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Properties of a Triangle P A Q B C mABC + mBCA + mCAB = 1800 (Internal angles of any triangle add up to 1800) mPAB + mQBA + mACR = 3600 (Exterior angles of any triangle add up to 3600) R Properties of a Triangle (Contd) 600 a a 600 600 a A triangle which has all three of its sides equal in length is called an equilateral triangle. All angles of an equilateral triangle are congruent and measure 600 each. Ø0 Ø0 A triangle which has two of its sides equal in length is called an isosceles triangle. The base angles of an isosceles triangle are always equal. Incenter of a Triangle The point where the three angle bisectors of a triangle meet. Circumcenter of a Triangle The point where the three perpendicular bisectors of a triangle meet. Centroid of a Triangle The point where the three medians of the triangle intersect. The 'center of gravity' of the triangle Orthocenter of a Triangle The point where the three altitudes of a triangle intersect. Properties of Equilateral Triangle a With an equilateral triangle, the radius of the incircle is exactly half the radius of the circumcircle. Congruence of Triangles - SSS Test Triangles are congruent if all three sides in one triangle are congruent to the corresponding sides in the other. Congruence of Triangles - SAS Test Triangles are congruent if any pair of corresponding sides and their included angles are equal in both triangles. Congruence of Triangles - ASA Test Triangles are congruent if any two angles and their included side are equal in both triangles. Congruence of Triangles - AAS Test Triangles are congruent if two pairs of corresponding angles and a pair of opposite sides are equal in both triangles. Congruence of Triangles - HL Test Two right triangles are congruent if the hypotenuse and one corresponding leg are equal in both triangles. Pythagoras Theorem A B C In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In ABC if mABC = 900 then, l(AC)2 = l(AB)2 + l(BC)2 30°- 60°- 90° Triangle A 600 2 Units 1 Unit 900 B 300 3 Units C In a 30°- 60°- 90° Triangle, the hypotenuse is double the side opposite to 30° angle and the side opposite to 60° angle is Sqrt(3) times the side opposite to 30° angle. 45°- 45°- 90° Triangle A 450 2 1 Unit 900 B Units 450 1 Unit C In a 45°- 45°- 90° Triangle, sides opposite to 450 angles are of equal length, and, Hypotenuse is sqrt(2) times either side.