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By Haworth A.H.

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A learner working individually can be too easily self-convinced! Plenary (15 min) There are 23 = 8 possible three-letter triads of the letters A and S, and they have the following properties: • AAA ⇒ similarity • AAS, ASA, SAA ⇒ congruence • SAS ⇒ congruence • SSA, ASS ⇒ two possibilities • SSS ⇒ congruence When you know any two angles of a triangle, you effectively know all three angles, since you can calculate the third angle by using the fact that the sum of the interior angles of a triangle is 180°.

3 + π) + (3 – π) = 6). , π – π = 0). , 2 8 = 16 = 4). , 12 = 4 = 2). 1 a is always irrational, provided a ≠ 0 (which it cannot be, since a is irrational), because if it were 2 3 1 m a n rational then it could be expressed as = , where m and n are integers with n ≠ 0, and inverting n m 1 a both sides of this would give a = , so a would be rational, which is a contradiction. So cannot be rational if a is irrational. This is another ‘proof by contradiction’ argument, similar to the one above. , ( 3 ) = 3).

This observation is TEACH ER SH EET Resources for Teaching Mathematics: 14–16 critical for proving the circle theorems. Plenary (20 min) This phase of the lesson would work well as a collecting and summarizing exercise, with some emphasis on proof, depending on how you gauge the lesson to have proceeded. Theorems likely to be encountered are ‘the angle in a semicircle is 90°’, ‘the angle at the centre is twice the angle at the circumference’, ‘angles in the same segment are equal’ and ‘a radius and a tangent meet at 90°’.

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Synopsis plantarum succulentarum by Haworth A.H.

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