- 1. Lines &Angles By - NM Spirit T- 901-568-0202 Email- nm.sspirit@gmail.com
- 2. Point An exact location on a plane is called a point. Line Line segment Ray A straight path on a plane, extending in both directions with no endpoints, is called a line. A part of a line that has two endpoints and thus has a definite length is called a line segment. A line segment extended indefinitely in one direction is called a ray. Recap Geometrical Terms
- 4. RAY: A part of a line, with one endpoint, that continues without end in one direction LINE: A straight path extending in both directions with no endpoints LINE SEGMENT: A part of a line that includes two points, called endpoints, and all the points between them
- 5. INTERSECTING LINES: Lines that cross PARALLEL LINES: Lines that never cross and are always the same distance apart
- 6. Common endpoint B C B A Ray BC Ray BA Ray BA and BC are two non-collinear rays When two non-collinear rays join with a common endpoint (origin) an angle is formed. What Is An Angle ? Common endpoint is called the vertex of the angle. B is the vertex of ABC. Ray BA and ray BC are called the arms of ABC.
- 7. To name an angle, we name any point on one ray, then the vertex, and then any point on the other ray. For example: ABC or CBA We may also name this angle only by the single letter of the vertex, for example B. A B C Naming An Angle
- 8. An angle divides the points on the plane into three regions: A B C F R P T X Interior And Exterior Of An Angle • Points lying on the angle (An angle) • Points within the angle (Its interior portion. ) • Points outside the angle (Its exterior portion. )
- 9. Right Angle: An angle that forms a square corner Acute Angle: An angle less than a right angle Obtuse Angle: An angle greater than a right angle
- 10. Two angles that have the same measure are called congruent angles. Congruent angles have the same size and shape. A B C 300 D E F 300 D E F 300 Congruent Angles
- 11. Pairs Of Angles : Types • Adjacent angles • Vertically opposite angles • Complimentary angles • Supplementary angles • Linear pairs of angles
- 12. Adjacent Angles Two angles that have a common vertex and a common ray are called adjacent angles. C D B A Common ray Common vertex Adjacent Angles ABD and DBC Adjacent angles do not overlap each other. D E F A B C ABC and DEF are not adjacent angles
- 13. Vertically Opposite Angles Vertically opposite angles are pairs of angles formed by two lines intersecting at a point. APC = BPD APB = CPD A DB C P Four angles are formed at the point of intersection. Point of intersection ‘P’ is the common vertex of the four angles. Vertically opposite angles are congruent.
- 14. If the sum of two angles is 900, then they are called complimentary angles. 600 A B C 300 D E F ABC and DEF are complimentary because 600 + 300 = 900 ABC + DEF Complimentary Angles
- 15. If the sum of two angles is 1800 then they are called supplementary angles. PQR and ABC are supplementary, because 1000 + 800 = 1800 RQ P A B C 1000 800 PQR + ABC Supplementary Angles
- 16. Two adjacent supplementary angles are called linear pair of angles. A 600 1200 PC D 600 + 1200 = 1800 APC + APD Linear Pair Of Angles
- 17. A line that intersects two or more lines at different points is called a transversal. Line L (transversal) BA Line M Line N DC P Q G F Pairs Of Angles Formed by a Transversal Line M and line N are parallel lines. Line L intersects line M and line N at point P and Q. Four angles are formed at point P and another four at point Q by the transversal L. Eight angles are formed in all by the transversal L.
- 18. Pairs Of Angles Formed by a Transversal • Corresponding angles • Alternate angles • Interior angles
- 19. Corresponding Angles When two parallel lines are cut by a transversal, pairs of corresponding angles are formed. Four pairs of corresponding angles are formed. Corresponding pairs of angles are congruent. GPB = PQE GPA = PQD BPQ = EQF APQ = DQF Line M BA Line N D E L P Q G F Line L
- 20. Alternate Angles Alternate angles are formed on opposite sides of the transversal and at different intersecting points. Line M BA Line N D E L P Q G F Line L BPQ = DQP APQ = EQP Pairs of alternate angles are congruent. Two pairs of alternate angles are formed.
- 21. The angles that lie in the area between the two parallel lines that are cut by a transversal, are called interior angles. A pair of interior angles lie on the same side of the transversal. The measures of interior angles in each pair add up to 1800. Interior Angles Line M BA Line N D E L P Q G F Line L 600 1200 1200 600 BPQ + EQP = 1800 APQ + DQP = 1800
- 22. Theorems of Lines and Angles
- 23. If a ray stands on a line, then the sum of the adjacent angles so formed is 180°. PX Y QGiven: The ray PQ stands on the line XY. To Prove: ∠QPX + ∠YPQ = 1800 Construction: Draw PE perpendicular to XY. Proof: ∠QPX = ∠QPE + ∠EPX = ∠QPE + 90° ………………….. (i) ∠YPQ = ∠YPE − ∠QPE = 90° − ∠QPE …………………. (ii) (i) + (ii) ⇒ ∠QPX + ∠YPQ = (∠QPE + 90°) + (90° − ∠QPE) ∠QPX + ∠YPQ = 1800 Thus the theorem is proved E
- 24. Vertically opposite angles are equal in measure To Prove : = C and B = D A + B = 1800 ………………. Straight line ‘l’ B + C = 1800 ……………… Straight line ‘m’ + = B + C A = C Similarly B = D A B C D l m Proof: Given: ‘l’ and ‘m’ be two lines intersecting at O as. They lead to two pairs of vertically opposite angles, namely, (i) ∠ A and ∠ C (ii) ∠ B and ∠ D.
- 25. If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other. Given: Transversal PS intersects parallel lines AB and CD at points Q and R respectively. To Prove: ∠ BQR = ∠ QRC and ∠ AQR = ∠ QRD Proof : ∠ PQA = ∠ QRC ------------- (1) Corresponding angles ∠ PQA = ∠ BQR -------------- (2) Vertically opposite angles from (1) and (2) ∠ BQR = ∠ QRC Similarly, ∠ AQR = ∠ QRD. Hence the theorem is proved BA C D p S Q R
- 26. If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel. Given: The transversal PS intersects lines AB and CD at points Q and R respectively where ∠ BQR = ∠ QRC. To Prove: AB || CD Proof: ∠ BQR = ∠ PQA --------- (1) Vertically opposite angles But, ∠ BQR = ∠ QRC ---------------- (2) Given from (1) and (2), ∠ PQA = ∠ QRC But they are corresponding angles. So, AB || CD BA C D P S Q R
- 27. Lines which are parallel to the same line are parallel to each other. Given: line m || line l and line n || line l. To Prove: l || n || m Construction: BA C D P S R l n m E F Let us draw a line t transversal for the lines, l, m and n t Proof: It is given that line m || line l and line n || line l. ∠ PQA = ∠ CRQ -------- (1) ∠PQA = ∠ ESR ------- (2) (Corresponding angles theorem) ∠ CRQ = ∠ ESR Q But, ∠ CRQ and ∠ESR are corresponding angles and they are equal. Therefore, We can say that Line m || Line n
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