1. Rectangles, Squares and Rhombuses

2. Rectangles
A rectangle is a parallelogram with four interior right angles.
Here are the properties of a rectangle:
Properties
All the interior angles are right angles.
Diagonals are equal.
Diagonals bisect each other into four equal parts.
All properties of a parallelogram A B C D
AC = BD
Abbreviation
Property of rectangle

3. (property of rectangle)
In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at O. A B C D a O b
30
∵ ABCD is a rectangle.
∴ BCD = 90
∴ OCD
= 90 – 30
= 60
∵ OC = OD
∴ a = OCD
(property of rectangle)
= BCD – BCA
(property of rectangle)
(base s, isos. △)

4. In the figure, ABCD is a rectangle
In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at O. A B C D a O b
30
Consider △OCD.
∴ a + b + OCD = 180
60 + b + 60 = 180
b = 60
( sum of △)

5. Follow-up question
In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at E. Find the unknowns.
b cm A B C D E
a cm
15 cm
8.5 cm
Solution
∵ CE = BE
∴ a = 8.5
∵ DE = BE = 8.5 cm
∴ BD = 2 × 8.5 cm
= 17 cm
(property of rectangle)
(property of rectangle)

6. Follow-up question (cont’d)
In the figure, ABCD is a rectangle. The diagonals AC and BD intersect at E. Find the unknowns.
b cm A B C D E
a cm
15 cm
8.5 cm
Solution
Consider △BCD.
∵ BCD = 90 ∴
(property of rectangle) 2 - = BC BD CD
(Pyth. theorem) 15 17 2 - = b 64 = 8 =

7. Squares
A square is a parallelogram with four equal interior right angles and four equal sides.
Here are the properties of a square:
Properties
All sides are equal.
Diagonals are perpendicular to each other.
Angle between each diagonal and a side is 45.
All properties of a rectangle
45
45°
Property of square
Abbreviation

8. In the figure, ABCD is a square. ∵ AC is a diagonal.
60 a b
In the figure, ABCD is a square.
∵ AC is a diagonal.
∴ BAO = 45 (property of square)
Consider △ABO.
BAO + ABO + AOB = 180 ( sum of △)
45 + 60 + a = 180
a = 75
ABC = 90 (property of square)
60 + b = 90
b = 30

9. Follow-up question A B C D
4 cm
a cm b c O
d cm
In the figure, ABCD is a square. The diagonals AC and BD intersect at O. Find the unknowns.
Solution
∵ ABCD is a square.
∴ CD = AD
a = 4
(property of square)
∵ AC is a diagonal.
∴ b = 45
(property of square)

10. Follow-up question (cont’d)A B C D
4 cm
a cm b c O
d cm
In the figure, ABCD is a square. The diagonals AC and BD intersect at O. Find the unknowns.
Solution
∵ The diagonals of a square are perpendicular to each other.
∴ c = 90
(property of square)
Consider △BCD.
∵ ∠BCD = 90
(property of square) 2 + = CD BC BD
(Pyth. theorem) 4 2 + = cm 32 = cm 2 4
(or
cm)

11. Follow-up question (cont’d)A B C D
4 cm
a cm b c O
d cm
In the figure, ABCD is a square. The diagonals AC and BD intersect at O. Find the unknowns.
Solution 2 1 = BD
(property of square) ∴ OD d 2
(or ) 32 =

12. Rhombuses A rhombus is a parallelogram with four equal sides.
Here are the properties of a rhombus:
Properties
All sides are equal.
Diagonals are perpendicular to each other.
Interior angles are bisected by the diagonals.
All properties of a parallelogram
Property of rhombus
Abbreviation

13. ∵ BC = AD (property of rhombus) ∴ a = 3
In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O.
3 cm b c O
25 d
a cm
∵ BC = AD (property of rhombus)
∴ a = 3
∵ ABO = ADO (property of rhombus)
∴ c = 25
Consider △ABO.
AOB = 90 (property of rhombus)
b + c + AOB = 180 ( sum of △)
b + 25 + 90 = 180
b = 65

14. d = b (property of rhombus) = 65A B C D
In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O.
3 cm b c O
25 d
a cm
d = b (property of rhombus)
= 65

15. Follow-up question
In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O. Find the unknowns.
(p – 2) cm A B C D O
(10 – p ) cm 2q 3q
Solution
AD = CD (property of rhombus)
10 – p = p – 2
2p = 12
p = 6

16. Follow-up question (cont’d)
In the figure, ABCD is a rhombus. The diagonal AC and BD intersect at O. Find the unknowns.
(p – 2) cm A B C D O
(10 – p ) cm 2q 3q
Solution
Consider △AOD.
AOD = 90 (property of rhombus)
AOD + 3q + 2q = 180 ( sum of △)
5q + 90 = 180
5q = 90
q = 18