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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.

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  1. 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 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. 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.

  2. 4 5 3 1 2 Theorem 4: The measure of the three angles of a triangle sum to 1800 . 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 Menu

  3. 90 45 135 3 0 180 1 2 4 Theorem 6: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. 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. Menu Quit

  4. b c a d Theorem 9: The opposite sides and opposite sides of a parallelogram are respectively equal in measure. 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 Menu

  5. b a a c b c c c a b b a 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. 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 2 4 90o 1 3 Note must show that the angles in small square are 90o Ð1 + Ð2 = 90o …. Complimentary angles => Ð1 + Ð3 = 90o Ð2 = Ð3 …. Similar triangles => Ð4 = 90o Ð1 + Ð4 + Ð3= 180o …. Straight angle Menu

  6. a o r c b 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. 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 | Menu

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