2015-02-09 rigidity and flexibility (congruent triangles)

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NAME DATE BAND

RIGIDITY AND FLEXIBILITY ADV GEOMETRY | PACKER COLLEGIATE INSTITUTE

Warm Up II:

1. (a) With a ruler and protractor, draw as many quadrilaterals as you can (up to 3! we don’t have all day

for you to draw a zillion!) with four side lengths of 1 inch.

(b) With a ruler and protractor, draw as many quadrilaterals (up to 3! we don’t have all day for you to

draw a zillion!) as you can with four side lengths of 1 inch, and one interior angle of 60o

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(c) With a ruler and protractor, draw as many triangles (up to 3! we don’t have all day for you to draw a

zillion!) with lengths 4, 5, and 10 (in cm).

An Interesting Aside (on the right side):

There is something interesting about part (c)… What are allowable side lengths for a triangle? Think about it

and answer for the following:

Side 1 Side 2 Side 3 Allowable? Yes or No

2 inches 3 inches 20 inches Yes No

1 inch 1 inch 1 inch Yes No

1 inch 1 inch 2 inches Yes No

5 centimeters 5 centimeters 6 centimeters Yes No




249 centimeters 412 centimeters 712 centimeters Yes No

Home Enjoyment:

Jurgensen

p. 221: #7-16,

p. 222-223: Written Exercises: #1-6, 18

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Section 1: The Big Question, and Why We Care

The big goal for this unit is to determine, given some information about two triangles, if the two triangles are

congruent.

Of course, this is easy if we know everything about the triangles…

But we don’t always need to know that all three sides and all three angles are congruent.

For example, we already have stated that if we have two right triangles where the legs are congruent, then

the two triangles are congruent. And that turned out to be super useful, because it let us make a hugely

important conclusion about perpendicular bisectors.

Pop quiz:

What are the congruent triangles? ________________

What was the hugely important conclusion? ______________________________________________

__________________________________________________________________________________

_________________________________________________________________________________.

Neat. From just knowing that the legs of the right triangles were the same, we could conclude that the two

triangles were congruent. And that helped us see the hypotenuses were also congruent. We went from a little

bit of information, and were able to figure out a lot more.

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Section 2: Drawing Challenges

1. Given: One side of a triangle has length 2 inches, and the angle created at one of those end points is 40

degrees. Can you draw a triangle that fits this information? Can you draw a different triangle that fits

this information? Any more?

If you could only draw one triangle with the information given, then every triangle with a side length of 2 inches,

with an angle of 40 degrees at one of those endpoints must look the same. There are no other triangles that you

could draw with that information. In other words, any two triangles with that information must be congruent.

On the other hand, if you could draw multiple triangles with the information given, then you cannot conclude

that every triangle with a side length of 2 inches and an angle of 40 degrees at one of those endpoints are all the

same. You do not have enough information to conclude that all triangles with those properties are congruent.

So… So… is every triangle with a side length of 2 inches and an angle of 40 degrees at one of those endpoints

congruent to every other triangle with a side length of 2 inches and an angle of 40 degrees at one of those

endpoints? Or can they look different?

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2. Given: One side length of a triangle is 2 inches, and the two angles created at the endpoints of this side

are 40 degrees and 60 degrees.

Can you draw a triangle that fits this information? Can you draw a different triangle that fits this

information? Any more?

So… So… is every triangle with one side length of 2 inches, and the two angles created at the endpoints of this

side are 40 degrees and 60 degrees congruent to every other triangle with that property? Or can they look

different?

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3. Given: A triangle with a side length of 4 inches. Off of one endpoint of this 4” side, you have another

side length of 3 inches. Off of the other endpoint of this 4” side, you have an angle of 30 degrees. (New

mathematical phrasing: The 3 inch side is opposite the 30 degree angle.)1



So… So… if you have two triangles with the information given above, do you know that the two triangles are

congruent? Or can they look different?

1 Another example of this term “opposite”: “In all right triangles, the hypotenuse is opposite the right angle.”

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4. Given: A triangle with three side lengths of 5 cms, 4 cms, and 7 cms, and the angle opposite the 7 cm

side is 78.5 degrees.





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5. Given: A triangle with angles of 20 degrees and 40 degrees.





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6. Given: You have a triangle with two sides of length 3 inches and 4 inches, and an angle between them of

45 degrees.





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7. Given: You have a triangle with two sides of length 3 inches and 4 inches.





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8. Given: You have a triangle with sides of lengths 2 inches, 3 inches, and 4 inches.



So… So… if you have two triangles with the information given above, do you know that the two triangles

are congruent? Or can they look different?

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9. Given: You have a triangle with angles of 50 degrees, 70 degrees, and 60 degrees, and the side opposite

the smallest angle is 4 cm.





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10. Given: You have a triangle with two angles of 60 degrees and 50 degrees, and the side opposite the 60

degrees has length 3 inches.





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Section 3: Summarizing your Findings 1. You just drew a bunch of triangles based on some given information. We’d like you to summarize your findings in the table below. The first column shows a pictorial representation of the information you were given about the measures of the triangle’s sides and/or angles. Match this picture with the questions number in Section 2, state if you found one or many triangles with this description, and then answer if you were forced to draw only one triangle. Given Information Which # (in

Section 2) One or many? Were you

forced to draw one triangle?

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The following pages provide room for you to take notes and make diagrams to help us answer this question: “What is the minimum amount of information needed to know if two triangles are congruent?”

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2. Class Notes A

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3. Class Notes B

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4. Class Notes C

2015-02-09 rigidity and flexibility (congruent triangles)

Documents

congruent triangles

right triangles

multiple triangles

different triangle

interior angle

important conclusion

big goal

big question