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TMUA 2023 · Paper 2 · Question 9 of 20

TMUA 2023 Paper 2 Question 9

Logic and arguments — Converse and contrapositive. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Logic and argumentsConverse and contrapositive8 options
Consider the following statement about a pentagon P:

(×) If at least one of the interior angles in P is 108∘, then all the interior angles in P form an arithmetic sequence.

Which of the following is/are true?

I   The statement (×)
II  The contrapositive of (×)
III The converse of (×)

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · D
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
The interior angles of a pentagon total 540∘. The statement itself is false: the angles 108∘,  100∘,  110∘,  110∘,  112∘ sum to 540∘ and include a 108∘ angle, but form no arithmetic sequence. A contrapositive is equivalent to its statement, so II is false too. The converse is true: five terms in arithmetic progression can be written a−2d,  a−d,  a,  a+d,  a+2d, summing to 5a = 540, so the middle term is 108∘.