1. Proving Properties of Parallelograms
Adapted from Walch Education

2. 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

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

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
1.10.1: Proving Properties of Parallelograms

5. 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

6. 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

7. 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

8. 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

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

10. 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

11. 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

12. 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

13. 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

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

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

16. 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

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.
1.10.1: Proving Properties of Parallelograms

18. Thanks for Watching!
Ms. Dambreville