Introduction There are many kinds of quadrilaterals. Some quadrilaterals are parallelograms; some are not. For example, trapezoids and kites are special quadrilaterals, but they are not parallelograms. Some parallelograms are known as special parallelograms. What makes a parallelogram a more specialized parallelogram? Rectangles, rhombuses, and squares are all special parallelograms with special properties. They have all the same characteristics that parallelograms have, plus more. 1.10.2: Proving Properties of Special Quadrilaterals

Key Concepts 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

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

Key Concepts, continued 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. 1.10.2: Proving Properties of Special Quadrilaterals

Key Concepts, continued 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

Key Concepts, continued The diagonals of a rhombus also form four right angles where they intersect. 1.10.2: Proving Properties of Special Quadrilaterals

Key Concepts, 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

Key Concepts, continued 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

Key Concepts, continued Properties of Squares 1.10.2: Proving Properties of Special Quadrilaterals

Key Concepts, continued 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

Key Concepts, continued Properties of Trapezoids and are the legs. and are the bases. 1.10.2: Proving Properties of Special Quadrilaterals

Key Concepts, continued 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

Key Concepts, continued Properties of Isosceles Trapezoids and are the legs. and are the bases. 1.10.2: Proving Properties of Special Quadrilaterals

Key Concepts, continued 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

Key Concepts, continued Properties of Kites 1.10.2: Proving Properties of Special Quadrilaterals

Key Concepts, continued Quadrilaterals can be grouped according to their properties. This kind of grouping is called a hierarchy. In the hierarchy of quadrilaterals on the following slide, you can see that all quadrilaterals are polygons but that not all polygons are quadrilaterals. The arrows connecting the types of quadrilaterals indicate a special version of the category above each quadrilateral type. For example, parallelograms are special quadrilaterals. Rectangles and rhombuses are special parallelograms, and squares have all the properties of rectangles and rhombuses. 1.10.2: Proving Properties of Special Quadrilaterals

Hierarchy of Quadrilaterals 1.10.2: Proving Properties of Special Quadrilaterals

Common Errors/Misconceptions assuming both pairs of opposite sides are parallel after determining that one pair of opposite sides is parallel mistakenly classifying a rhombus as a square since all the sides are congruent confusing the properties among the special quadrilaterals not understanding that rectangles have the same properties as a parallelogram plus additional properties not understanding the hierarchy of quadrilaterals 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice Example 1 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

Guided Practice: Example 1, continued Graph the quadrilateral. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 1, continued Calculate the slopes of the sides to determine if opposite sides are parallel. If opposite sides are parallel, the quadrilateral is a parallelogram. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 1, continued The first pair of opposite sides is parallel: The second pair of opposite sides is parallel: Therefore, the quadrilateral is a parallelogram. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 1, continued Examine the slopes of the consecutive sides to determine if they intersect at right angles. 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

Guided Practice: Example 1, continued 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

Guided Practice: Example 1, continued You could also determine if the diagonals are congruent by calculating the length of each diagonal using the distance formula, If the diagonals are congruent, then the parallelogram is a rectangle or square. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 1, continued The diagonals are congruent: . The parallelogram is a rectangle. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 1, continued Calculate the lengths 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

Guided Practice: Example 1, continued 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 1, continued Opposite sides are congruent, which is consistent with a parallelogram, but all sides are not congruent. 1.10.2: Proving Properties of Special Quadrilaterals

✔ Guided Practice: Example 1, continued Summarize your 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

Guided Practice: Example 1, continued http://www.walch.com/ei/00179 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice Example 2 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

Guided Practice: Example 2, continued Graph the quadrilateral. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued Calculate the slopes of the sides to determine if opposite sides are parallel. If opposite sides are parallel, the quadrilateral is a parallelogram. is opposite . is opposite . 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued The opposite sides are parallel: and . . Therefore, the quadrilateral is a parallelogram. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued Examine the slopes of consecutive sides to determine if the sides are perpendicular. If the slopes of consecutive sides are opposite reciprocals of each other, then the sides intersect at right angles. If the sides intersect at right angles, then the parallelogram is a rhombus or square. Let’s use consecutive sides and . 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued The slopes are not opposite reciprocals, so the parallelogram is not a rectangle or a square. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued Calculate the lengths of the sides. In a rhombus, the sides are congruent. Use the distance formula to calculate the lengths of the sides. 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued The sides are all congruent. 1.10.2: Proving Properties of Special Quadrilaterals

✔ Guided Practice: Example 2, continued Summarize your findings. The quadrilateral has opposite sides that are parallel and all four sides are congruent, but the sides are not perpendicular. Therefore, the quadrilateral is a parallelogram and a rhombus, but not a square. ✔ 1.10.2: Proving Properties of Special Quadrilaterals

Guided Practice: Example 2, continued http://www.walch.com/ei/00180 1.10.2: Proving Properties of Special Quadrilaterals