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LinesAnglesClass9
- 1. Lines And Angles Class-9 Done By : Smrithi Jaya
- 2. Types of lines ▪ Line a line can be defined as a straight one- dimensional figure that has no thickness and extends endlessly in both directions Line Segment A line segment can be defined as a line with 2 end points Ray A ray can be defined as a line with one end point ▪ Collinear points : points that lie on the same line Non collinear points : points that do not lie on tha same line
- 3. Angles ▪ When two rays originate from the same end point an angle is formed. ▪ The rays are called arms and the endpoint is called vertex
- 4. Types of Angles ▪ AcuteAngle - 0° − 90° ▪ Obtuse Angle - 90° − 180° ▪ Right Angle - 90° ▪ Straight angle - 180° ▪ Complete angle - 360° ▪ Reflex angle - 180° − 360° In the given figure : 𝒚 = 𝟑𝟎𝟎° 𝒊. 𝒆 𝒚 𝒊𝒔 𝒕𝒉𝒆 𝒓𝒆𝒇𝒍𝒆𝒙 𝒂𝒏𝒈𝒍𝒆 𝒙 = 𝟑𝟔𝟎 − 𝒚 = 𝟑𝟔𝟎 − 𝟑𝟎𝟎 = 𝟔𝟎°
- 6. Adjacent angles ▪ Two angles are said to be adjacent if they have ▪ A common vertex ▪ A common arm ▪ Two non common arms on different sides of the common arm • Point B is the common vertex • BD is the common arm • BA and BC are non- common arms • Therefore < 1 𝑖𝑠 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 2
- 7. Complementary and supplementary angles ▪ Complementary angles ▪ Two angles whose sum is 90° are called complementary angles ▪ SupplementaryAngles ▪ Two angles whose sum is 180° are called supplementary angles
- 8. Linear pairs and vertically opposite angles ▪ Linear pair property or staraight line property ▪ Vertically opposite angles If AC is a straight line, then 𝑥 + 𝑦 = 180° or < 𝐴𝐵𝐷 and < 𝐶𝐵𝐷 is a linear pair When AB and CD intersect at O two pairs of vertically opposite angles are formed and are equal i.e. < 1 = < 2 < 3 = < 4
- 9. Intersecting and Non-intersecting lines In this figure PQ and RS are Intersecting lines PQ and RS are parallel or non - intersecting lines
- 11. Axioms and Theorems AXIOMS THEOREMS The axiom is a statement which is self evident theorem is a statement which is not self evident Axiom cannot be proven by any kind of mathematical representation. Theorem can be proved by mathematical representation
- 12. AXIOMS ▪ Axiom 6.1 : if ray stands on a line the sum of two angles formed is 180° ▪ 𝑥 + 𝑦 = 180° ▪ Axiom 6.2 : if sum of two angles is 180° then the two non common arms form a line ▪ i.e AC is a line
- 13. AXIOMS ▪ Axiom 6.3 or corresponding angles axiom: if a transversal intersects two parallel lines, then each pair of corresponding angles is equal. i.e. < 1 =< 5 , < 2 = < 6 , < 3 = < 7, < 4 =< 8 Axiom 6.4 : If a transversal intersects two lines such that angles formed are corresponding then the two lines are said to be parallel.
- 14. THEOREMS ▪ Theorem 1 : vertically opposite angles are congruent ▪ Theorem 2: if a transversal intersects two parallel lines , then each pair of alternate interior angles are equal ▪ Theorem 3 : If a transversal intersects two parallel lines such that a pair of alternate interior angles is equal then the two lines are parallel. ▪ Theorem 4 : If a transversal intersects two parallel lines then each pair of co-interior angles are supplementary.
- 15. THEOREMS ▪ Theorem 5 : If a transversal intersects two parallel lines such that a pair of co interior angles are supplementary then the two lines are parallel. ▪ Theorem 6 : Lines which are parallel to the same line are parallel to each other. ▪ Theorem 7:The sum of the angles of a triangle is 180°. ▪ Theorem 8 or Exterior angle theorem : If a side of a triangle is produced then the exterior angle so formed is equal to sum of interior opposite angles.