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Theorems: vertically opposite angles

Theorems: vertically opposite angles. Draw one straight line crossing another straight line. Measure all 4 angles. C A B D Angle A = Angle B Angle C = Angle D Theorem

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Theorems: vertically opposite angles

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  1. Theorems: vertically opposite angles • Draw one straight line crossing another straight line. Measure all 4 angles. • C A B D Angle A = Angle B Angle C = Angle D Theorem • Vertically opposite angles are equal

  2. Isosceles triangles • Draw an isosceles triangle Theorem • In isosceles triangle the angles opposite the equal sides are equal. • Conversely if 2 angles of a triangle are equal the triangle is isosceles. • Angle A = Angle B A B

  3. Transversals Theorem • If a transversal makes equal alternate angles on 2 lines the lines are parallel. • Conversely if 2 lines are parallel then a transversal makes equal alternate angles • Angle 4 = Angle 6 etc.

  4. Transversals • Theorem • Two lines are parallel if, and only if, for any transversal the corresponding angles are equal. • Eg angle 3 = angle 7 (corresponding angles)

  5. Triangle angles • Draw a triangle. • Using your protractor measure all angles Theorem • All angles of a triangle add up to 180o

  6. Exterior angles Theorem • The exterior angle in a triangle is equal to the opposite interior angles. • ie angle d = angle a + angle c • The exterior angle is d

  7. Properties of Parallelograms Theorem • In a parallelogram opposite sides are equal and opposite angles are equal. • Conversely if the opposite angles and opposite sides of a quadrilateral are the same the shape is a parallelogram. • eg

  8. More properties of Parallelograms Theorem • The diagonals of a parallelogram bisect each other.

  9. 3 parallel lines Theorem • If three parallel lines cut off equal segments on some transversal they will cut off equal segments on any other transversal. • If lDEl = lFEl then lCBl = lBAl

  10. Triangles Theorem • Let ABC be a triangle. If a line is parallel to BC and cuts AB in a ratio s:t then it also cuts AC in the same ratio. • so if lBDl is half the size of lADl then lCEl is half the size of lEAl The converse is true

  11. More triangles • Theorem • If 2 triangles are similar the their sides are proportional, in order.

  12. Pythagoras theorem • In a right angled triangle the square of the hypotenuse is the sum of the squares o the other 2 sides. • Right angled triangle

  13. Converse of Pythagoras theorem Theorem • If the square of one side of a triangle is the sum of the squares on the other 2 sides then the angle opposite the first side is a right angle. • Right angled triangle

  14. Circle • The angle at the centre of a circle standing on a given arc is twice the angle at an point of the circle standing on the same arc.

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