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**Menu**Theorem 1 Vertically opposite angles are equal in measure. Theorem 2 The measure of the three angles of a triangle sum to 180o . 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.**90**90 45 45 135 135 0 0 180 180 Theorem 1: Vertically opposite angles are equal in measure 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. Menu**4**5 3 1 2 Theorem 2: The measure of the three angles of a triangle sum to 1800 . 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**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. To Prove:Ð1 = Ð3 + Ð4 Proof:Ð1 + Ð2 = 1800 …………..Straight line Ð2 + Ð3 + Ð4 = 1800 ………….. Triangle. Þ Ð1 + Ð2 = Ð2 + Ð3 + Ð4 Þ Ð1 = Ð3 + Ð4 Q.E.D. Menu**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. 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. Menu**b**c a d Theorem 5: The opposite sides and opposite angles of a parallelogram are respectively equal in measure. 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**c**b a d x Theorem 6: A diagonal bisects the area of a parallelogram 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 Menu Q.E.D**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. 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**L**a o r 90 o b Theorem 8: A line through the centre of a circle perpendicular to a chord bisects the chord. 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 Menu**|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. 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. Menu**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. 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 = c2Q.E.D. 3 4 1 2 Must prove that it is a square. i.e. Show that │∠1 │= 90o │∠1│+ │∠2│ =│∠3│+│∠4│ (external angle…) ⇒│∠1│=│∠4│= 90o QED But │∠2│=│∠3│ (Congruent triangles) Menu