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Proving Properties of Parallelograms Adapted from Walch Education

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Polygons A quadrilateral is a polygon with four sides. A convex polygon is a polygon with no interior angle greater than 180 and all diagonals lie inside the polygon. A diagonal of a polygon is a line that connects nonconsecutive vertices. 1.10.1: Proving Properties of Parallelograms 2 Convex polygon

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Polygons, continued Convex polygons are contrasted with concave polygons. A concave polygon is a polygon with at least one interior angle greater than 180 and at least one diagonal that does not lie entirely inside the polygon. 1.10.1: Proving Properties of Parallelograms 3 Concave polygon

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Parallelogram A parallelogram is a special type of quadrilateral with two pairs of opposite sides that are parallel. By definition, if a quadrilateral has two pairs of opposite sides that are parallel, then the quadrilateral is a parallelogram. Parallelograms are denoted by the symbol. 1.10.1: Proving Properties of Parallelograms 4 Parallelogram

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Parallelogram, continued If a polygon is a parallelogram, there are five theorems associated with it. In a parallelogram, both pairs of opposite sides are congruent. Parallelograms also have two pairs of opposite angles that are congruent. 1.10.1: Proving Properties of Parallelograms 5

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6 Theorem If a quadrilateral is a parallelogram, opposite sides are congruent. The converse is also true. If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. A D C B

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1.10.1: Proving Properties of Parallelograms 7 Theorem If a quadrilateral is a parallelogram, opposite angles are congruent. The converse is also true. If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. A D C B

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Parallelogram, continued Consecutive angles are angles that lie on the same side of a figure. In a parallelogram, consecutive angles are supplementary; that is, they sum to 180. The diagonals of a parallelogram bisect each other. 1.10.1: Proving Properties of Parallelograms 8

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9 Theorem If a quadrilateral is a parallelogram, then consecutive angles are supplementary. A D C B

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1.10.1: Proving Properties of Parallelograms 10 Theorem The diagonals of a parallelogram bisect each other. The converse is also true. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. A D C B P

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1.10.1: Proving Properties of Parallelograms 11 Theorem The diagonal of a parallelogram forms two congruent triangles. A D C B

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Practice Use the parallelogram to verify that the opposite angles in a parallelogram are congruent and consecutive angles are supplementary given that and. 1.10.1: Proving Properties of Parallelograms 12

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Step 1 Extend the lines in the parallelogram to show two pairs of intersecting lines and label the angles with numbers. 1.10.1: Proving Properties of Parallelograms 13

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Step 2 Prove 1.10.1: Proving Properties of Parallelograms 14 andGiven Alternate Interior Angles Theorem Vertical Angles Theorem Alternate Interior Angles Theorem Transitive Property

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Step 3 Prove 1.10.1: Proving Properties of Parallelograms 15 andGiven Alternate Interior Angles Theorem Vertical Angles Theorem Alternate Interior Angles Theorem Transitive Property

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Step 4 Prove that consecutive angles of a parallelogram are supplementary. 1.10.1: Proving Properties of Parallelograms 16 andGiven 4 and 14 are supplementary. Same-Side Interior Angles Theorem 14 and 9 are supplementary. Same-Side Interior Angles Theorem 9 and 7 are supplementary. Same-Side Interior Angles Theorem 7 and 4 are supplementary. Same-Side Interior Angles Theorem

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We have proven consecutive angles in a parallelogram are supplementary using the Same-Side Interior Angles Theorem of a set of parallel lines intersected by a transversal. 1.10.1: Proving Properties of Parallelograms 17

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THANKS FOR WATCHING! Ms. Dambreville