2. Menu Theorem 4 The measure of the three angles of a triangle sum to 180 degrees. Theorem 6 An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. Theorem 9 The opposite sides and opposite sides of a parallelogram are respectively equal in measure. Theorem 19 The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference standing on the same arc. Theorem 14 In a right-angled triangle, the square of the length of the side opposite to the right angle is equal to the sum of the squares of the other two sides.

3. Proof:  3  3 +  4  4 +  5  5 = 180 0 Straight line  1  1 =  4  4 and  2  2 = 5555Alternate angles   3  3 +  1  1 +  2  2 = 180 0  1  1 +  2  2 +  3  3 = 180 0 Q.E.D. 45 Given: Given:Triangle 12 3 Construction: Construction:Draw line through  3 parallel to the base Theorem 4:The measure of the three angles of a triangle sum to 180 0. To Prove: To Prove:  1 +  2 +  3 = 180 0 Menu Use mouse clicks to see proof

4. 0 180 90 45 135 Theorem 6: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. opposite angles in measure. 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 Quit

5. 2 3 1 4 Given: Given:Parallelogram abcd cb ad Construction: Construction: Draw the diagonal |ac| Theorem 9: The opposite sides and opposite sides of a parallelogram are respectively equal in measure. 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 Use mouse clicks to see proof

6. Given: Given:Triangle abc Proof:Area of large sq. = area of small sq. + 4(area )))) (a + b) 2 b) 2 = c 2 c 2 + 4(½ab) a2 a2 a2 a2 + 2ab +b 2 +b 2 = c 2 c 2 + 2ab a2 a2 a2 a2 + b2 b2 b2 b2 = c2c2c2c2 a b c a b c a b c a b c Construction: Construction:Three right angled triangles as shown Theorem 14:In a right-angled triangle, the square of the length of the side opposite to the right angle is equal to the sum of the squares of the other two sides. To Prove: To Prove:a 2 + b 2 = c 2 Menu Note must show that the angles in small square are 90 o 1 90 o 4 3 2  1 +  2 = 90 o …. Complimentary angles  1 +  4 +  3 = 180 o …. Straight angle  2 =  3 …. Similar triangles  1 +  3 = 90 o  4 = 90 o

7. Theorem 19: The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference standing on the same arc. measure of the angle at the circumference 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