Quadrilaterals

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## Quadrilaterals

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**Today’s Learning Goals**• 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) Definitions • 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.**Parallelograms**• 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? Parallelograms • 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 Parallelograms 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 Parallelograms 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 Parallelograms • 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 Parallelograms • 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) Other Quadrilaterals • 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) Other Quadrilaterals • 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) Other Quadrilaterals • 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!**Quadrilateral Construction**• 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.**Quadrilateral Construction**• 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.**Quadrilateral Construction**• 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.**Quadrilateral Construction**• 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.**Partner Work**• You have 20 minutes to work on the following problems with your partner.**For those that finish early**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**Big Idea from Today’s Lesson**• 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.**Homework**• Pgs. 225 – 226 (6 – 11, 14, 15)