Theorems Dealing with Rectangles, Rhombuses, Squares
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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.