Lines and Angles

Lines and Angles Fundamental concepts of Geometry: Point: It is an exact location. It is a fine dot which has neither length nor breadth nor thickness but has position i.e., it has no magnitude. Line segment: The straight path joining two points A and B is called a line segment AB . It has and points and a definite length. Ray: A line segment which can be extended in only one direction is called a ray. Intersecting lines: Two lines having a common point are called intersecting lines. The common point is known as the point of intersection. Concurrent lines: If two or more lines intersect at the same point, then they are known as concurrent lines. Angles: When two straight lines meet at a point they form an angle. In the figure above, the angle is represented as ∠AOB. OA and OB are the arms of ∠AOB. Point O is the vertex of ∠AOB. The amount of turning from one arm (OA) to other (OB) is called the measure of the angle (ÐAOB). Right angle: An angle whose measure is 90o is called a right angle. Acute angle: An angle whose measure is less then one right angle (i.e., less than 90o), is called an acute angle. 1 Obtuse angle: An angle whose measure is more than one right angle and less than two right angles (i.e., less than 180o and more than 90o) is called an obtuse angle. Reflex angle: An angle whose measure is more than 180o and less than 360o is called a reflex angle. Complementary angles: If the sum of the two angles is one right angle (i.e., 90o), they are called complementary angles. Therefore, the complement of an angle θ is equal to 90° − θ. Supplementary angles: Two angles are said to be supplementary, if the sum of their measures is 180o. Example: Angles measuring 130o and 50o are supplementary angles. Two supplementary angles are the supplement of each other. Therefore, the supplement of an angle θ is equal to 180° − θ. 2 Vertically opposite angles: When two straight lines intersect each other at a point, the pairs of opposite angles so formed are called vertically opposite angles. In the above figure, ∠1 and ∠3 and angles ∠2 and ∠4 are vertically opposite angles. Note: Vertically opposite angles are always equal. Bisector of an angle: If a ray or a straight line passing through the vertex of that angle, divides the angle into two angles of equal measurement, then that line is known as the Bisector of that angle. A point on an angle is equidistant from both the arms. In the figure above, Q and R are the feet of perpendiculars drawn from P to OB and OA. It follows that PQ = PR. 3 Parallel lines: Two lines are parallel if they are coplanar and they do not intersect each other even if they are extended on either side. Transversal: A transversal is a line that intersects (or cuts) two or more coplanar lines at distinct points. In the above figure, a transversal t is intersecting two parallel lines, l and m, at A and B, respectively. Angles formed by a transversal of two parallel lines: In the above figure, l and m are two parallel lines intersected by a transversal PS. The following properties of the angles can be observed: ∠3 = ∠5 and ∠4 = ∠6 [Alternate angles] ∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8, ∠3 = ∠7 [Corresponding angles] ∠4 + ∠5 = ∠3 + ∠6 = 180° [Supplementary angles] In the figure given below, which of the lines are parallel to each other? Answer: As 67° + 113° = 180°, lines P and S, R and S, and S and U are parallel. Therefore, lines P, R, S and U are parallel to each other. Similarly, lines Q and T are parallel to each other. 4 In the figure given below, PQ and RS are two parallel lines and AB is a transversal. AC and BC are angle bisectors of ∠BAQ and ∠ABS, respectively. If ∠BAC = 30°, find ∠ABC and ∠ACB. Answer: ∠BAQ + ∠ABS = 180° [Supplementary angles] ∠BAQ ∠ABS 180° + = = 90° ⇒ ∠BAC + ∠ABC = 90° 2 2 2 Therefore, ∠ABC = 60° and ∠ACB = 90°. ⇒ For what values of x in the figure given below are the lines P-A-Q and R–B-S parallel, given that AD and BD intersect at D? Answer: We draw a line DE, parallel to RS, as shown in the figure below: In the above figure, ∠CDE = ∠RBD = 10x + 5 ⇒ ∠CDA = 10x + 5 −30 = 10x − 25. Let the line PQ and RS be parallel. Therefore, PQ // DE. ⇒ ∠EDA = ∠CAD = 10x − 25 = 6x − 5 ⇒ x = 5. In the figure given below, lines AB and DE are parallel. What is the value of ∠CDE? Answer: We draw a line CF // DE at C, as shown in the figure below. 5 ∠BCF = ∠ABC = 55° ⇒ ∠DCF = 30°. ⇒ CDE = 180° − 30° = 150°.