Section 5 4 Special Parallelograms RECTANGLE A rectangle

RECTANGLE • A rectangle is a quadrilateral with four right angles. Therefore, every rectangle

Rectangle • Why? D C A B • Both pairs of opposite angles are

Rhombus • A rhombus is a quadrilateral with four congruent sides. Therefore, every rhombus

rhombus • Why? D A C B • Both pairs of opposite sides are

Square • A square is a quadrilateral with four right angles and four congruent

Square • Why? D C A B • Both pairs of opposite angles (and

Conclusion: Since rectangles, rhombuses and squares are all parallelograms, they have all the properties

Theorem 5 -12 • The diagonals of a rectangle are congruent. D C A

Theorem 5 -13 • The diagonals of a rhombus are perpendicular. D A C

Theorem 5 -14 • Each diagonal of a rhombus bisects two angles of the

Theorem 5 -15 • The midpoint of the hypotenuse of a right triangle is

Theorem 5 -16 • If an angle of a parallelogram is a right angle,

Theorem 5 -17 • If two consecutive sides of a parallelogram are congruent, then

Theorem 5 -17 Thus, ABCD is a Rhombus D A C B

Always, Sometimes or Never? Always a rhombus? • A square is _____ • The

Always, Sometimes or Never? • The diagonals of a rhombus are Sometimes ______ congruent?

Always, Sometimes or Never? • The diagonals of a rhombus Always ______ bisect each

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Section 5 -4 Special Parallelograms

RECTANGLE • A rectangle is a quadrilateral with four right angles. Therefore, every rectangle is a parallelogram.

Rectangle • Why? D C A B • Both pairs of opposite angles are congruent.

Rhombus • A rhombus is a quadrilateral with four congruent sides. Therefore, every rhombus is a parallelogram.

rhombus • Why? D A C B • Both pairs of opposite sides are congruent.

Square • A square is a quadrilateral with four right angles and four congruent sides. Therefore, every square is a rectangle, a rhombus, and a parallelogram.

Square • Why? D C A B • Both pairs of opposite angles (and sides) are congruent.

Conclusion: Since rectangles, rhombuses and squares are all parallelograms, they have all the properties of a parallelogram.

Theorem 5 -12 • The diagonals of a rectangle are congruent. D C A B

Theorem 5 -13 • The diagonals of a rhombus are perpendicular. D A C B

Theorem 5 -14 • Each diagonal of a rhombus bisects two angles of the D rhombus. A C B

Theorem 5 -15 • The midpoint of the hypotenuse of a right triangle is equidistant from the vertices. A D B C

Theorem 5 -16 • If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle. D C Angle A is a Rt. Angle Thus, ABCD is a Rectangle! A B

Theorem 5 -17 • If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Theorem 5 -17 Thus, ABCD is a Rhombus D A C B

Always, Sometimes or Never? Always a rhombus? • A square is _____ • The diagonals of a parallelogram ______ Sometimes bisect the angles of the parallelogram? • A quadrilateral with one pair of sides congruent and one pair parallel is ______ Sometimes a parallelogram?

Always, Sometimes or Never? • The diagonals of a rhombus are Sometimes ______ congruent? • A rectangle Sometimes _____ has consecutive sides congruent? • A rectangle _____ Sometimes has perpendicular diagonals?

Always, Sometimes or Never? • The diagonals of a rhombus Always ______ bisect each other? • The diagonals of a parallelogram are _____ Sometimes perpendicular bisectors of each other?