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Proofs of Theorems and Glossary of Terms. Just Click on the Proof Required. Menu. 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.

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slide2

Just Click on the Proof Required

Menu

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.

Go to JC Constructions

theorem 4 three angles in any triangle add up to 180 c

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

Use mouse clicks to see proof

Given: Triangle

Proof:Ð3 + Ð4 + Ð5 = 1800Straight line

Ð1 = Ð4 and Ð2 = Ð5 Alternate 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

Constructions

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slide4

90

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135

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180

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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

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.

Constructions

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theorem 9 in a parallelogram opposite sides are equal and opposite angle are equal

b

c

a

d

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

Use mouse clicks to see proof

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

Constructions

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slide6

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

Use mouse clicks to see proof

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.

Constructions

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slide7

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.

Use mouse clicks to see proof

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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