6.6 Special Quadrilaterals Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Goals: Identify special quadrilaterals based on limited information. Prove that a quadrilateral is a special type, such as a rhombus or trapezoid. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Parallelogram Rhombus Rectangle Square Trapezoid Kite Isosceles Trapezoid Each shape has all the properties of all of the shapes above it. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Example 1 Which of these have at least one pair of congruent sides? Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid Kite Isosceles Trapezoid       February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Example 2 Which of these have at least one pair of congruent angles? Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid Kite Isosceles Trapezoid       February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Example 3 Which of these have two pairs of congruent angles? Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid Kite Isosceles Trapezoid      February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Example 4 In which of these figures are diagonals perpendicular? Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid Kite Isosceles Trapezoid    February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Example 5 In which of these is the sum of the angles 360?  Quadrilateral Parallelogram Rhombus Rectangle Square Trapezoid Kite Isosceles Trapezoid        All of these have angle sums of 360 -- they are all quadrilaterals. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Identifying Quadrilaterals Be as specific as possible.

Geometry 6.6 Special Quadrilaterals What is this? What we know: Diagonals bisect each other. Diagonals congruent. Must be a… Rectangle. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals What is this? What we know: Diagonals bisect each other. Diagonals congruent. Diagonals perpendicular. Must be a… Square. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals What is this? What we know: One pair of opposite sides parallel. Base angles congruent. Must be an… Isosceles Trapezoid. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals What is this? What we know: Both pair of opposite angles congruent. And that’s all. Must be a… Parallelogram. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Why? 1 + 2 + 3 + 4 = 360 1  3; 2  4 2 1 + 2 2 = 360 1 + 2 = 180 1 and 2 are supplementary. The same is true for all consecutive pairs. Opposite sides are parallel. 1 2 3 4 February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals What is this? What we know: Four angles congruent Must be a… Rectangle. February 17, 2019 Geometry 6.6 Special Quadrilaterals

What kinds of quadrilaterals meet the conditions shown? Rectangle Square February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals What kind of quadrilateral has at least one pair of opposite sides congruent? Parallelogram Rhombus Rectangle Square Isosceles Trapezoid February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals In the following examples, which two segments or angles must be congruent to enable you to prove ABCD is the given quadrilateral? February 17, 2019 Geometry 6.6 Special Quadrilaterals

Example 1: show ABCD is a rectangle. Which two segments or angles must be congruent to enable you to prove ABCD is a rectangle? Possible Solutions: Show A  B B  C C  D D  A or AC  BD A B C D February 17, 2019 Geometry 6.6 Special Quadrilaterals

Example 2: show ABCD is a parallelogram. Which two segments or angles must be congruent to enable you to prove ABCD is a parallelogram? Possible Solutions: AD  BC A C & B  D A B D C February 17, 2019 Geometry 6.6 Special Quadrilaterals

Your Turn. Show ABCD is an Isosceles Trapezoid. Possible Answers AD  BC A  B D  C A B D C February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Example 3 Mark these points on the graph and connect them to form a quadrilateral. P(-2, 4) Q(3, 5) R(4, 2) S(-1, 1) February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals What kind of quadrilateral is PQRS? Prove your conjecture. P(-2, 4) R(4, 2) S(-1, 1) Probably a parallelogram. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals To prove PQRS is a parallelogram: Show opposite sides parallel, or Show opposite sides congruent, or Show one pair of opposite sides congruent and parallel. Q(3, 5) P(-2, 4) R(4, 2) S(-1, 1) February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals 1) Show opposite sides parallel. P(-2, 4) Q(3, 5) R(4, 2) S(-1, 1) Find slopes of PQ and SR. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals 1) Show opposite sides parallel. P(-2, 4) Q(3, 5) R(4, 2) S(-1, 1) Find slopes of PS and QR. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals 1) Show opposite sides parallel. P(-2, 4) Q(3, 5) R(4, 2) S(-1, 1) We know that PQ || SR and PS || QR. PQRS is a parallelogram by definition. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Another Example Graph the points P(-1, 2) Q(4, 4) R(2, -1) S(-1, -1) and connect them. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Conjecture: PQRS is a kite. Q(4, 4) To prove this: Would showing SQ  PR be enough? NO. Why? The diagonals of a rhombus are perpendicular, also. P(-1, 2) R(2, -1) S(-1, -1) February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Conjecture: PQRS is a kite. Q(4, 4) P(-1, 2) R(2, -1) S(-1, -1) A kite has two pairs of consecutive & congruent sides. Show PQ  RQ and PS  RS. February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals Conjecture: PQRS is a kite. Find PQ, RQ, PS, and RS. KITE February 17, 2019 Geometry 6.6 Special Quadrilaterals

Geometry 6.6 Special Quadrilaterals As you are doing these problems tonight, be sure to have enough information to justify your conjecture. February 17, 2019 Geometry 6.6 Special Quadrilaterals