JMATHSLEARNING

Given               A △ABC To Prove          ∠1 + ∠2 + ∠3 = 180° Construction   Through A, draw a line DE parallel to BC. Proof               DE ॥ BC and AB is the transversal.                         ∴    ∠1 = ∠4   (Alt. Int. angle)   ............ (i)                         Again, DE ॥ BC  and AC is the transversal.                         ∴    ∠2 = ∠5   (Alt. Int. angle)   ............ (ii)                         On adding the corresponding sides of  (i) and (ii)                         ∴    ∠1 + ∠2  = ∠4 + ∠5                            On adding ∠3 on both sides                         So, ∠1 + ∠2 + ∠3 = ∠4 + ∠5 + ∠3                         But, ∠4 + ∠5 + ∠3 = ∠ DAE = 180°                                                              ( ∵  ∠DAE is a straight angle)                         ∴    ∠1 + ∠2 + ∠3 = 180°                         Hence, the sum of the angles of a triangle is  180° . 

Median   The median of a triangle corresponding to any side is the segment joining the midpoint of that side with the opposite vertex.             In the given figure, D, E, F are the respective midpoints of sides BC, CA and AB of triangle ABC. ∴    AD is the median, corresponding to side BC. BE is the median, corresponding to side CA. CF is the median, corresponding to side AB. The medians of a triangle are concurrent , i.e., they interest each other at the same point. Cendroid : The point of intersection of all the three medians of a triangle is called its centroid . In the above figure, the medians AD, BE and CF of triangle ABC interest at the point G. ∴      G is the centroid of triangle ABC. Altitudes The altitude of a triangle corresponding to any side is the length of perpendicular drawn from the opposite vertex to that side. The side on which the perpendicular is being drawn, is called its base .        In the given figure, AL ⟘  BC; BM  ⟘  CA and

Triangle A plane figure bounded by three line segments is called triangle .  We denote a triangle by the symbol△  . A  △ABC  has: (i)           Three vertices, namely A, B and C. (ii)          Three sides, namely AB, BC and CA. (iii)         Three angles, namely ∠A,  ∠B and  ∠C.   A triangle has six parts— three sides and three angles Type of Triangles :     1.        On the basis of sides : (i). Equilateral Triangle : A triangle having all sides equal is called an equilateral triangle . In the given figure, ABC is a triangle in which AB = BC = CA. ∴     △ABC  is an equilateral triangle. (ii).  Isosceles Triangle : A triangle having two sides equal is called an isosceles triangle . In the given figure, ABC is a triangle in which AB = AC. ∴     △ABC  is an isosceles triangle. (iii). Scalene Triangle : A triangle in which all the sides are of different length is called a scalene triangle .

Angle   Two rays starting from a common point form an angle. In ∠ AOB, O is the vertex, OA  and OB  are the two rays . − → . Measure of a angle   The  amount of turning from OA to OB is called the measure of ∠ AOB , written as m∠ AOB.  An angle is measured is degrees denoted by ‘°’. Types of angles (i)                   Acute angle: An angle whose measure is greater than 0° but less than 90° is called an acute angle . Example: 15° ,30° ,60° ,75° ,etc.   ∠ AOB  = 30° is an acute angle.                       (ii)                 Right angle: An angle whose measure is 90° is called a right angle .          ∠AOB  = 90°is a right angle. (iii)                Obtuse angle: An angle whose measure is greater than 90° and less than 180° is called an obtuse angle . Example: 100°, 110°, 120°, 140°, etc.   ∠AOB  = 110° is an obtuse angle.              (iv)               Straight angle: When the rays of an angle are opposite rays forming a stra

Point     A point is an exact location.                              A fine dot represents a point.                             It is denoted by capital letter – A, B, P, Q, etc. In the given figure, P is a point. Line Segment   The straight path between two points A and B is called the line segment  AB The points A and B are called the end points of the line segment  AB  .   A line segment has a definite length.                                                         Ray  A portion of a line which starts at a point and goes off in a particular direction to infinity is known as ray. Ray has one end point A. A Ray has no definite length. We write a ray  Line  A line segment  when extended indefinitely in both the directions is called the line . A line has no end points. A line has no definite length.  It is denoted by small letters l, m, n, etc. Incidence Axioms on lines     A line contains infinitely many points.