Name: Memorandum
Gr 10
CAPS Reference
Topic
2.1
𝐺𝐺
Time:
Euclidean Geometry
Properties of Quadrilaterals
𝐽𝐽
2
1
Date:
𝑂𝑂
𝐻𝐻
60°
1
Statement
Reason
2.1.1 𝑂𝑂𝑂𝑂 = 4𝑐𝑐𝑐𝑐
𝐺𝐺𝐺𝐺 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝐻𝐻𝐻𝐻 𝑜𝑜𝑜𝑜 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
2.1.2 𝐼𝐼�1 = 60°
𝐺𝐺𝐺𝐺 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏 𝐼𝐼̂ 𝑜𝑜𝑜𝑜 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
2.1.3
In ∆𝐻𝐻𝐻𝐻𝐻𝐻 and ∆𝐽𝐽𝐽𝐽𝐽𝐽
1) 𝐻𝐻𝐻𝐻 = 𝑂𝑂𝑂𝑂 = 4𝑐𝑐𝑐𝑐
2) 𝐻𝐻𝐻𝐻 = 𝐽𝐽𝐽𝐽
3) 𝑂𝑂𝑂𝑂 = 𝑂𝑂𝑂𝑂
∴ ∆𝐻𝐻𝐻𝐻𝐻𝐻 ≡ ∆𝐽𝐽𝐽𝐽𝐽𝐽
Proven at 2.1.1
Given
Common
𝑠𝑠, 𝑠𝑠, 𝑠𝑠
𝐼𝐼
𝟏𝟏
𝟏𝟏
𝑯𝑯𝑯𝑯
𝟐𝟐
2.2
𝑂𝑂
𝐴𝐴
𝐵𝐵
𝐷𝐷
𝑃𝑃
Statement
Reason
2.2 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵
Given ABCD is a parallelogram
But 𝐴𝐴𝐴𝐴 = 𝑂𝑂𝑂𝑂
∴ 𝐴𝐴𝐴𝐴 + 𝑂𝑂𝑂𝑂 = 𝐴𝐴𝐴𝐴
And 𝐵𝐵𝐵𝐵 = 𝑃𝑃𝑃𝑃
∴ 𝐵𝐵𝐵𝐵 + 𝑃𝑃𝑃𝑃 = 𝐵𝐵𝐵𝐵
∴ 𝑂𝑂𝑂𝑂 = 𝑃𝑃𝑃𝑃
And OD//PC
∴ 𝑂𝑂𝑂𝑂𝑂𝑂𝑂𝑂 is a parallelogram
𝐶𝐶
Given
Given
Given
One pair of opposite sides are equal and parallel.
2.3
𝐴𝐴
𝐵𝐵
1
1
2
2
2
1
1
2
𝐷𝐷
𝐶𝐶
Statement
Reason
In ∆𝐴𝐴𝐴𝐴𝐴𝐴 and ∆𝐶𝐶𝐶𝐶𝐶𝐶
1) 𝐴𝐴𝐴𝐴 = 𝐶𝐶𝐶𝐶
2) 𝐵𝐵𝐵𝐵 = 𝐵𝐵𝐵𝐵
3) 𝐴𝐴𝐴𝐴 = 𝐵𝐵𝐵𝐵
Opposite sides of a parallelogram 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 are equal
Common
Given
∴ ∆𝐴𝐴𝐴𝐴𝐴𝐴 ≡ ∆𝐶𝐶𝐶𝐶𝐶𝐶
𝑠𝑠, 𝑠𝑠, 𝑠𝑠
� = 90°
∴ 𝐶𝐶̂ = 𝐵𝐵� = 𝐴𝐴̂ = 𝐷𝐷
Interior <s of a quadrilaterals is equal to 360°
�
∴ 𝐶𝐶̂ = 𝐵𝐵� and 𝐴𝐴̂ = 𝐷𝐷
∴ 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 a rectangle
2.4
𝐻𝐻
𝐷𝐷
𝐺𝐺
2
1
𝑂𝑂
𝐸𝐸
Statement
Reason
2.4.1 𝐷𝐷𝑂𝑂�𝐺𝐺 = 90°
𝐷𝐷𝐷𝐷 ⊥ 𝐸𝐸𝐸𝐸 𝑜𝑜𝑜𝑜 𝑘𝑘𝑘𝑘𝑘𝑘𝑘𝑘 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷
2.4.3 𝐷𝐷𝐷𝐷 = 𝐻𝐻𝐻𝐻
𝑂𝑂𝑂𝑂 = 𝐷𝐷𝐷𝐷
∴ 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 is a rectangle
Given
Given
2.4.2 𝐷𝐷𝐷𝐷 2 = 42 + 32
= 16 + 9
𝐷𝐷𝐷𝐷 = √25
𝐷𝐷𝐷𝐷 = 5 𝑐𝑐𝑐𝑐
(Pyth)
Opposite of rectangle are equal
𝐹𝐹
2.5
𝑃𝑃
1
𝑆𝑆
2
𝑂𝑂
1
2
2
𝑄𝑄
Statement
2.5
In ∆𝑃𝑃𝑃𝑃𝑃𝑃 and ∆𝑄𝑄𝑄𝑄𝑄𝑄
1) 𝑃𝑃𝑃𝑃 = 𝑄𝑄𝑄𝑄
�2 = 𝑅𝑅
�2
2) 𝑃𝑃
�1 + 𝑃𝑃
�2 = 𝑇𝑇
�2 + 𝑇𝑇
�2 = 180°
3) 𝑃𝑃
�
�
∴ 𝑃𝑃1 = 𝑇𝑇1
∴ ∆𝑃𝑃𝑃𝑃𝑃𝑃 ≡ ∆𝑄𝑄𝑄𝑄𝑄𝑄
∴ ∆𝑃𝑃𝑃𝑃𝑃𝑃 ≡ ∆𝑆𝑆𝑆𝑆𝑆𝑆
∴ 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 is a parallelogram
1
𝑇𝑇
2
Reason
Given
Alternating <s PS//QR
<s on a straight line
𝑆𝑆, 𝐴𝐴, 𝐴𝐴
1
𝑅𝑅