constructing a triangle with given side lengths.

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Give each student a piece of dry spaghetti. They should measure and break the spaghetti so that they have three pieces that are the same length as the sides of the triangle in the exercise. Instruct them to make a triangle with these pieces on top of the triangle they drew. Have them compare the triangles and explain why they are congruent. Then ask them to consider whether it is possible to create a triangle from the spaghetti pieces that is not congruent to the triangle they drew in the exercise.

EXPLAIN 1 Justifying SSS Triangle Congruence

INTEGRATE MATHEMATICAL PRACTICESFocus on CommunicationMP.3 Some justifications in this exercise may be difficult for some students to understand. Have students model each step with the triangles they have drawn on tracing paper. Encourage students to restate the steps in their own words. Students should fully understand each step before moving on to thenext one.

QUESTIONING STRATEGIESDraw two segments that share an endpoint. How many different segments could be drawn

to turn this figure into a triangle? There is only one

segment.

PROFESSIONAL DEVELOPMENT

SSS Triangle Congruence 256


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ATransform △ABC by a translation along ‾ ⇀ AD

followed by a rotation about point D, so that _ AB and

_ DE coincide. The segments coincide

because they are the same length.

Does a reflection across _ AB map point C to Therefore, point D lies on the perpendicular

point F? To show this, notice that DC = DF, bisector of _ CF by the converse of the

which means that point D is equidistant from perpendicular bisector theorem. Because EC = EF, point C and point F. point E also lies on the perpendicular bisector of

_ CF .

Since point D and point E both lie on the perpendicular bisector of

_ CF and there

is a unique line through any two points, ‹ −

› DE is the perpendicular bisector of _ CF .

By the definition of reflection, the image of point C must be point F. Therefore, △ABC is mapped onto △DEF by a translation, followed by a rotation, followed by a reflection, and the two triangles are congruent.

B Show that △ABC ≅ △PQR.

Triangle ABC is transformed by a sequence of rigid motions to form the figure shown below. Identify the sequence of rigid motions. (You will complete the proof on the following page.)

1.

2.

3.

A BD E

F

C

A BD E

F

C

A BD E

F

C

A

B

P

Q

R

C

A

B

P

Q

R C

Translation along ‾ ⇀ AP

Reflection across _ PQ .

Rotation about P so that _ PQ and

_ AB

coincide.


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COLLABORATIVE LEARNING

Peer-to-Peer ActivityGive each student five index cards. Students should prepare a card that shows two labeled triangles for each of the following situations:

• The triangles can be proved congruent by SSS.• The triangles can be proved congruent by SAS.• The triangles can be proved congruent by ASA.• The triangles can be proved noncongruent.• There is not enough information to determine congruence.

Collect and shuffle them. Divide students into pairs and give each pair ten cards. Have them sort their cards into the above categories.

257 Lesson 5 . 4


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Complete the explanation by filling in the blanks with the name of a point, line segment, or geometric theorem.

Because _ QR ≅ , point Q is equidistant from and . Therefore,

by the converse of the Theorem, point Q lies on the

of _ RC . Similarly,

_ PR ≅ . So point lies on

the perpendicular bisector of . Because two points determine a line, the line ‹ −

› PQ is

the .

By the definition of reflection, the image of point C must be point . Therefore,

△ABC ≅ △PQR because △ABC is mapped to by a translation, a rotation,

and a .

Reflect

3. Can you conclude that two triangles are congruent if two pairs of corresponding sides are congruent? Explain your reasoning and include an example.

Your Turn

4. Use rigid motions and the converse of the perpendicular bisector theorem to explain why △ABC ≅ △ADC.

A

B

D

C

_ QC

_ PC

_ RC

P

R

△PQR

R (or C ) C (or R)

Perpendicular Bisector

perpendicular bisector

reflection

perpendicular bisector of _ RC

No; you need a third piece of information to ensure a rigid motion maps

one triangle to the other, such as congruent included angles or another

pair of congruent sides.

_ AB ≅ _ AD and

_ CB ≅ _ CD , so A is equidistant

from B and D, and C is equidistant from B and D. By the converse of the Perpendicular Bisector Theorem,

‹ − › AC is the perpendicular

bisector of _ BD . By the definition of a

reflection, point D is the image of point B reflected across

_ AC . The reflection also maps _

AC onto _ AC , so △ABC ≅ △ADC.




GE_MNLESE385795_U2M05L4 258 6/3/14 8:16 AM

DIFFERENTIATE INSTRUCTION

ManipulativesHave students create a triangle and a quadrilateral using strips of construction paper or tagboard and brass fasteners. Then have students attempt to change the angles in each without bending the strips of paper. Students should notice that the triangle always remains the same but that they can create many different quadrilaterals. Discuss how this activity illustrates the Side-Side-Side Triangle Congruence Theorem.






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Explain 2 Proving Triangles Are Congruent Using SSS Triangle Congruence

You can apply the SSS Triangle Congruence Theorem to confirm that triangles are congruent. Remember, if any one pair of corresponding parts of two triangles is not congruent, then the triangles are not congruent.

Example 2 Prove that the triangles are congruent or explain why they are not congruent.

AB = DE = 1.7 m, so _ AB ≅ _ DE .

BC = EF = 2.4 m, so _ BC ≅

_ EF .

AC = DF = 2.3 m, so _ AC ≅ _ DF .

The three sides of △ABC are congruent to the three sides of △DEF.

△ABC ≅ △DEF by the SSS Triangle Congruence Theorem.

DE = = 20 cm, so .

DH = = 12 cm, so .

EH = = 24 cm, so .

The three sides of △DEH are congruent to

the three sides of , so the two triangles are

congruent by .

Your Turn

Prove that the triangles are congruent or explain why they are not congruent.

5. 6.

AB

C DE

F

2.3 m

2.3 m

1.7 m

1.7 m

2.4 m

2.4 m

D

EFH

G

20 cm20 cm

24 cm

24 cm

12 cm

12 cm

M N

P QR

S

54 in.54 in.

28 in.

38 in.

32 in.

38 in.K

G

LJ

the SSS Triangle Congruence Theorem

FG _ DE ≅

_ FG

_ DH ≅

_ FH

_EH ≅

_GH

FH

GH

△FGH

The corresponding sides _ MN and

_ QR are It is given that

_ GK ≅ _ GL and

_ JK ≅ _ JL ,

not congruent.

and _ GJ ≅ _ GJ by the Reflexive Property. not congruent. Therefore, the triangles are




LANGUAGE SUPPORT

Connect ContextThe phrase Given Three Side Lengths in the title of the Explore section may confuse some English Learners. Explain that it indicates that you will be constructing triangles when you are given the lengths of the sides. Remind students that in math, they are often given partial information to help solve a problem, and that the information is called a given.

EXPLAIN 2 Proving Triangles Are Congruent Using SSS Triangle Congruence

INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 Although the example emphasizes the SSS Triangle Congruence Theorem, it is important for students to keep in mind that the triangles are congruent because one can be mapped onto the other by one or more rigid motions. Maintain this connection by asking which segments of one triangle map onto certain segments of another triangle.

QUESTIONING STRATEGIESWhen do you use the SSS Triangle Congruence Theorem instead of the ASA or

SAS Triangle Congruence Theorems to determine whether two triangles are congruent? When you

know three pairs of corresponding congruent sides

and no pairs of corresponding congruent angles,

you cannot use a theorem that involves an angle.

LANGUAGE SUPPORT Have students work in small groups. Have them complete a chart like the following, highlighting the sides and angles that are congruent in each pair of triangles.

Triangle Congruence TheoremsTheorem Definition Picture

259 Lesson 5 . 4


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Explain 3 Applying Triangle CongruenceYou can use the SSS Triangle Congruence Theorem and other triangle congruence theorems to solve many real-world problems that involve congruent triangles.

Example 3 Find the value of x for which you can show the triangles are congruent.

Lexi bought matching triangular pendants for herself and her mom in the shapes shown. For what value of x can you use a triangle congruence theorem to show that the pendants are congruent? Which triangle congruence theorem can you use? Explain.

_ AB ≅

_ JK and

_ AC ≅

_ JL , because they have the same measure. So, if

_ BC ≅

_ KL ,

then △ABC ≅ △JKL by the SSS Triangle Congruence Theorem. Write an equation setting the lengths equal and solve for x. 4x - 6 = 3x - 4; x = 2

Adeline made a design using triangular tiles as shown. For what value of x can you use a triangle congruence theorem to show that the tiles are congruent? Which triangle congruence theorem can you use? Explain.

Notice that _ PQ ≅ _ MN and ≅

_ MO , because they have the same measure.

If _ NO ≅ _ QR , then △MNO ≅ by the Triangle Congruence

Theorem.

Write an equation setting the lengths equal and solve for x.

v Your Turn

7. Craig made a mobile using geometric shapes including triangles shaped as shown. For what value of x and y can you use a triangle congruence theorem to show that the triangles are congruent? Which triangle congruence theorem can you use? Explain.

AK L

JBC

3 cm 3 cm3 cm 3 cm

(4x - 6) cm

(3x - 4) cm

M

N OP

QR

4 in. 4 in. 4 in.4 in.

4 in.

(3x - 11) in.

F

G

H

T

U

V60º

30º

(8x - 12) cm(7y + 4)º 12 cm

_ PR

△PQR SSS

m∠H = 30° by the Triangle Sum Theorem, so ∠H ≅ ∠V. ∠G ≅ ∠U, they are

right angles. If ̄ GH ≅ ̄ UV , then △FGH ≅ △TUV by the ASA. x = 3; y = 8

3x - 11 = 4, 3x = 15, x = 5


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GE_MNLESE385795_U2M05L4 260 10/14/14 7:14 PM

EXPLAIN 3 Applying Triangle Congruence

AVOID COMMON ERRORSBecause students are concentrating on sides when using the SSS Triangle Congruence Theorem, they may not write corresponding angles in the correct order in the congruence statement. Discuss methods they can use to make sure they get the order right.

QUESTIONING STRATEGIESHow do you know whether angles in one triangle are congruent to the angles in the

other triangle? You know the triangles are

congruent by SSS, so you know that corresponding

angles are also congruent because of CPCTC.

CONNECT VOCABULARY Have students collaboratively write a definition for a triangle, providing supports. For example: “A triangle is a __________, It has ___________ sides.It has 3 ________. A right triangle has a _________.”



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Elaborate

8. An isosceles triangle has two sides of equal length. If we ask everyone in class to construct an isosceles triangle that has one side of length 8 cm and another side of length 12 cm, how many sets of congruent triangles might the class make?

9. Essential Question Check-In How do you explain the SSS Triangle Congruence Theorem?

Use a compass and a straightedge to complete the drawing of △DEF so that it is congruent to △ABC.

1.

On a separate piece of paper, use a compass and a ruler to construct two congruent triangles with the given side lengths. Label the lengths of the sides.

2. 3 in., 3.5 in., 4 in. 3. 3 cm, 11 cm, 12 cm

Evaluate: Homework and Practice• Online Homework• Hints and Help• Extra Practice

A

B

D

F

CE

11 cm

12 cm 12 cm 3 cm3 cm

11 cm

4 in.

4 in.

3 in.3 in.

3.5 in.

3.5 in.

The class could make at most 2 sets of congruent triangles. One set would

have sides of length 8 cm, 8 cm, and 12 cm. The second set would have

sides of length 12 cm, 12 cm, and 8 cm.

Possible answer: Two triangles are congruent if a series of rigid motion

transformations maps one triangle onto the other. For two triangles that

have pairs of congruent sides, begin by translating and then rotating one

triangle so that it shares one side with the other triangle. Then apply the

converse of the Perpendicular Bisector Theorem to show that a reflection

completes the mapping.


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ELABORATE QUESTIONING STRATEGIES

Two triangles appear to be congruent. You know that three pairs of corresponding parts

are congruent, but you have no information about the other corresponding parts. How can you determine whether the triangles really are congruent? Sample

answer: If the congruent parts are two pairs of

corresponding angles and the included side, or two

pairs of corresponding sides and the included angle,

you can use the ASA or SAS Triangle Congruence

Theorem. If the congruent parts are three pairs of

corresponding sides, you can use the SSS Triangle

Congruence Theorem. Otherwise, check whether

there is a sequence of rigid motions that map one

triangle onto the other.

SUMMARIZE THE LESSONWhen do you use the SSS Triangle Congruence Theorem? What information do

you need in order to use this theorem? You use the

SSS Triangle Congruence Theorem to prove that two

triangles are congruent by using three pairs of

congruent sides. You need to know that all three

pairs of corresponding sides are congruent to use it.

261 Lesson 5 . 4





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Identify a sequence of rigid motions that maps one side of △ABC onto one side of △DEF.

4.

5.

6.

7.

In each figure, identify the perpendicular bisector and the line segment it bisects, and explain how to use the information to show that the two triangles are congruent.

8.

9.

A

BD

E

F

C

A

B

F

E

D

C

A

B

FE

DC

A

B

F

E

D

C

A

B

D

CD

E

F

G

Possible answer: A translation along ‾ ⇀ CF ,

and then a clockwise rotation about

F so that ― CB coincides with

― FE .

Possible answer: A translation along ‾ ⇀ AD ,

and then a clockwise rotation about D so

that ― AB coincides with

― DE .

Possible answer: A translation along ‾ ⇀ AD ,

and then a counterclockwise rotation

about D so that ― AB coincides with

― DE .

Possible answer: A translation along ‾ ⇀ CF ,

and then a counterclockwise rotation

about F so that ― CA coincides with

― FD .

‹ −

› BC is the perpendicular bisector of ― AD .

This shows that A maps to D by a

reflection across ‹ −

› BC .

‹ −

› DF is the perpendicular bisector of ― EG .

This shows that E maps to G by a

reflection across ‹ −

› DF .




GE_MNLESE385795_U2M05L4.indd 262 21/03/14 11:50 AMExercise Depth of Knowledge (D.O.K.)COMMONCORE Mathematical Practices

1–3 2 Skills/Concepts MP.4 Modeling

4–9 2 Skills/Concepts MP.1 Problem Solving

10–13 2 Skills/Concepts MP.2 Reasoning

14–20 3 Strategic Thinking MP.1 Problem Solving

21–27 2 Skills/Concepts MP.3 Logic

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreConstructing Triangles Given Side Lengths

Exercises 1–3

Example 1Justifying SSS Triangle Congruence

Exercises 4–9

Example 2Proving Triangles Are Congruent Using SSS Triangle Congruence

Exercises 10–13

Example 3Applying Triangle Congruence

Exercises 14–20

AVOID COMMON ERRORSSome students may have trouble applying SSS to adjacent triangles. Adjacent triangles share a side, so you can apply the Reflexive Property to get a pair of congruent sides.






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Prove that the triangles are congruent or explain why this is not possible.

10.

11.

12.

13.

14. Carol bought two chairs with triangular backs. For what value of x can you use a triangle congruence theorem to show that the triangles are congruent? Which triangle congruence theorem can you use? Explain.

R Q

S T

6 m6 m

9 m

9 mY N

X M

Z P

51º

50º50º

50º

G

(9x - 21) in.H

25 in. 25 in.

I

T

U

25 in. 25 in.

15 in. V

JK

W X

YL

12 cm

10 cm8 cm

8 cm

J

K W X

YL

9 cm 10 cm

7 cm 8 cm

Congruent, by SSS congruence; ― RS ≅

― TQ

because they have the same measure,

― RT ≅

― RT by the reflexive property, and

_ ST ≅

_ QR because they have the same measure.

Not congruent, because only one

triangle has an interior angle of 51° so

there is no rigid motion that will map

one to the other.

Possibly congruent, depending on the

lengths of the unlabeled sides.

Not congruent. The diagrams show a

total of 4 different side lengths, but

two congruent triangles have only 3

different side lengths.

X = 4; _ GH ≅

_ TU and

_ GI ≅

_ TV , because they have the same measure.

So, if _ HI ≅

_ UV , then ▵GHI ≅ △TUV by the SSS Triangle Congruence Theorem.




COMMUNICATE MATH Introduce the word criterion and its plural, criteria. Ask students if they can define criterion and give an example of its use. For instance, you might discuss a college’s criteria for accepting students (a completed application, a minimum grade-point average, a minimum SAT score, and so on). Tell students that they have developed three criteria for determining whether two triangles are congruent, the ASA, SAS, and SSS Triangle Congruence Theorems. Discuss how they can determine whether two given triangles meet any of these criteria.

263 Lesson 5 . 4





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15. For what values of x and y can you use a triangle congruence theorem to show that the triangles are congruent? Which triangle congruence theorem can you use? Explain.

Find all possible solutions for x such that △ABC is congruent to △DEF. One or more of the problems may have no solution.

16. △ABC: sides of length 6, 8, and x.△DEF: sides of length 6, 9, and x - 1.

17. △ABC: sides of length 3, x + 1, and 14.△DEF: sides of length 13, x - 9, and 2x - 6

18. △ABC: sides of length 17, 17, and 2x + 1.△DEF: sides of length 17, 17, and 3x - 9

19. △ABC: sides of length 19, 25, and 5x - 2.△DEF: sides of length 25, 28, and 4 - y

20. △ABC: sides of length 8, x - y, and x + y△DEF: sides of length 8, 15, and 17

21. △ABC: sides of length 9, x, and 2x - y△DEF: sides of length 8, 9, and 2y - x

22. These statements are part of an explanation for the SSS Triangle Congruence Theorem. Write the numbers 1 to 6 to place these strategies in a logical order. The statements refer to triangles ABC and DEF shown here.

Rotate the image of △ABC about E, so that the image of

_ BC coincides with

_ EF .

Apply the definition of reflection to show D is the reflection of A across

‾ ⇀ EF .

Conclude that △ABC ≅ △DEF because a sequence of rigid motions maps one triangle onto the other.

Translate △ABC along ‾ ⇀ BE .

Define ‾ ⇀ EF as the perpendicular bisector of the

line connecting D and the image of A.

Identify E, and then F, as equidistant from D and the image of A.

9 cm(14x - 47) cm

60º (15y)º

30º

C

E

GF

30ºA

B

A

E

D

F

B

C

x = 4, y = 4; ∠A ≅ ∠E,

∠C ≅ ∠G, and ― AC ≅

_ EG , ASA

x = 9 no solution

x = 10 x = 6, y = -15

x = 16, y = 1; x = 16, y = -1 Possible solution: x = 8, y = 8

2

5

6

1

4

3





INTEGRATE TECHNOLOGYStudents may find it useful to use geometry software to model exercises they are struggling with.






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23. Determine whether the given information is sufficient to guarantee that two triangles are congruent. Select the correct answer for each lettered part.A. The triangles have three pairs of congruent

corresponding angles. sufficient not sufficientB. The triangles have three pairs of congruent

corresponding sides. sufficient not sufficientC. The triangles have two pairs of congruent

corresponding sides and one pair of congruent corresponding angles. sufficient not sufficient

D. The triangles have two pairs of congruent corresponding angles and one pair of congruent corresponding sides. sufficient not sufficient

E. Two angles and the included side of one triangle are congruent to two angles and the included side of the other triangle. sufficient not sufficient

F. Two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. sufficient not sufficient

24. Make a Conjecture Does a version of SSS congruence apply to quadrilaterals? Provide an example to support your answer.

25. Are two triangles congruent if all pairs of corresponding angles are congruent? Support your answer with an example.

A. the triangles have the same shape but not necessarily the same size.

B. SSS Triangle Congruence Theorem

C. These conditions do not produce a unique triangle.

D. The Triangle Sum Theorem will show that the third pair of angles is congruent, so the triangles are congruent by the ASA Triangle Congruence Theorem.

E. ASA Triangle Congruence Theorem

F. SAS Triangle Congruence Theorem

No; Possible example: a variety of non-congruent rhombuses, including

No; At least one pair of sides must be congruent. For example, two

a square, may have 4 sides of the same length.

triangles that have three 60 degree angles may not have at least one pair

of congruent corresponding sides, such as a triangle with side lengths of

4 inches and another triangle with side lengths of 8 inches.




JOURNALHave students compare and contrast the SSS and SAS theorems and support their answers with a sketch that illustrates each theorem.

265 Lesson 5 . 4


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H.O.T. Focus on Higher Order Thinking

26. Explain the Error Ava wants to know the distance JK across a pond. She locates points as shown. She says that the distance across the pond must be 160 ft by the SSS Triangle Congruence Theorem. Explain her error.

27. Analyze Relationships Write a proof. Given: ∠BFC ≅ ∠ECF, ∠BCF ≅ ∠EFC

_ AB ≅ _ DE , _ AF ≅

_ DC

Prove: △ABF ≅ △DEC

Statements Reasons

Lesson Performance TaskMike and Michelle each hope to get a contract with the city to build benches for commuters to sit on while waiting for buses. The benches must be stable so that they don’t collapse, and they must be attractive. Their designs are shown. Judge the two benches on stability and attractiveness. Explain your reasoning.

L

160 ft

210 ft

210 ft190 ft

190 ft

M

N

J

K

A B

F E D

C

Mike Michelle

The distance is 160 ft, but the justification should be

the SAS Triangle Congruence Theorem.

1. ∠BFC ≅ ∠ECF, ∠BCF ≅ ∠EFC

2. ― FC ≅

― FC

3. △BFC ≅ △ECF

6. △ABF ≅ △DEC

4. ― FB ≅

― CE

5. ― AB ≅

― DE , ― AF ≅

― DC

1. Given

2. Reflexive Property of Congruence

3. ASA Triangle Congruence Theorem

4. CPCTC

5. Given

6. SSS Triangle Congruence Theorem

Answers will vary. While Mike's bench will probably be chosen as the more attractive, students should note that Michelle's is more stable. There are many quadrilaterals with the same side lengths as Mike's bench. But, by the SSS Congruence Theorem, the triangles in Michelle's design cannot change shape.


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EXTENSION ACTIVITY

Have students construct models of Mike and Michelle’s benches from sturdy pieces of cardboard or photo-print paper cut into long, narrow strips and attached at the corners with brads. Students can explore further by making other polygons, checking their stability, and explaining why, for example, there is no SSSSS Pentagon Congruence Theorem. A 5-sided polygon is not stable. This means

that there are many different shapes that can be constructed from the same

5 sides of a pentagon.

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Ask students to consider the changes in Mike’s bench if it were to transform from the original shape to complete collapse. Which properties of the bench would change and how would they change? Which properties would not change? Sample

answer: Change: During collapse, the area inside the

parallelogram would decrease continually; the

measures of the angles would change, with the

measures of one pair of opposite angles increasing

continually and the measures of the other pair

decreasing continually.

Not change: The lengths of the sides and the

perimeter of the parallelogram would not change.

AVOID COMMON ERRORSStudents may argue that Mike’s bench may be solidly constructed with screws, nails, and glue, so that it would not collapse. Stress that the bench is a real-world object introduced here to model an abstract geometrical principle, and that it is not a perfect model. The important conclusion to draw is the one relating to quadrilaterals, not to benches.

Scoring Rubric2 points: Student correctly solves the problem and explains his/her reasoning.1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning.0 points: Student does not demonstrate understanding of the problem.



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