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Parallelograms and Rectangles

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Quadrilateral Definitions Parallelogram: opposite sides are parallel Rectangle: adjacent sides are perpendicular

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the first proof…

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Prove: If it is a parallelogram, then the opposite sides are equal. By definition, a parallelogram has opposite sides that are parallel. Construct a segment:

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We may use the properties of parallel lines to show certain angle congruencies. As they are alternate interior angles, and using the reflexive property, we know

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Therefore, we know that the following triangles are congruent because of ASA Since these are congruent triangles, we may assume that Therefore, if it is a parallelogram, then the opposite sides are equal.

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Prove: If the opposite sides are equal, then it is a parallelogram. Given: Construct segment

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Using the reflexive property, we can say Therefore, using SSS we know

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As the triangles are congruent, we know that corresponding angles are congruent. Therefore, If the alternate interior angles are congruent, then segments

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Therefore, if the opposite segments are equal, then it is a parallelogram.

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Therefore… It is a parallelogram, if and only if the opposite sides are equal.

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the second proof...

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Prove: If it is a parallelogram, then the diagonals bisect each other. Given parallelogram ABCD, Using the property proven in the previous proof,

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Construct segment BD This forms two congruent triangles, because of SSS, as the following segments are congruent : This implies corresponding angles are congruent

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Construct segment AC This also forms two congruent triangles Because of SSS, as the following sides are congruent This implies that corresponding angles are congruent

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Look at both diagonals and the created triangles With both diagonals displayed, we may conclude that we have two sets of congruent triangles, based upon ASA. For example, since

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Since we have congruent triangles We can then say that Therefore, the diagonals of the parallelogram bisect each other since the segments are congruent.

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Prove:If the diagonals bisect each other, then it is a parallelogram. Since the diagonals bisect each other, we know certain segments are congruent. We may also say that vertical angles are congruent

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Using SAS, we may say there are two sets of congruent triangles Therefore, we may say Therefore, since the diagonals bisect each other, then the opposite sides are congruent. From the previous proof, we know that it is a parallelogram

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therefore, It is a parallelogram, if and only if the diagonals bisect each other.

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the third proof…

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Prove: If it is a rectangle, then it is a parallelogram and the diagonals are equal. By definition, a rectangle has adjacent sides that are perpendicular. Since segment BC and segment AD are both perpendicular to segment AB, we may conclude that segment BC and segment AD are parallel. The same may be concluded about segments AB and DC.

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Since opposite sides are parallel, we may conclude that the rectangle is also a parallelogram. Since it is a parallelogram, then we know that opposite sides are congruent.

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Construct Segments AC and BD Since the rectangle is also a parallelogram, then we may say, With the constructed segments, the congruent sides, and the right angles, we have 4 congruent triangles (by SAS):

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With 4 congruent triangles, we know corresponding sides are congruent. Therefore, we may state that:

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Hence, if it is a rectangle, then it is a parallelogram and the diagonals are equal.

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Prove: If it is a parallelogram and the diagonals are equal, then it is a rectangle. Given: Opposite sides of a parallelogram are both parallel and congruent. Given: The diagonals are equal. Using SSS, we know the 4 following triangles are congruent:

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If the four triangles are congruent, then corresponding angles are congruent. The sum of the angles in the parallelogram (or any quadrilateral for that matter) must be 360 degrees, and all of the interior angles must be congruent.

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If the interior angles are 90 degrees, then we can say that the adjacent sides are perpendicular. Therefore, it is a rectangle.

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THEREFORE… It is a rectangle, if and only if it is a parallelogram and the diagonals are equal.

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Parallelograms, Trapezoids, Rectangles, Rhombi, Kites, and Squares….Oh MY!

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