Geometry Day 44

1. Geometry Day 44 Other Types of Quadrilaterals: Rectangles, Rhombi, Squares Trapezoids, Kites

2. Agenda • Proving the properties of rectangles, rhombi, and squares. • Proving properties of trapezoids and kites.

3. Review • A parallelogram is a quadrilateral with two pairs of parallel sides. • The opposite sides of a parallelogram are congruent. • The opposite angles of a parallelogram are congruent. • Any two consecutive angles of a parallelogram are supplementary. • The two diagonals of a parallelogram bisect each other.

4. Review • To prove a quadrilateral is a parallelogram, we can use the following methods: • If the opposite sides are parallel. • If the opposite sides are congruent. • If the opposite angles are congruent. • If the diagonals bisect each other. • If one pair of sides are both parallel and congruent.

5. Rectangles • A rectangle is an equiangular quadrilateral. Each angle is 90. • Can you prove that a rectangle is also a parallelogram? • Because every rectangle is a parallelogram, that means that all the properties of a parallelogram also apply to rectangles. Thus: • A rectangle’s opposite sides are parallel. • A rectangle’s opposite sides are congruent. • A rectangle’s opposite angles are congruent. • A rectangle’s diagonals bisect each other. • We don’t need to prove these because: a) we’ve proven them for parallelograms and b) we’ve proven that a rectangle is a parallelogram.

6. Rectangles • A rectangle has a unique property, that parallelograms in general don’t share. • The diagonals of a rectangle are congruent. • AC  BD • Since we know that thediagonals of a parallelogrambisect each other, this meansthat: • AX  BX  CX  DX • What can we state about the four triangles formed? • The converse of the above is true: If a parallelogram has congruent diagonals, it is a rectangle. A D X B C

7. Rhombi • A rhombus is an equilateral quadrilateral. • Can you prove that a rhombus is also a parallelogram? • Because every rhombus is a parallelogram, that means that all the properties of a parallelogram also apply to rhombi. • Again, we don’t need to prove these.

8. Rhombi A D • Rhombi have unique properties as well. • The diagonals bisect theangles they pass through. • DAX  BAX • ABX  CBX • etc. • The diagonals of a rhombus are perpendicular. • AC  BD • What can we state about the four triangles formed? • Converses: • If the diagonals of a parallelogram are perpendicular, it is a rhombus. • If the diagonals of a parallelogram bisects the angles, it is a rhombus. • Also, if two consecutive sides of a parallelogram are congruent, it is a rhombus. X B C

9. Squares • Squares are regular quadrilaterals. • By definition, squares are rectangles (equal angles) and rhombi (equal sides). • This means that squares are also parallelograms. • Squares have all of the properties of parallelograms, rectangles, and rhombi.

10. The Parallelogram Family • Let’s look at the relationship between parallelograms, rectangles, rhombi, and squares in a Venn diagram.

11. Proofs • We need to prove the following: • The diagonals of a rectangle are congruent. • If the diagonals of a parallelogram are congruent, then it is a rectangle. • The diagonals of a rhombus are perpendicular. • The diagonals of a rhombus each bisect a pair of opposite angles. • If the diagonals of a parallelogram are perpendicular, then it is a rhombus. • If the diagonals of a parallelogram each bisect a pair of opposite angles, then it is a rhombus. • If two consecutive sides of a parallelogram are congruent, then it is a rhombus. • Break into groups. Each group will prove one of the first six (the last is trivial).

12. Trapezoids B A • A trapezoid is a quadrilateral with a single pair of parallel sides. • The parallel sides are calledthe bases. • AB and CD • The non-parallel sides are legs. • AD and BC • Base angles are formed by a base and a leg. There are two pairs of base angles. • A and B • C and D. • Because of the parallel sides, two pairs of consecutive angles (not the base angle pairs) are supplementary. • mA + mD = 180 • mB + mC = 180 D C

13. Midsegments B A • The midsegment of atrapezoid connects the midpoints of the legs. • If AE = ED and BF = FC,then EF is a midsegment. • Theorem: The midsegment of a trapezoid is parallel to both of its bases, and is the average of the length of the bases. • EF ║ AB ║ CD • EF = ½(AB + CD) • In groups, write a coordinate proof for this theorem. F E D C

14. Trapezoids • A trapezoid with congruent legs is called an isosceles trapezoid. • AD  BC • If a trapezoid is isosceles, eachpair of base angles are congruent • A  B; C  D • Converse: • If a trapezoid has one pair of congruent base angles, it is isosceles. • Biconditional: • A trapezoid is isosceles if and only if its diagonals are congruent. • We need to prove both parts of a biconditional separately. A B D C

15. Kite • Kites have exactly two pair of consecutive congruent sides. • The properties of a kite are similarto that of a rhombus, except “halved.” • Notice that one diagonal creates twoisosceles triangles. • The other diagonal creates two congruenttriangles (SSS). • ONE diagonal bisects theangles it passes through. • ONE diagonal is bisected. • ONE pair of opposite angles are congruent. • The diagonals of a kite are perpendicular.

16. Outstanding Proofs • Here are the theorems that require proof: • If a trapezoid is isosceles, each pair of base angles are congruent • If a trapezoid has one pair of congruent base angles, it is isosceles. • If a trapezoid is isosceles, then its diagonals are congruent. • If a trapezoid has diagonals that are congruent, then it’s isosceles. • Exactly one diagonal of a kite bisects a pair of opposite angles. • A kite has exactly one pair of opposite angles that are congruent. • The diagonals of a kite are perpendicular. • You know what to do.

17. Homework 26 • Workbook, pp. 78, 79, 81-82