Angles and properties for class VII by G R Ahmed
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The document defines different types of angles and their properties. It explains what an angle is, how angles are named and measured. It discusses right angles, straight angles, acute angles, obtuse angles and reflex angles. It describes angles on a straight line, angles at a point, intersecting lines, parallel lines, perpendicular lines, vertically opposite angles, corresponding angles and alternate angles. It presents proofs of several theorems regarding the sum of interior and exterior angles of triangles.
Angles and properties for class VII by G R Ahmed
- 1. Angle and Properties For class VII Prepared by GOlam Robbani Ahmed
- 2. Angles An angle is formed when two lines meet. The size of the angle measures the amount of space between the lines. In the diagram the lines ba and bc are called the ‘arms’ of the angle, and the point ‘b’ at which they meet is called the ‘vertex’ of the angle. An angle is denoted by the symbol .An angle can be named in one of the three ways: a c b . .Amount of space Angle
- 3. 1. Three letters a b c . . Using three letters, with the centre at the vertex. The angle is now referred to as : abc or cba.
- 4. 2. A number c b . .1 a Putting a number at the vertex of the angle. The angle is now referred to as 1.
- 5. 3. A capital letter b . .B a c Putting a capital letter at the vertex of the angle. The angle is now referred to as B.
- 6. Right angle A quarter of a revolution is called a right angle. Therefore a right angle is 90. Straight angle A half a revolution or two right angles makes a straight angle. A straight angle is 180. Measuring angles We use the symbol to denote a right angle.
- 7. Acute, Obtuse and reflex Angles Any angle that is less than 90 is called an acute angle. An angle that is greater than 90 but less than 180 is called an obtuse angle. An angle greater than 180 is called a reflex angle.
- 8. Angles on a straight line Angles on a straight line add up to 180. A + B = 180 . Angles at a point Angles at a point add up to 360. A+ B + C + D + E = 360 A B A B D E C
- 9. Pairs of lines: Consider the lines L and K : .p L K Intersecting
- 10. Parallel lines L K L is parallel to K Written: LK Parallel lines never meet and are usually indicated by arrows. Parallel lines always remain the same distance apart.
- 11. Perpendicular L is perpendicular to K Written: L K The symbol is placed where two lines meet to show that they are perpendicular L K
- 12. Parallel lines and Angles 1.Vertically opposite angles When two straight lines cross, four angles are formed. The two angles that are opposite each other are called vertically opposite angles. Thus a and b are vertically opposite angles. So also are the angles c and d. From the above diagram: AB C D A+ B = 180 …….. Straight angle B + C = 180 ……... Straight angle A + C = B + C ……… Now subtract c from both sides A = B
- 13. 2. Corresponding Angles The diagram below shows a line L and four other parallel lines intersecting it. The line L intersects each of these lines. L All the highlighted angles are in corresponding positions. These angles are known as corresponding angles. If you measure these angles you will find that they are all equal.
- 14. In the given diagram the line L intersects two parallel lines A and B. The highlighted angles are equal because they are corresponding angles. The angles marked with are also corresponding angles . A B L . . Remember: When a third line intersects two parallel lines the corresponding angles are equal.
- 15. 3. Alternate angles The diagram shows a line L intersecting two parallel lines A and B. The highlighted angles are between the parallel lines and on alternate sides of the line L. These shaded angles are called alternate angles and are equal in size. Remember the Z shape. A B L
- 16. Theorem: Vertically opposite angles are equal in measure. Given: To prove : Construction: Proof: Straight angle Straight angle 1=2 Label angle 3 1=2 Intersecting lines L and K, with vertically opposite angles 1 and 2. 1+3=180 2+3=180 Q.E.D. L K 1 2 1+3=3+2 .....Subtract 3 from both sides 3
- 17. Theorem: The measure of the three angles of a triangle sum to 180. Given: To Prove: 1+2+3=180 Construction: Proof: 1=4 and 2=5 Alternate angles 1+2+3=4+5+3 But 4+5+3=180 Straight angle 1+2+3=180 The triangle abc with 1,2 and 3. 4 5 a b c 1 2 3 Q.E.D. Draw a line through a, Parallel to bc. Label angles 4 and 5.
- 18. Theorem: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. Given: A triangle with interior opposite angles 1 and 2 and the exterior angle 3. To prove: 1+ 2= 3 Construction: Label angle 4 Proof: 1+ 2+ 4=180 3+ 4=180 Three angles in a triangle 1+ 2+ 4= 3+ 4 Straight angle 1+ 2= 3 a b c 3 1 2 4 Q.E.D.