2. Proofs of Theorems and Glossary of Terms

3. 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 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 Just Click on the Proof Required Go to JC Constructions

4. Proof:  3 +  4 +  5 = 180 0 Straight line  1 =  4 and  2 = 55 Alternate angles  3 +  1 +  2 = 180 0  1 +  2 +  3 = 180 0 Q.E.D. 45 Given: Given:Triangle 12 3 Construction: 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 = 180 0 Menu Constructions Quit Use mouse clicks to see proof

5. 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:  1 +  2 = 180 0 ………….. Straight line  2 +  3 +  4 = 180 0 ………….. Theorem 2.  1 +  2 =  2 +  3 +  4  1 =  3 +  4 Q.E.D. 1 2 3 4 Menu Constructions Quit Use mouse clicks to see proof

6. 2 3 1 4 Given: Given:Parallelogram abcd cb ad 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|  abc =  adc and  abc =  adc Proof:In the triangle abc and the triangle adc  1 =  4 …….. Alternate angles |ac| = |ac| …… Common  2 =  3 ……… Alternate angles  ……… ASA = ASA.  The triangle abc is congruent to the triangle adc  ……… ASA = ASA.  |ab| = |cd| and |ad| = |bc|  abc =  adc and  abc =  adc Q.E.D Menu Constructions Quit Use mouse clicks to see proof

7. Given: Given:Triangle abc Proof:** Area of large sq. = area of small sq. + 4(area )) (a + b) 2 = c 2 + 4(½ab) a2 a2 + 2ab +b 2 = c 2 + 2ab a2 a2 + b2 b2 = c2c2 Q.E.D. a b c a b c 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:a 2 + b 2 = c 2 Menu Constructions Quit Use mouse clicks to see proof

8. 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:|  boc | = 2 |  bac | To Prove: |  boc | = 2 |  bac | Construction: Construction: Join a to o and extend to r r Proof:In the triangle aob a b c o 1 3 2 4 5 | oa| = | ob | …… Radii  |  2 | = |  3 | …… Theorem 4 |  1 | = |  2 | + |  3 | …… Theorem 3 |  1 | = |  2 | + |  3 | …… Theorem 3  |  1 | = |  2 | + |  2 |  |  1 | = 2|  2 |  Similarly |  4 | = 2|  5 |  |  boc | = 2 |  bac | Q.E.D Menu Constructions Quit Use mouse clicks to see proof