Proving Properties of Parallelograms Adapted from Walch Education

Polygons 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. Polygons Convex polygon 1.10.1: Proving Properties of Parallelograms

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. Polygons, continued Concave polygon 1.10.1: Proving Properties of Parallelograms

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 1.10.1: Proving Properties of Parallelograms

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. Parallelogram, continued 1.10.1: Proving Properties of Parallelograms

If a quadrilateral is a parallelogram, opposite sides are congruent. 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 B D C 1.10.1: Proving Properties of Parallelograms

If a quadrilateral is a parallelogram, opposite angles are congruent. 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 B D C 1.10.1: Proving Properties of Parallelograms

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 180º. The diagonals of a parallelogram bisect each other. Parallelogram, continued 1.10.1: Proving Properties of Parallelograms

Theorem If a quadrilateral is a parallelogram, then consecutive angles are supplementary. A B D C 1.10.1: Proving Properties of Parallelograms

The diagonals of a parallelogram bisect each other. 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 B P D C 1.10.1: Proving Properties of Parallelograms

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

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

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

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

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

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

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

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