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Proofs of Theorems and Glossary of Terms

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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Proofs of Theorems and Glossary of Terms

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  1. Proofs of Theorems and Glossary of Terms

  2. 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

  3. 4 5 3 1 2 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 Menu Quit

  4. 90 45 135 3 0 180 1 2 4 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 Menu Quit

  5. 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 3 4 Construction:Draw the diagonal |ac| 1 Proof: In the triangle abc and the triangle adc 2 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 Menu Quit

  6. b a a c b c c c a b b a 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 Menu Quit

  7. 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 | 5 2 Construction:Join a to o and extend to r Proof: In the triangle aob 4 1 3 | 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 | Constructions Menu Quit

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