Rhombuses, Rectangles, and Squares Chapter 6 Section 6.4 Rhombuses, Rectangles, and Squares

| | A rhombus is a parallelogram with four congruent sides. PROPERTIES OF SPEC IAL PARALLELOGRAMS A rhombus is a parallelogram with four congruent sides. A rectangle is a parallelogram with four right angles. | | A square is a parallelogram with four congruent sides and four right angles.

Always Sometimes Sometimes Always Always Always PROPERTIES OF SPEC IAL PARALLELOGRAMS Always Sometimes Sometimes Always Always Always

mECD = 32 mADC = 2mBDC mADC = 86 = mCBA mDAB = 2mEAB PROPERTIES OF SPEC IAL PARALLELOGRAMS mECD = 32 mADC = 2mBDC mADC = 86 = mCBA mDAB = 2mEAB mDAB = 114 mDAB + mADC = 180 114 + mADC = 180 mADC = 66

Diagonals are  3x – 15 = 90 3x = 105 x = 35 5x – 8 = 3x + 24 PROPERTIES OF SPEC IAL PARALLELOGRAMS Diagonals are  3x – 15 = 90 3x = 105 x = 35 Alt. Int Angle Thm mADE = m CBE 5x – 8 = 3x + 24 2x – 8 = 24 2x = 32 x = 16

Consecutive ’s Supplementary PROPERTIES OF SPEC IAL PARALLELOGRAMS Consecutive ’s Supplementary 4x + 14 + 2x + 10 = 180 6x +24 = 180 6x = 156 x = 26

Using Properties of Special Parallelograms COROLLARIES ABOUT SPECIAL QUADRILATERALS 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 a rhombus and a rectangle. You can use these corollaries to prove that a quadrilateral is a rhombus, rectangle, or square without proving first that the quadrilateral is a parallelogram.

If the Sides are , then it is a rectangle Proving a Parallelogram is a Special Kind If the Sides are , then it is a rectangle P(-2, 3) S(2, 3) Q(-2, -4) R(2, -4) And A parallelogram with 4 right angles is a Rectangle

If the Sides are , then it is a rhombus Proving a Parallelogram is a Special Kind If the Sides are , then it is a rhombus Q(3, 6) R(-1, -1) P(7, -1) S(3, -8)

Since PQRS is a parallelogram opposite sides are congruent Proving a Parallelogram is a Special Kind Since PQRS is a parallelogram opposite sides are congruent Q(3, 6) R(-1, -1) S(3, -8) All 4 Sides  P(7, -1) A parallelogram with 4  sides is a Rhombus

If diagonals are , then it is a rectangle Proving a Parallelogram is a Special Kind If diagonals are , then it is a rectangle Q(3, 7) R(6, 4) P(-4, 0) S(-1, -3)

If diagonals are , then it is a rectangle Proving a Parallelogram is a Special Kind If diagonals are , then it is a rectangle R(6, 4) P(-4, 0) Q(3, 7) S(-1, -3) PQRS is a rectangle #16

Need to show: All Sides  OR 2. Diagonals  Def.  ’s Proving a Parallelogram is a Special Kind Need to show: All Sides  OR 2. Diagonals  Def.  ’s

HW #71 Pg 353-355 44-62, 66-68, 74-82 Even, 83   

Diagonals Bisect Each Other Opp. Sides  Parallelograms Opp. ’s  Opp. Sides || Cons.  Supplementary Diagonals Bisect Each Other Rhombus Squares Rectangles 4  Sides Diagonals are  4 Right ’s Diagonals Bisect the Opp ’s Diagonals are 

Cons. ’s Supplementary Opp. Sides || Opp. Sides  Opp. ’s  Parallelograms Quadrilateral Cons. ’s Supplementary Diagonals Bisect Each Other One Pair Opp Sides  and ||

4  Sides Rhombus Quadrilateral Rectangle 4 Right ’s Square

Both Diagonals Bisect each Opposite  Parallelogram Diagonals  Rhombus Both Diagonals Bisect each Opposite  2 Consecuitive Sides  Square Rectangle 1 Right 