PROOFS: RECTANGLES,
RHOMBUSES, & SQUARES
In this lesson you will:
Examine special ways to prove parallelograms
are rectangles, rhombuses, and squares.
Practice applying the characteristics and
theorems of parallelograms, rectangles,
rhombuses, and squares through writing proofs.
Given a quadrilateral, what additional information
would prove that the quadrilateral is a rectangle? A
rhombus? A square?
Rectangle:
 One right angle. (consecutive sides have reciprocal slopes)
 AND Congruent diagonals. (distance formula)
Rhombus:
 Four congruent sides. (distance formula)
Square:
 Both a rectangle and a rhombus.
Given a quadrilateral, what additional information
would prove that the quadrilateral is a rectangle? A
rhombus? A square?
Rectangle:
Rhombus:
Square:
 Both a rectangle and a rhombus.
Given a parallelogram, what additional
information would prove that the quadrilateral is a
rectangle? A rhombus? A square?
Rectangle:
 One right angle. (consecutive sides have reciprocal slopes)
 OR Congruent diagonals. (distance formula)
Rhombus:
 Two consecutive congruent sides. (distance formula)
Square:
 Both a rectangle and a rhombus.
Given a parallelogram, what additional
information would prove that the quadrilateral is a
rectangle? A rhombus? A square?
Rectangle:
Rhombus:
Square:
 Both a rectangle and a rhombus.
Example 1: What additional information do you need to
prove that PQRS is a square?
Q
P
R
S
Both pairs of opposite sides
are congruent therefore PQRS
is a parallelogram.
Since all four sides are congruent,
PQRS is also a rhombus.
We need one right angle or congruent
diagonals to show PQRS is
also a rectangle.
Q
Example 2: Given: Parallelogram PQRS
∆QTR ≅ ∆STR
Prove: PQRS is a rhombus
T
P
Statements
R
S
Reasons
1. Parallelogram PQRS,
∆QTR ≅ ∆STP
1. Given
2. QR ≅ SR
2. CPCTC
5.PQRS is a rhombus
5. If a parallelogram has two congruent
consecutive sides, then it is a rhombus.
Example 3: Given: Rectangle PQRS.
Prove: ∆PQT ≅ ∆RST
Q
R
T
P
Statements
S
Reasons
1. Rectangle PQRS
1. Given
2. QT ≅ TS; PT ≅ TR
2. If a quadrilateral is a parallelogram, then
its diagonals bisect each other.
3. <PTQ ≅ <RTS
3. Vertical Angles are Congruent
4. ∆PQT ≅ ∆RST
4. SAS
Example 4: It is given that ABCD is a parallelogram.
Decide whether it is a rectangle, square, rhombus, or
none of these. Justify your answer.
A(7, -1), B(3, 6), C(-1, -1), D(3, -8)
B
CB =
2
2
(3 − −1 ) + (6 − −1 )
CB =
42 +72
CB = 16+49
C
A
D
CB = 65
BA =
2
2
(3 − 7 ) + (6 − −1 )
BA =
(−4)2 +72
BA = 16+49
BA = 65
Since parallelogram ABCD has two consecutive sides congruent,
ABCD is a rhombus.
Example 5: It is given that ABCD is a parallelogram.
Decide whether it is a rectangle, square, rhombus, or
none of these. Justify your answer.
A(1, 1), B(-2, 4), C(-5, 1), D(-2, -2)
B
C
A
D
CB =
2
(−5 − −2 ) + (1 − 4)2
BA =
2
(−2 − 1 ) + (4 − 1)2
CB =
(−3)2 + (−3)2
BA =
(−3)2 + (3)2
CB = 9 + 9
BA = 9 + 9
CB = 18
BA = 18
1−4
−5 − (−2)
−3
Slope CB =
−3
4−1
−2 − 1
3
Slope BA =
−3
Slope CB =
Slope BA =
Slope CB = 1
Slope BA = -1
Since parallelogram ABCD has two consecutive congruent sides,
ABCD is a rhombus. Since parallelogram ABCD has one right angle,
ABCD is a rectangle. Since parallelogram ABCD is both a rhombus
and a rectangle, ABCD is a square.
PROOFS: RECTANGLES,
RHOMBUSES, & SQUARES
In this lesson you learned:
 How to prove a quadrilateral is a:
 Rectangle: Prove it has one right angle and
congruent diagonals.
 Rhombus: Prove it has four congruent sides.
 Square: Prove it is both a rectangle and a rhombus.
 How to prove a parallelogram is a:
 Rectangle: Prove it has one right angle or congruent
diagonals.
 Rhombus: Prove it has two congruent consecutive
sides.
 Square: Prove it is both a rectangle and a rhombus.