Specialist Maths Geometry Proofs Week 2

Parallel Lines Corresponding Angle Alternate Angles Allied or Co-interior Angles a a a a a b a + b = 180 0

Triangles a b c a + b + c = 180 o a b c Exterior angle of a triangle c = a + b a a b b Isosceles Triangle

Quadrilaterals ab cd a + b + c + d = 360 o a ab b a a a a b b b b

Congruent triangles SSSSAS AA cor S RHS a aa a b b

Similar Triangles a c b kc kb ka To prove triangles are similar, show two set of angles are equal, then the third set must also be equal.

Circle Theorems a 2aa a Angle in a semi circle Chord of a circle Tangent-radius Tangents from external point Angle at centre Angles subtended from same arc

Intersecting Chords A A A B B B C C D D X X X P AX.BX =(PX) 2 AX.BX = CX.DX

More Theorems A B M N Mid Point Theorem MN is parallel to AB and half its length a a Angle between tangent & chord b a a+ b = 180 o Angles in Cyclic Quadrilateral

Tests for Cyclic quadrilaterals or Concyclic Points a b b a Test 1 Show a + b = 180 o Show a = b Test 2

Example 6 Ex 5B1) D C B A E α α x cm4 cm 2 cm

Solution 6 C B A E α α x cm4 cm 2 cm D

Example 7 (Ex 5B1) O P Q R αβ (PQ is tangent)

Solution 7 (PQ is tangent) O P Q R αβ

Example 8 (Ex 5B1) P Y R M Q X

Solution 8 Y R M Q X P

Example 9 (Ex 5B2) A B C D E F α β

Solution 9 A B C D E F α β α β β

Example 10 (Ex 5B2)

Solution 10

Example 11(Ex 5B2) X Y Z A B

Solution 11

This Week Test book pages 184 to 202 Exercise 5B1 questions 1-14 Exercise 5B2 questions 1-20 Review Sets 5A – 5C