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.

Theorem 1: Vertically opposite angles are equal in measure 180 90 45 135 180 90 45 135 1 4 2 3 To Prove: Ð1 = Ð3 and Ð2 = Ð4 Proof: Ð1 + Ð2 = 1800 ………….. Straight line Ð2 + Ð3 = 1800 ………….. Straight line Þ Ð1 + Ð2 = Ð2 + Ð3 Þ Ð1 = Ð3 Similarly Ð2 = Ð4 Q.E.D. Menu

Theorem 2: The measure of the three angles of a triangle sum to 1800 . Given: Triangle To Prove: Ð1 + Ð2 + Ð3 = 1800 1 2 3 Construction: Draw line through Ð3 parallel to the base 4 5 Proof: Ð3 + Ð4 + Ð5 = 1800 Straight line Ð1 = Ð4 and Ð2 = Ð5 Alternate angles Þ Ð3 + Ð1 + Ð2 = 1800 Ð1 + Ð2 + Ð3 = 1800 Q.E.D. Menu

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

Theorem 4: If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. a b c d 3 4 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

Theorem 5: The opposite sides and opposite angles of a parallelogram are respectively equal in measure. Given: Parallelogram abcd c b a d 3 To Prove: |ab| = |cd| and |ad| = |bc| and Ðabc = Ðadc 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

Theorem 6: A diagonal bisects the area of a parallelogram x 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

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. a b c o To Prove: | Ðboc | = 2 | Ðbac | r 2 5 Construction: Join a to o and extend to r Proof: In the triangle aob 1 4 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 | Þ | Ðboc | = 2 | Ðbac | Q.E.D Menu

Theorem 8: A line through the centre of a circle perpendicular to a chord bisects the chord. L 90 o o a b r 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

Theorem 9: If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Given: Two Triangles with equal angles To Prove: |df| |ac| = |de| |ab| |ef| |bc| Construction: On ab mark off ax equal in length to de. On ac mark off ay equal in length to df a c b d f e 1 2 3 Proof: Ð1 = Ð4 Þ [xy] is parallel to [bc] |ay| |ac| = |ax| |ab| Þ As xy is parallel to bc x y 4 5 |df| |ac| = |de| |ab| Similarly |ef| |bc| Q.E.D. Menu

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 a b c a b c 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. a b c 3 a b c 4 1 2 Must prove that it is a square. i.e. Show that │∠1 │= 90o │∠1│+ │∠2│ =│∠3│+│∠4│ (external angle…) But │∠2│=│∠3│ (Congruent triangles) ⇒│∠1│=│∠4│= 90o QED Menu