Proofs of Theorems and Glossary
of Terms
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Just Click on the Proof Required
Theorem 4
Three angles in any triangle add up to 180°.
Theorem 6
Each exterior angle of a triangle is equal to the sum of the two interior opposite angles
Theorem 9
In a parallelogram opposite sides are equal and opposite angle are equal
Theorem 14 Theorem of Pythagoras : In a right angle triangle, the square of the hypotenuse is the sum
of the squares of the other two sides
Theorem 19 The angle at the centre of the circle standing on a given arc is twice the angle at any
point of the circle standing on the same arc.
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Theorem 4:
Three angles in any triangle add up to 180°C.
Use mouse clicks to see proof
Given:
Triangle
To Prove:
1 + 2 + 3 = 1800
Construction: Draw line through 3 parallel to the base
4 3 5
3 + 4 + 5 = 1800
Proof:
Straight line
1 = 4 and 2 = 5 Alternate angles

1
2
3 + 1 + 2 = 1800
1 + 2 + 3 = 1800
Q.E.D.
Constructions
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Theorem 6:
Each exterior angle of a triangle is equal to the sum of the two
interior opposite angles
Use mouse clicks to see proof
3
4
To Prove:
Proof:
1
2
1 = 3 + 4
1 + 2 = 1800
…………..
2 + 3 + 4 = 1800
Straight line
…………..
Theorem 2.
1 + 2 = 2 + 3 + 4
1 = 3 + 4
Q.E.D.
Constructions
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Theorem 9:
In a parallelogram opposite sides are equal and opposite angle
are equal
Use mouse clicks to see proof
b
c
3
Given:
Parallelogram abcd
To Prove:
|ab| = |cd| and |ad| = |bc|
4
and
Construction:
1
a
2
d
Proof:
abc = adc
Draw the diagonal |ac|
In the triangle abc and the triangle adc
1 = 4 …….. Alternate angles
2 = 3 ……… Alternate angles
|ac| = |ac| …… Common
The triangle abc is congruent to the triangle adc

………
ASA = ASA.
|ab| = |cd| and |ad| = |bc|
and
abc = adc
Q.E.D
Constructions
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Theorem
14:Theorem of Pythagoras : In a right angle triangle, the square of the
hypotenuse is the sum of the squares of the other two sides
Use mouse clicks to see proof
b
a
a
c
c
c
b
a
Given:
Triangle abc
To Prove:
a2 + b2 = c2
Construction: Three right angled triangles as shown
b
Proof:
** Area of large sq. = area of small sq. + 4(area D)
(a + b)2 = c2 + 4(½ab)
a2 + 2ab +b2 = c2 + 2ab
c
a
b
Constructions
a2 + b2 = c2
Q.E.D.
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Theorem 19: The angle at the centre of the circle standing on a given arc is twice
the angle at any point of the circle standing on the same arc.
Use mouse clicks to see proof
a
To Prove:
| boc | = 2 | bac |
Construction:
Join a to o and extend to r
Proof:
2 5
o
In the triangle aob
3
| oa| = | ob | …… Radii

1 4
r
| 2 | = | 3 | …… Theorem 4
c
b
| 1 | = | 2 | + | 3 | …… Theorem 3
 | 1 | = | 2 | + | 2 |
 | 1 | = 2| 2 |
Similarly
| 4 | = 2| 5 |
 | boc | = 2 | bac |
Constructions
Q.E.D
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