Other Types of Quadrilaterals: Rectangles, Rhombi, Squares Trapezoids, Kites

 Proving the properties of rectangles, rhombi, and squares.  Proving properties of trapezoids and kites.

 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.

 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.

 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.

 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 the diagonals of a parallelogram bisect each other, this means that:  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 B D C X

 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.

 Rhombi have unique properties as well.  The diagonals bisect the angles 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. A B D C X

 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.

 Let’s look at the relationship between parallelograms, rectangles, rhombi, and squares in a Venn diagram.

 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).

 A trapezoid is a quadrilateral with a single pair of parallel sides.  The parallel sides are called the 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  A D B C

 The midsegment of a trapezoid 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. A D B C E F

 A trapezoid with congruent legs is called an isosceles trapezoid.  AD  BC  If a trapezoid is isosceles, each pair 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 D B C

 Kites have exactly two pair of consecutive congruent sides.  The properties of a kite are similar to that of a rhombus, except “halved.”  Notice that one diagonal creates two isosceles triangles.  The other diagonal creates two congruent triangles (SSS).  ONE diagonal bisects the angles it passes through.  ONE diagonal is bisected.  ONE pair of opposite angles are congruent.  The diagonals of a kite are perpendicular.

 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.

 Workbook, pp. 78, 79, 81-82