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120 o

115 o. 95 o. 120 o. 50 o. 40 o. 34 o. 34 o. Revision Angle Properties. 65 o. Two angles making a straight line add to 180 o. 145 o. Angles round a point Add up to 360 o. 90 o. 146 o. 146 o. 3 angles in a triangle ALWAYS add up to 180 o. angles opposite each other

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120 o

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  1. 115o 95o 120o 50o 40o 34o 34o Revision Angle Properties 65o Two angles making a straight line add to 180o 145o Angles round a point Add up to 360o 90o 146o 146o 3 angles in a triangle ALWAYS add up to 180o. angles opposite each other at a cross are equal.

  2. RevisionAngle Properties d = 115o ao co bo ho eo go fo 60º 60º 60º * * ALL angles in an equilateral triangle are 60o Two angles in an isosceles are equal h is corresponding to d and must be 115o b is opposite to d and must be 115o c is must be 65o(straight line) e is alternate to c and must also be 65o

  3. B A C Angles in a Semi-Circle Key Point for Angles in a Semi-circle A triangle ABC within a semicircle with base the length of the diameter will ALWAYS be right angled at P on the circumference. Remember - Angles in any triangle sum to 180o

  4. Hints Angles in a Semi-Circle Example 1 : Sketch diagram and find all the missing angles. 20o 43o Look for right angle triangles Remember ! Angles in any triangle sum to 180o 47o 70o 4

  5. Angles in a Semi-Circle Example 2 : Sketch the diagram. (a) Right down two right angle triangles (a) Calculate all missing angles. C D 60o E 25o A B

  6. P 4cm 3cm A B a cm Angles in a Semi-Circle Pythagoras Theorem We have been interested in right angled triangles within a semi-circle. Since they are right angled we can use Pythagoras Theorem to calculate lengths. Example 1 : Calculate the value of a 5cm 6

  7. Angles in a Semi-Circle Pythagoras Theorem Example 2 : Calculate the length of XY Y 8cm 6cm cm X Z 10 cm 7

  8. Which of the lines are tangent to the circle? Angles in a Semi-Circle Tangent Line A tangent line is a line that touches a circle at only one point.

  9. Angles in a Semi-Circle Tangent Line The radius of the circle that touches the tangent line is called the point of contact radius. Special Property The point of contact radius is always perpendicular (right-angled) to the tangent line.

  10. Angles in a Semi-Circle Tangent Line • Sketch the diagram and find the size of the marked • angle in this diagram B aº 10 Angles in triangle = 180º So a = 180 – (90 + 46) = 44º 46º C 8 A

  11. Angles in a Semi-Circle Tangent Line find the missing angles b, c, d and e Angles in triangle = 180º So b = 180 – (90 + 36) = 54º C bº B 36º A

  12. Angles in a Semi-Circle Tangent Line • Since isosceles triangle, angle CAD MUST = angle CDA, • Therefore angle CAD = 30º • Angles in triangle = 180º • So d = 180 – (30 + 30) • = 120º B eº • Straight line at point C • so angle BCA = 180 – 120 = 60º A C dº • Angles in triangle = 180º • So e = 180 – (90 + 60) • = 30º D 30º

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