Proofs of Theorems and Glossary of Terms Menu

Menu Just Click on the Proof Required Theorem 4 Three angles in any triangle

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

Theorem 6: interior opposite angles Use mouse clicks to see proof 90 3 45

Theorem In 9: are equal Use mouse clicks to see proof b c 3

Theorem 14: Theorem of Pythagoras : In a right angle triangle, the square of

Theorem 19: The angle at the centre of the circle standing on a given

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Proofs of Theorems and Glossary of Terms

Menu Just Click on the Proof Required 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. Use mouse clicks to see proof 4 3 5 Given: Triangle To Prove: Ð 1 + Ð 2 + Ð 3 = 1800 Construction: Draw line through Ð 3 parallel to the base Proof: line Ð 3 + Ð 4 + Ð 5 = 1800 Straight Ð 1 = Ð 4 and Ð 2 = Ð 5 Alternate angles 1 2 Þ Ð 3 + Ð 1 + Ð 2 = 1800 Ð 1 + Ð 2 + Ð 3 = 1800 Q. E. D. Constructions Menu Quit

Theorem 6: interior opposite angles Use mouse clicks to see proof 90 3 45 4 0 135 1 2 180 Ð 1 = Ð 3 + Ð 4 To Prove: Ð 1 + Ð 2 = 1800 Proof: …………. . Ð 2 + Ð 3 + Ð 4 = 1800 Þ Þ Straight line …………. . Theorem 2. Ð 1 + Ð 2 = Ð 2 + Ð 3 + Ð 4 Ð 1 = Ð 3 + Ð 4 Q. E. D. Constructions Menu Quit

Theorem In 9: are equal Use mouse clicks to see proof b c 3 Given: Parallelogram abcd To Prove: |ab| = |cd| and |ad| = |bc| 4 and Construction: 1 a 2 Proof: d Ðabc = Ðadc Draw the diagonal |ac| In the triangle abc and the triangle adc Ð 1 = Ð 4 ……. . Alternate angles Ð 2 = Ð 3 ……… Alternate angles |ac| = |ac| …… Common Þ The triangle abc is congruent to the triangle adc Þ ……… ASA = ASA. |ab| = |cd| and |ad| = |bc| and Ðabc = Ðadc Q. E. D Constructions Menu Quit

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 b a a c c b Triangle abc To Prove: a 2 + b 2 = c 2 Construction: Three right angled triangles as shown Proof: ** Area of large sq. = area of small sq. + 4(area D) (a + b)2 = c 2 + 4(½ab) a 2 + 2 ab +b 2 = c 2 + 2 ab c a Given: a b Constructions a 2 + b 2 = c 2 Q. E. D. Menu Quit

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 a To Prove: | Ðboc | = 2 | Ðbac | Construction: Join a to o and extend to r Proof: 2 5 o In the triangle aob 3 | oa| = | ob | …… Radii Þ 1 4 r | Ð 2 | = | Ð 3 | …… Theorem 4 c b | Ð 1 | = | Ð 2 | + | Ð 3 | …… Theorem 3 Þ | Ð 1 | = | Ð 2 | + | Ð 2 | Þ | Ð 1 | = 2| Ð 2 | Similarly | Ð 4 | = 2| Ð 5 | Þ | Ðboc | = 2 | Ðbac | Constructions Q. E. D Menu Quit