Proofs of Theorems and Glossary of Terms

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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 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.

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

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Proof: 3 + 4 + 5 = 1800Straight line

1 = 4 and 2 = 5Alternate angles

3 + 1 + 2 = 1800

1 + 2 + 3 = 1800

Q.E.D.

4 5

Given:Given: Triangle

1 2

3Construction:Construction: Draw line through 3 parallel to the base

Theorem 4: Three angles in any triangle add up to 180°C.

To Prove:To Prove: 1 + 2 + 3 = 1800

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0 180

90

45 135

Theorem 6:Theorem 6: Each exterior angle of a triangle is equal to the sum of the two interior opposite angles

To Prove:To Prove: 1 = 3 + 4

Proof:Proof: 1 + 1 + 2 = 1802 = 1800 …………..0 ………….. Straight lineStraight line

2 + 2 + 3 + 3 + 4 = 1804 = 1800 ………….. 0 ………….. Theorem 2.Theorem 2.

1 + 1 + 2 = 2 = 2 + 2 + 3 + 3 + 4 4

1 = 1 = 3 + 3 + 4 4

Q.E.D.Q.E.D.

12

3

4

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2

3

1

4

Given:Given: Parallelogram abcdcb

a d

Construction:Construction: Draw the diagonal |ac|

Theorem 9: In a parallelogram opposite sides are equal and opposite angle are equal

To Prove:To Prove: |ab| = |cd| and |ad| = |bc|

and abc = abc = adcadc

Proof:Proof: In the triangle abc and the triangle adcIn the triangle abc and the triangle adc

1 = 1 = 4 4 …….. …….. Alternate anglesAlternate angles

|ac| = |ac| …… |ac| = |ac| …… CommonCommon

2 = 2 = 3 ……… 3 ……… Alternate anglesAlternate angles

The triangle abc is congruent to the triangle adc……… ……… ASA = ASA.ASA = ASA.

|ab| = |cd| and |ad| = |bc||ab| = |cd| and |ad| = |bc|

and abc = abc = adcadc

Q.E.DQ.E.D

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Given:Given: Triangle abc

Proof: ** Area of large sq. = area of small sq. + 4(area )

(a + b)2 = c2 + 4(½ab)

a2 + 2ab +b2 = c2 + 2ab

a2 + b2 = c2

Q.E.D.

a

b

c

a

bc

a

b

c

a

b c

Construction:Construction: Three right angled triangles as shown

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

To Prove:To Prove: a2 + b2 = c2

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Theorem 19: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.

To Prove:To Prove: | | boc | = 2 | boc | = 2 | bac | bac |

Construction:Construction: Join a to o and extend to r

r

Proof:Proof: In the triangle aobIn the triangle aob

a

b c

o

13

2

4

5

| oa| = | ob | …… | oa| = | ob | …… RadiiRadii

| | 2 | = | 2 | = | 3 | …… 3 | …… Theorem 4Theorem 4

| | 1 | = | 1 | = | 2 | + | 2 | + | 3 | …… 3 | …… Theorem 3Theorem 3

| | 1 | = | 1 | = | 2 | + | 2 | + | 2 |2 |

| | 1 | = 2| 1 | = 2| 2 |2 |

SimilarlySimilarly | | 4 | = 2| 4 | = 2| 5 |5 |

| | boc | = 2 | boc | = 2 | bac | bac | Q.E.DQ.E.D

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