1. Proving Properties of Special Quadrilaterals
Adapted from Walch Education

2. Rectangle A rectangle has four sides and four right angles.
A rectangle is a parallelogram, so opposite sides are parallel, opposite angles are congruent, and consecutive angles are supplementary.
The diagonals of a rectangle bisect each other and are also congruent.
1.10.2: Proving Properties of Special Quadrilaterals

3. Rectangle
Theorem
If a parallelogram is a rectangle, then the diagonals are congruent.
1.10.2: Proving Properties of Special Quadrilaterals

4. Rhombus
A rhombus is a special parallelogram with all four sides congruent.
Since a rhombus is a parallelogram, opposite sides are parallel, opposite angles are congruent, and consecutive angles are supplementary.
The diagonals bisect each other; additionally, they also bisect the opposite pairs of angles within the rhombus.
The diagonals of a rhombus also form four right angles where they intersect.
1.10.2: Proving Properties of Special Quadrilaterals

5. Rhombus
Theorem
If a parallelogram is a rhombus, the diagonals of the rhombus bisect the opposite pairs of angles.
1.10.2: Proving Properties of Special Quadrilaterals

6. Rhombus, continued Theorem
If a parallelogram is a rhombus, the diagonals are perpendicular.
The converse is also true. If the diagonals of a parallelogram intersect at a right angle, then the parallelogram is a rhombus.
1.10.2: Proving Properties of Special Quadrilaterals

7. Square A square has all the properties of a rectangle and a rhombus.
Squares have four congruent sides and four right angles.
The diagonals of a square bisect each other, are congruent, and bisect opposite pairs of angles.
The diagonals are also perpendicular.
1.10.2: Proving Properties of Special Quadrilaterals

8. Square Properties of Squares
1.10.2: Proving Properties of Special Quadrilaterals

9. Trapezoid
Trapezoids are quadrilaterals with exactly one pair of opposite parallel lines.
Trapezoids are not parallelograms because they do not have two pairs of opposite lines that are parallel.
The lines in a trapezoid that are parallel are called the bases, and the lines that are not parallel are called the legs.
1.10.2: Proving Properties of Special Quadrilaterals

10. Trapezoid Properties of Trapezoids and are the legs.
and are the bases.
1.10.2: Proving Properties of Special Quadrilaterals

11. Isosceles Trapezoid
Isosceles trapezoids have one pair of opposite parallel lines. The legs are congruent.
Since the legs are congruent, both pairs of base angles are also congruent, similar to the legs and base angles in an isosceles triangle.
The diagonals of an isosceles trapezoid are congruent.
1.10.2: Proving Properties of Special Quadrilaterals

12. Isosceles Trapezoid Properties of Isosceles Trapezoids
and are the legs.
and are the bases.
1.10.2: Proving Properties of Special Quadrilaterals

13. Kite
A kite is a quadrilateral with two distinct pairs of congruent sides that are adjacent.
Kites are not parallelograms because opposite sides are not parallel.
The diagonals of a kite are perpendicular.
1.10.2: Proving Properties of Special Quadrilaterals

14. Kite Properties of Kites
1.10.2: Proving Properties of Special Quadrilaterals

15. Hierarchy of Quadrilaterals
1.10.2: Proving Properties of Special Quadrilaterals

16. Practice
Quadrilateral ABCD has vertices A (–6, 8), B (2, 2), C (–1, –2), and D (–9, 4). Using slope, distance, and/or midpoints, classify as a rectangle, rhombus, square, trapezoid, isosceles trapezoid, or kite.
1.10.2: Proving Properties of Special Quadrilaterals

17. Graph the Quadrilateral
1.10.2: Proving Properties of Special Quadrilaterals

18. Calculate the slopes of the sides
If opposite sides are parallel, the quadrilateral is a parallelogram.
The first pair of opposite sides is parallel:
1.10.2: Proving Properties of Special Quadrilaterals

19. Calculate the slopes of the sides
The second pair of opposite sides is parallel:
Therefore, the quadrilateral is a parallelogram.
1.10.2: Proving Properties of Special Quadrilaterals

20. Examine the slopes of the consecutive sides
If the slopes are opposite reciprocals, the lines are perpendicular and therefore form right angles. If there are four right angles, the quadrilateral is a rectangle or a square.
1.10.2: Proving Properties of Special Quadrilaterals

21. Examine the slopes of the consecutive sides
is the opposite reciprocal of
The slopes of the consecutive sides are perpendicular: and
There are four right angles at the vertices.
The parallelogram is a rectangle or a square.
1.10.2: Proving Properties of Special Quadrilaterals

22. Determining whether the diagonals are congruent
If the diagonals are congruent, then the parallelogram is a rectangle or square.
determine if the diagonals are congruent by calculating the length of each diagonal using the distance formula,
1.10.2: Proving Properties of Special Quadrilaterals

23. The parallelogram is a rectangle.
The diagonals are congruent:
1.10.2: Proving Properties of Special Quadrilaterals

24. Calculate the length of the sides
If all sides are congruent, the parallelogram is a rhombus or a square.
Since we established that the angles are right angles, the rectangle can be more precisely classified as a square if the sides are congruent.
If the sides are not congruent, the parallelogram is a rectangle.
Use the distance formula to calculate the lengths of the sides.
1.10.2: Proving Properties of Special Quadrilaterals

25. 1.10.2: Proving Properties of Special Quadrilaterals

26. Opposite sides are congruent, which is consistent with a parallelogram, but all sides are not congruent.
1.10.2: Proving Properties of Special Quadrilaterals

27. Summarizing the findings
The quadrilateral has opposite sides that are parallel and four right angles, but not four congruent sides. This makes the quadrilateral a parallelogram and a rectangle.
1.10.2: Proving Properties of Special Quadrilaterals

28. Try this one…
Quadrilateral ABCD has vertices A (0, 8), B (11, 1), C (0, –6), and D (–11, 1). Using slope, distance, and/or midpoints, classify as a rectangle, rhombus, square, trapezoid, isosceles trapezoid, or kite.
1.10.2: Proving Properties of Special Quadrilaterals

29. Ms. Dambreville
Thanks for watching!