6-4 & 6-5 Rectangles, Rhombi and Squares The student will be able to: 1. Recognize and apply properties of rectangles, rhombi, and squares. 2. Determine whether parallelograms are rectangles. 3. Determine whether quadrilaterals are rectangles, rhombi, or

Rectangle – a quadrilateral with four right angles. A rectangle is a special type of parallelogram, so all the properties of parallelograms apply. All four angles are right angles. Opposite sides are parallel and congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. If a parallelogram is a rectangle, then its diagonals are congruent. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. In rectangle GHJK,

A rectangle has 4 right angles. Quadrilateral ABCD is a rectangle. and Find x, y, and the measure of all angles and the diagonals. 2y + 12 A rectangle has 4 right angles. Since a rectangle is a parallelogram, All properties of a parallelogram apply. 3x - 5 2x + 3 Opposite sides are parallel, so the diagonal is a transversal. y + 13 What do we know about the angles of a rectangle? They all equal 90°. What do we know about interior angles of parallel lines cut by a transversal? Alternate interior angles are congruent. What do we know about diagonals of a rectangle? They are congruent.

Quadrilateral ABCD is a rectangle Quadrilateral ABCD is a rectangle. and Find x, y, and the measure of all listed angles and the diagonals. 2y + 12 y + 13 2y + 12 + y + 13 = 90 3y + 25 = 90 -25 3x - 5 2x + 3 3y = 65 y = 21 ⅔ y + 13 2x+ 3 = 3x – 5 = 21 ⅔ + 13 +5 -2x = 2(21 ⅔) + 12 = 43 ⅓ + 12 = 34 ⅔ 8 = x = 55 ⅓ = 2{2(8) + 3} = 2{3(8) - 5} = 2(16 + 3) = 2(24 - 5) = 2(19) = 2(19) = 38 = 38

Rhombus – a parallelogram with all four sides congruent. A rhombus has all the properties of a parallelogram along with the following: If a parallelogram is a rhombus, then its diagonals are perpendicular. A parallelogram with diagonals that are perpendicular is a rhombus. If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. A parallelogram with diagonals that bisect a pair of opposite angles is a rhombus.

Example 2: The diagonals of rhombus FGHJ intersect at K. A. If and , find . B. If and find y. What do we know about all four of the angles with K as the vertex? They all equal 90°. If we know the length of and how can we find ? All sides of a rhombus are congruent, so What do we know about the diagonals of a rhombus? a2 + b2 = c2 They bisect each angle, so 52 + b2 = 132 25 + b2 = 169 +5 -6y -25 12= 3y b2 = 144 4 = y b = 12

Square – a parallelogram with four congruent sides and four right angles. If a parallelogram with four right angles is a rectangle and a parallelogram with four congruent sides is a rhombus, then a parallelogram with four right angles and four congruent sides is a square.

Example 3: Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain. If the diagonals are perpendicular, then ABCD is either a rhombus or a square. If the diagonals are congruent and perpendicular, then ABCD is a square. Use the distance formula to compare the lengths of the diagonals. Use slope to determine whether the diagonals are perpendicular. What do you notice about the slopes? Opposite signs & fractions are flipped. What type of parallelogram is it? Rectangle, Rhombus & Square

When determining whether a quadrilateral is a rhombus, rectangle, or square. You must: * Determine if the quadrilateral is a parallelogram. Find the lengths of 2 pair of opposite sides (4 lengths) or Find the slopes of 2 pair of opposite sides (4 slopes) or Find the lengths and slopes of 1 pair of opposite sides (2 lengths and 2 slopes) * Find the lengths of the diagonals. The same length tells us it is a rectangle AND * Find the slopes of the diagonals or * Find the slopes of 2 consecutive sides Perpendicular slopes (opposite sign and flipped fraction)tell us it is a rhombus. * If a parallelogram is a rhombus and a rectangle, it is a square.