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)