Proving Properties of Parallelograms. Adapted from Walch Education
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1 Proving Properties of Parallelograms Adapted from Walch Education
2 A quadrilateral is a polygon with four sides. A convex polygon is a polygon with no interior angle greater than 180º and all diagonals lie inside the polygon. A diagonal of a polygon is a line that connects nonconsecutive vertices. Convex polygon Polygons 2
3 Convex polygons are contrasted with concave polygons. A concave polygon is a polygon with at least one interior angle greater than 180º and at least one diagonal that does not lie entirely inside the polygon. Concave polygon Polygons, continued 3
4 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. Parallelogram Parallelogram 4
5 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. Parallelogram, continued 5
6 Theorem If a quadrilateral is a parallelogram, opposite sides are congruent. A B D DC BC The converse is also true. If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. C 6
7 Theorem If a quadrilateral is a parallelogram, opposite angles are congruent. A B ÐC ÐD D The converse is also true. If the opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. C 7
8 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 180º. The diagonals of a parallelogram bisect each other. Parallelogram, continued 8
9 Theorem If a quadrilateral is a parallelogram, then consecutive angles are supplementary. A B D C mða + mðb = 180 mðb + mðc = 180 mðc + mðd = 180 mðd + mða = 180 9
10 Theorem The diagonals of a parallelogram bisect each other. A B P PC PD D C The converse is also true. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. 10
11 Theorem The diagonal of a parallelogram forms two congruent triangles. A B D C 11
12 Use the parallelogram to verify that the opposite angles in a parallelogram are congruent and consecutive angles are supplementary given that and. Practice 12
13 Extend the lines in the parallelogram to show two pairs of intersecting lines and label the angles with numbers. Step 1 13
14 Prove and Ð13 Ð16 Ð9 Ð9 Given Alternate Interior Angles Theorem Vertical Angles Theorem Alternate Interior Angles Theorem Transitive Property Step 2 14
15 Prove and Ð10 Ð11 Ð11@ Ð14 Ð14 Given Alternate Interior Angles Theorem Vertical Angles Theorem Alternate Interior Angles Theorem Transitive Property Step 3 15
16 Prove that consecutive angles of a parallelogram are supplementary. Step 4 and 4 and 14 are supplementary. 14 and 9 are supplementary. 9 and 7 are supplementary. 7 and 4 are supplementary. Given Same-Side Interior Angles Theorem Same-Side Interior Angles Theorem Same-Side Interior Angles Theorem Same-Side Interior Angles Theorem 16
17 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. 17
18 THANKS FOR WATCHING! Dr. Dambreville
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