Theorems Dealing with Rectangles, Rhombuses, Squares Topic Index | Geometry Index | Regents Exam Prep Center

Definition:   A rectangle is a parallelogram with four right angles.

 *Rectangle I have all of the properties of the parallelogram PLUS - 4 right angles - diagonals congruent

Using the definition, the properties of the rectangle
can be "proven" true and become theorems.

 When dealing with a rectangle, the definition and theorems are stated as ...

 1.  A rectangle is a parallelogram with four right angles. While the definition states "parallelogram", it is sufficient to say:  "A quadrilateral is a rectangle if and only if it has four right angles.", since any quadrilateral with four right angles is a parallelogram. 2.  If a parallelogram has one right angle it is a rectangle. 3.  A parallelogram is a rectangle if and only if its diagonals are congruent.
 Construction workers use the fact that the diagonals of a rectangle are congruent (equal) when attempting to build a "square" footing for a building, a patio, a fenced area, a table top, etc.  Workers measure the diagonals.  When the diagonals of the project are equal the building line is said to be square.

Definition:  A rhombus is a parallelogram with four congruent sides.

 *Rhombus I have all of the properties of the parallelogram PLUS - 4 congruent sides - diagonals bisect angles - diagonals perpendicular

Using the definition, the properties of the rhombus
can be "proven" true and become theorems.

 When dealing with a rhombus, the definition and theorems are stated as ...

 1. A rhombus is a parallelogram with four congruent sides. While the definition states "parallelogram", it is sufficient to say: "A quadrilateral is a rhombus if and only if it has four congruent sides.", since any quadrilateral with four congruent sides is a parallelogram. 2. If a parallelogram has two consecutive sides congruent, it is a rhombus. 3. A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. 4. A parallelogram is a rhombus if and only if the diagonals are perpendicular. (Proof of theorem appears further down page.)

Definition:  A square is a parallelogram with four congruent sides and four right angles.

 *Square Hey, look at me! I have all of the properties of the parallelogram AND the rectangle AND the rhombus. I have it all!

Using the definition, the properties of the rhombus
can be "proven" true and become theorems.

 When dealing with a square, the definition is stated as ...

 A square is a parallelogram with four congruent sides and four right angles. This definition may also be stated as A quadrilateral is a square if and only if it is a rhombus and a rectangle.

Proof of Theorem:  If a parallelogram is a rhombus, then the diagonals are perpendicular.
(Remember:  when attempting to prove a theorem to be true,
you cannot use the theorem as a reason in your proof.)

 STATEMENTS REASONS 1 1 Given 2 Draw segment from  A to C 2 Two points determine exactly one line. 3 3 A rhombus is a parallelogram with four congruent sides. 4 4 A rhombus is a parallelogram with four congruent sides. 5 5 If a quadrilateral is a parallelogram, the diagonals bisect each other. 6 6 A bisector of a segment intersects the segment at its midpoint. 7 7 Midpoint of a line segment is the point on that line segment that divides the segment two congruent segments. 8 8 Reflexive Property - A quantity is congruent to itself. 9 9 SSS - If three sides of one triangle are congruent to three sides of a second triangle, the triangles are congruent. 10 10 CPCTC - Corresponding parts of congruent triangles are congruent. 11 11 If 2 congruent angles form a linear pair, they are right angles. 12 12 Perpendicular lines meet to form right angles.

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