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

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

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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.

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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.

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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.

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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

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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.

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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

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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.

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

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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).

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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

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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

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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

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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.

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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.

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Workbook, pp. 78, 79, 81-82

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