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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

1 Three letters 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.

A number 2. A number c b.. 1 a Putting a number at the vertex of the angle. The angle is now referred to as  1.

A capital letter 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.

Right angle right angle. A quarter of a revolution is called a right angle. Therefore a right angle is 90 . Straight angle 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.

Acute, Obtuse and reflex Angles acute angle Any angle that is less than 90  is called an acute angle. obtuse angle. An angle that is greater than 90  but less than 180  is called an obtuse angle. reflex angle An angle greater than 180  is called a reflex angle.

Parallel lines L K L is parallel to K Written: L  K Parallel lines never meet and are usually indicated by arrows. Parallel lines always remain the same distance apart.

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

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: A B C D A+ B = 180  …….. Straight angle B + C = 180  ……... Straight angle A + C = B + C ……… Now subtract c from both sides A = B

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.

S UPPLEMENTARY A NGLES / L INEAR P AIR Two angles that form a line (sum=180  ) t  5+  6=180  6+  8=180  8+  7=180  7+  5=180  1+  2=180  2+  4=180  4+  3=180  3+  1=180

A LTERNATE I NTERIOR A NGLES Two angles that lie between parallel lines on opposite sides of the transversal t  3   6  4  

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 angle  1+  2+  3=  4+  5+  3 But  4+  3  5 =180  Linear pair   1+  2+  3=180  The triangle abc with 1,2 and a b c Draw a line through a, Parallel to bc. Label angles 4 and 5. 5

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