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created by chris markstrum © 2005 6.4 Rhombi, Rectangles, and Squares California State Standards for Geometry 4: Prove basic theorems involving congruence & similarity. 7: Prove & use theorems involving parallel lines and properties of quadrilaterals. 12: Find & use side and angle measures of triangles and polygons 17: Prove theorems using coordinate geometry definitions Rhombus A parallelogram with all sides congruent Rectangle A parallelogram with all angles congruent (4 right angles) Square A parallelogram with all sides congruent and all angles congruent created by chris markstrum 2005 created by chris markstrum 2005 Quadrilaterals parallelograms squares theorems about rhombi, rectangles, and squares Rhombus Corollary A quadrilateral is a rhombus if and only if it has four congruent sides. Rectangle Corollary A quadrilateral is a rectangle if and only if it has four right angles. Square Corollary A quadrilateral is a square if and only if it is both a rectangle and a rhombus. created by chris markstrum 2005 created by chris markstrum 2005 theorem A parallelogram is a rhombus if and only if its diagonals are perpendicular. If in ABCD B then it is a rhombus AC BD A C D theorem created by chris markstrum 2005 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. If in ABCD B then it is a rhombus AC bisects A and C and BD bisects B and D A C D theorem created by chris markstrum 2005 A parallelogram is a rectangle if and only if its diagonals are congruent. If in ABCD B A AC BD AC BD then it is a rectangle D C Given: Prove: 1. A AC BD ABCD is a rectangle ABCD, AC BD 1. Given 2. AB CD, AD BC 2. ACD BDC 3. 3. D opp. sides SSS 4. ADC BCD 4. C.P.C.T.C. 5. DAB BCD, 5. ADC CBA 6. A B C D 7. ABCD is a rectangle opp. ' s 6. Transitive Prop 7. Def rectangle B created by chris markstrum 2005 C Given : AC BD ABCD is a rectangle Prove : Statement 1. 2. ABCD , AC BD AB CD, AD BC created by chris markstrum 2005 A B D C Reason 1. 2. Given opp. sides 3. ACD BDC 3. SSS 4. ADC BCD 4. C.P.C.T.C. 5. DAB BCD, ADC CBA 5. 6. A B C D 6. Transitive 7. ABCD is a rectangle 7. Def rectangle opp. ' s Example created by chris markstrum 2005 A rectangle is a square. Sometimes, Always, or Never? Quadrilaterals parallelograms squares Example created by chris markstrum 2005 A square is a rhombus Sometimes, Always, or Never? Quadrilaterals parallelograms squares Example created by chris markstrum 2005 A rhombus is a rectangle Sometimes, Always, or Never? Quadrilaterals parallelograms squares Example created by chris markstrum 2005 A parallelogram is a rectangle Sometimes, Always, or Never? Quadrilaterals parallelograms squares Example created by chris markstrum 2005 rectangle EFGH, EG = 8z – 16 Find EK and GK E K EK 12 EG EK 12 8z 16 EK 4 z 8 GK EK GK 4 z 8 F H The diagonals of a rectangle bisect each other. G Example Rectangle ABCD created by chris markstrum 2005 mB 8 x 26 Find the value of x A B D C By definition of a rectangle, B is a right angle. mB 90 8 x 26 90 x8 Example Rectangle MATH Perimeter MAT 30 created by chris markstrum 2005 M MA AT 4 x 7 MT 2 x 1 H Find AH and the perimeter of rect. MATH (MA AT ) MT 30 (4 x 7) 2 x 1 30 x6 A T Example created by chris markstrum 2005 Rectangle MATH Perimeter MAT 30 M MA AT 4 x 7 MT 2 x 1 H Find AH and the perimeter of rect. MATH x6 AH MT 2 6 1 AH 13 A T Example created by chris markstrum 2005 M Rectangle MATH Perimeter MAT 30 MA AT 4 x 7 MT 2 x 1 H Find AH and the perimeter of rect. MATH x6 perimeter of rect. MATH = 2( MA AT ) 2(4 x 7) 2(4 6 7) 34 A T Summary What are the three ways to show that a parallelogram is a rhombus What are the two ways to show that a parallelogram is a rectangle? How do you show that a parallelogram is a square? created by chris markstrum 2005