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**Select the proof required then click**mouse key to view proof. Menu Theorem 1 Vertically opposite angles are equal in measure. Theorem 2 The measure of the three angles of a triangle sum to 1800 . Theorem 3 An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. Theorem 4 If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. Theorem 5 The opposite sides and opposite sides of a parallelogram are respectively equal in measure. Theorem 6 A diagonal bisects the area of a parallelogram Theorem 7 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 8 A line through the centre of a circle perpendicular to a chord bisects the chord. Theorem 9 If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Theorem 10 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. Constructions Sketches Quit**90**90 45 45 135 135 0 0 180 180 Theorem 1: Vertically opposite angles are equal in measure Use mouse clicks to see proof 1 4 2 3 To Prove:Ð1 = Ð3 and Ð2 = Ð4 Proof:Ð1 + Ð2 = 1800 …………..Straight line Ð2 + Ð3 = 1800 ………….. Straight line ÞÐ1 + Ð2 = Ð2 + Ð3 Þ Ð1 = Ð3 Similarly Ð2 = Ð4 Q.E.D. Constructions Sketches Menu Quit**4**5 3 1 2 Theorem 2: 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 Constructions Sketches Menu Quit**90**45 135 3 0 180 1 2 4 Theorem 3: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. 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 Sketches Menu Quit**a**c b d Theorem 4: If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. Use mouse clicks to see proof 4 3 Given:Triangle abc with |ab| = |ac| To Prove:Ð1 = Ð2 2 1 Construction:Construct ad the bisector of Ðbac Proof: In the triangle abd and the triangle adc Ð3 = Ð4 …………..Construction |ab| = |ac|………….. Given. |ad| = |ad|………….. Common Side. Þ The triangle abd is congruent to the triangle adc……….. SAS = SAS. Þ Ð1 = Ð2 Q.E.D. Constructions Sketches Menu Quit**b**c a d Theorem 5: 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 Constructions Sketches Menu Quit**c**b a d x Theorem 6: A diagonal bisects the area of a parallelogram Use mouse clicks to see proof Given: Parallelogram abcd To Prove:Area of the triangle abc = Area of the triangle adc Construction:Draw perpendicular from b to ad Proof: Area of triangle adc = ½ |ad| x |bx| Area of triangle abc = ½ |bc| x |bx| As |ad| = |bc| …… Theorem 5 Area of triangle adc = Area of triangle abc Þ The diagonal ac bisects the area of the parallelogram Q.E.D Constructions Sketches Menu Quit**a**o r c b Theorem 7: 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. 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 Sketches Menu Quit**L**a o r 90 o b Theorem 8: A line through the centre of a circle perpendicular to a chord bisects the chord. Use mouse clicks to see proof Given: A circle with o as centre and a line L perpendicular to ab. To Prove:| ar | = | rb | Construction:Join a to o and o to b Proof: In the triangles aor and the triangle orb Ðaro = Ðorb ………….90 o |ao| = |ob|………….. Radii. |or| = |or|………….. Common Side. Þ The triangle aor is congruent to the triangle orb……… RSH = RSH. Þ |ar| = |rb| Q.E.D Constructions Sketches Menu Quit**|ab|**|ab| |ab| |ac| |ac| |ac| To Prove: = = = a |bc| |bc| d |de| |ax| |ay| |df| |de| |df| = = |ef| |ef| 2 2 1 3 4 5 x y e f Þ As xy is parallel to bc 1 3 Similarly b c Theorem 9: If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Use mouse clicks to see proof Given: Two Triangles with equal angles Construction: On ab mark off ax equal in length to de. On ac mark off ay equal in length to df Proof:Ð1 = Ð4 Þ[xy] is parallel to [bc] Q.E.D. Constructions Sketches Menu Quit**b**a a c b c c c a b b a Theorem 10: 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. 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 Sketches Menu Quit