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Quadrilaterals

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Quadrilaterals

- We will learn that quadrilaterals are NOT stable figures that keep their shape under stress.
- We will understand the quadrilateral inequality – the sum of the lengths of any three sides of a quadrilateral is greater than the length of the fourth side.
- We will determine the sum of the interior angles for any quadrilateral.

a)

b)

c)

d)

e)

- A quadrilateral is a closed, four-sided 2-D figure with straight sides that do not overlap.

- Which of the following is a quadrilateral? Explain how you know.

Good…b), d), and e) are quadrilaterals. a) is not because the sides overlap and c) is not because one side is not straight.

- A parallelogram is a quadrilateral where both pairs of opposite sides are parallel.

- Below are several examples of parallelograms.

- What do you notice is true about opposite sides other than they are parallel?

Yes…opposite sides look like they are the same length for parallelograms.

B

C

A

D

- To prove that opposite sides have the same length, then construct AC.

- When we construct AC, two triangles are created. How could we name the two triangles?

- Consider the following parallelogram:

Nice…ABC and ADC.

- Consider the highlighted angle. Since AD || BC and AC is a transversal, which numbered angle is equal to the highlighted angle?

B

C

A

D

3

2

4

1

Great…m(1) = m(3) because they are alternate interior angles.

- Now, consider the blue highlighted angle. Knowing that CD || AB and AC is a transversal, which numbered angle is equal to the blue highlighted angle?

B

C

A

D

3

2

4

1

Great…m(4) = m(2) because they are also alternate interior angles.

B

C

3

B

2

A

D

4

3

A

C

4

1

D

2

1

C

A

- Let’s break the parallelogram up into the two triangles.

B

C

3

B

2

A

D

4

3

A

C

4

1

D

2

1

C

A

Beautiful…AB = CD and BC = AD

- From ASA, we see that ABC ADC.

- If ABC ADC, then what sides are equal in the parallelogram?

- So, we just proved that opposite sides of a parallelogram are equal for any parallelogram.

a)

b)

c)

d)

- A rectangle is a quadrilateral with exactly four right angles.

- Which of the following are rectangles?

Yes…a, b, and c are all rectangles because they are quadrilaterals with four right angles.

- Notice that a square is a rectangle because it satisfies the definition of a rectangle!

a)

b)

c)

d)

- A rhombus is a quadrilateral with all four sides having the same length.

- Which of the following are rhombi?

Great…b, c, and d are all rhombi because they all have four side lengths with the same measurement.

- Notice that a square is also a rhombus because it satisfies the definition of a rhombus!

a)

b)

c)

d)

- A trapezoid is a quadrilateral with exactly one pair of parallel sides.

- Which of the following are trapezoids?

Nice…d is the only trapezoid. The other shapes all have two pairs of parallel sides.

- Notice that trapezoids are not parallelograms while rectangles, squares, and rhombi are parallelograms!

- If you were given four different side lengths, would you always be able to make a quadrilateral?

Okay…some people think you would be able to and some think you might not.

- Today, we are going to try to make quadrilaterals using metal polystrips with different side lengths.

- Let’s try to make a quadrilateral with our metal polystrips with lengths of 8, 8, 8, and 8 units.

- What shape is made from four 8 side lengths?

Nice…it could be a square, rectangle, rhombus, and/or a parallelogram.

- Notice how a quadrilateral is NOT rigid likethe triangle was.

- How could we make the quadrilateral rigid?

Great…put a length across the diagonal to make two triangles from the quadrilateral.

- Now, let’s try to make a quadrilateral with side lengths of 4, 4, 6, and 17 units by putting the 17 length on the bottom.

- How come we could not make a quadrilateral with side lengths of 4, 4, 6, and 17?

Yes…4 + 4 + 6 < 17 so the sides will never meet to make a quadrilateral.

- Some of you thought that a quadrilateral could be made with any side lengths. We just saw an example of four side lengths that did not make a quadrilateral.

- Try to make more quadrilaterals with different side lengths greater than 3 and less than 18.

- You have 20 minutes to work on the following problems with your partner.

Determine which set or sets of side lengths below can make the following shapes.

i) A quadrilateral with all angles the same size.

ii) A parallelogram.

iii) A quadrilateral that is not a parallelogram.

a) 5, 5, 8, 8

b) 5, 5, 6, 14

c) 8, 8, 8, 8

d) 4, 3, 5, 14

- A quadrilateral can only be made if the longest side is shorter than the sum of the other three sides (Quadrilateral Inequality)!
- A quadrilateral is NOT rigid…more than 1 quadrilateral can be made from four given sides by pressing on one of the corners.
- The sum of the interior angles is 360° for ANY quadrilateral.
- A rhombus is a parallelogram with four sides of equal length.
- A square is a parallelogram with four sides of equal length and four right angles.
- A rectangle is a parallelogram with four right angles.
- A trapezoid is a quadrilateral with one pair of parallel sides.

- Pgs. 225 – 226 (6 – 11, 14, 15)