· web view2015/02/09 · warm up ii: (a) with a ruler and protractor, draw as many...

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RIGIDITY AND FLEXIBILITYADV 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

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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

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were also congruent. We went from a little bit of information, and were able to figure out a lot more.

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.

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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?

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?

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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?

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


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

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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?

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.


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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

· web view2015/02/09 · warm up ii: (a) with a ruler and protractor, draw as many...

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