1. Introduction
What does it mean to be opposite? What does it mean to be consecutive? Think about a rectangular room. If you put your back against one corner of that room and looked directly across the room, you would be looking at the opposite corner. If you looked to your right, that corner would be a consecutive corner. If you looked to your left, that corner would also be a consecutive corner.
1.10.1: Proving Properties of Parallelograms

2. Introduction, continued
The walls of the room could also be described similarly. If you were to stand with your back at the center of one wall, the wall straight across from you would be the opposite wall. The walls next to you would be consecutive walls. There are two pairs of opposite walls in a rectangular room, and there are two pairs of opposite angles. Before looking at the properties of parallelograms, it is important to understand what the terms opposite and consecutive mean.
1.10.1: Proving Properties of Parallelograms

3. Key Concepts 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
1.10.1: Proving Properties of Parallelograms

4. Key Concepts, continued
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
1.10.1: Proving Properties of Parallelograms

5. Key Concepts, continued
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
1.10.1: Proving Properties of Parallelograms

6. Key Concepts, continued
If a polygon is a parallelogram, there are five theorems associated with it.
In a parallelogram, both pairs of opposite sides are congruent.
1.10.1: Proving Properties of Parallelograms

7. Key Concepts, continued
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

8. Key Concepts, continued
Parallelograms also have two pairs of opposite angles that are congruent.
1.10.1: Proving Properties of Parallelograms

9. Key Concepts, continued
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

10. Key Concepts, 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.
1.10.1: Proving Properties of Parallelograms

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

12. Key Concepts, continued
The diagonals of a parallelogram have a relationship. They bisect each other.
1.10.1: Proving Properties of Parallelograms

13. Key Concepts, continued
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

14. Key Concepts, continued
Notice that each diagonal divides the parallelogram into two triangles. Those two triangles are congruent.
1.10.1: Proving Properties of Parallelograms

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

16. Common Errors/Misconceptions
thinking that all angles in a parallelogram are congruent even if the parallelogram isn’t a rectangle or square
misidentifying opposite pairs of sides
misidentifying opposite pairs of angles and consecutive angles
1.10.1: Proving Properties of Parallelograms

17. Guided Practice Example 1
Quadrilateral ABCD has the following vertices: A (–4, 4), B (2, 8), C (3, 4), and D (–3, 0). Determine whether the quadrilateral is a parallelogram. Verify your answer using slope and distance to prove or disprove that opposite sides are parallel and opposite sides are congruent.
1.10.1: Proving Properties of Parallelograms

18. Guided Practice: Example 1, continued Graph the figure.
1.10.1: Proving Properties of Parallelograms

19. Guided Practice: Example 1, continued
Determine whether opposite pairs of lines are parallel.
Calculate the slope of each line segment.
is opposite ; is opposite
1.10.1: Proving Properties of Parallelograms

20. Guided Practice: Example 1, continued
Calculating the slopes, we can see that the opposite sides are parallel because the slopes of the opposite sides are equal. By the definition of a parallelogram, quadrilateral ABCD is a parallelogram.
1.10.1: Proving Properties of Parallelograms

21. Guided Practice: Example 1, continued
Verify that the opposite sides are congruent.
Calculate the distance of each segment using the distance formula.
1.10.1: Proving Properties of Parallelograms

22. Guided Practice: Example 1, continued
1.10.1: Proving Properties of Parallelograms

23. Guided Practice: Example 1, continued
1.10.1: Proving Properties of Parallelograms

24. ✔ Guided Practice: Example 1, continued
From the distance formula, we can see that opposite sides are congruent. Because of the definition of congruence and since AB = DC and BC = AD, then
and ✔
1.10.1: Proving Properties of Parallelograms

25. Guided Practice: Example 1, continued
1.10.1: Proving Properties of Parallelograms

26. Guided Practice Example 2
Use the parallelogram from Example 1 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

27. Guided Practice: Example 2, continued
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

28. Guided Practice: Example 2, continued
1.10.1: Proving Properties of Parallelograms

29. Guided Practice: Example 2, continued Prove .
We have proven that one pair of opposite angles in a parallelogram is congruent.
and
Given
Alternate Interior Angles Theorem
Vertical Angles Theorem
Transitive Property
1.10.1: Proving Properties of Parallelograms

30. Guided Practice: Example 2, continued Prove .
We have proven that both pairs of opposite angles in a parallelogram are congruent.
and
Given
Alternate Interior Angles Theorem
Vertical Angles Theorem
Transitive Property
1.10.1: Proving Properties of Parallelograms

31. Guided Practice: Example 2, continued
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.
1.10.1: Proving Properties of Parallelograms

32. ✔ Guided Practice: Example 2, continued
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

33. Guided Practice: Example 2, continued
1.10.1: Proving Properties of Parallelograms