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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 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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4

5

3

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2

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Given:Triangle

Proof:Ð3 + Ð4 + Ð5 = 1800Straight line

Ð1 = Ð4 and Ð2 = Ð5Alternate angles

ÞÐ3 + Ð1 + Ð2 = 1800

Ð1 + Ð2 + Ð3 = 1800

Q.E.D.

To Prove:Ð1 + Ð2 + Ð3 = 1800

Construction:Draw line through Ð3 parallel to the base

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90

45

135

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0

180

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

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To Prove:Ð1 = Ð3 + Ð4

Proof:Ð1 + Ð2 = 1800 …………..Straight line

Ð2 + Ð3 + Ð4 = 1800 ………….. Theorem 2.

Þ Ð1 + Ð2 = Ð2 + Ð3 + Ð4

Þ Ð1 = Ð3 + Ð4

Q.E.D.

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b

c

a

d

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Given:Parallelogram abcd

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

and Ðabc = Ðadc

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Construction:Draw the diagonal |ac|

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Proof:In the triangle abc and the triangle adc

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Ð1 = Ð4 …….. Alternate angles

Ð2 = Ð3 ……… Alternate angles

|ac| = |ac| …… Common

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

Þ|ab| = |cd| and |ad| = |bc|

and Ðabc = Ðadc

Q.E.D

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b

a

a

c

b

c

c

c

a

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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

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Given:Triangle abc

To Prove:a2 + b2 = c2

Construction:Three right angled triangles as shown

Proof:**Area of large sq. = area of small sq. + 4(area D)(a + b)2 = c2 + 4(½ab)

a2 + 2ab +b2 = c2 + 2ab

a2 + b2 = c2

Q.E.D.

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a

o

r

c

b

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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To Prove:| Ðboc | = 2 | Ðbac |

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Construction:Join a to o and extend to r

Proof:In the triangle aob

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| oa| = | ob | …… Radii

Þ | Ð2 | = | Ð3 | …… Theorem 4

| Ð1 | = | Ð2 | + | Ð3 | …… Theorem 3

Þ | Ð1 | = | Ð2 | + | Ð2 |

Þ | Ð1 | = 2| Ð2 |

Similarly| Ð4 | = 2| Ð5 |

Q.E.D

Þ | Ðboc | = 2 | Ðbac |

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