Appendix 1: Proofs in Mathematics | A1.2

Question 5

Given that ABCD is a parallelogram and ∠B=80∘\angle B = 80^\circ. What can you conclude about the other angles of the parallelogram?

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Solution
Understand the Question
  • In a parallelogram:
    • Opposite angles are equal: ∠D=∠B\angle D = \angle B and ∠C=∠A\angle C = \angle A.
    • Adjacent (consecutive) angles are supplementary: They add up to 180∘180^\circ (e.g., ∠A+∠B=180∘\angle A + \angle B = 180^\circ).
  • Given ∠B=80∘\angle B = 80^\circ, we can find the remaining angles directly using these two geometric properties.

Step 1 · Find ∠D\angle D

In a parallelogram, opposite angles are equal.Diagram 1

Given ∠B=80∘\angle B = 80^\circ ∠D=∠B=80∘\angle D = \angle B = 80^\circ

Step 2 · Find ∠A\angle A

Consecutive angles in a parallelogram are supplementary:

∠A+∠B=180∘∠A+80∘=180∘∠A=180∘−80∘=100∘\begin{aligned} \angle A + \angle B &= 180^\circ \\ \angle A + 80^\circ &= 180^\circ \\ \angle A &= 180^\circ - 80^\circ \\ &= 100^\circ \end{aligned}

Step 3 · Find ∠C\angle C

Since opposite angles are equal: ∠C=∠A=100∘\angle C = \angle A = 100^\circ

Answer

∠A=100∘\angle A = 100^\circ, ∠C=100∘\angle C = 100^\circ, and ∠D=80∘\angle D = 80^\circ

Common Mistakes
  • Opposite vs. Adjacent Angle Rules: Confusing opposite angles (which are equal) with consecutive adjacent angles (which sum to 180∘180^\circ).
  • Incorrect Angle Sum Assumption: Assuming all angles in a parallelogram must be equal to 90∘90^\circ, which is only true for rectangles and squares.

More questions in A1.2

Q1

Given that all women are mortal, and suppose that A is a woman, what can we conclude about A?

Q2

Given that the product of two rational numbers is rational, and suppose aa and bb are rationals, what can you conclude about abab?

Q3

Given that the decimal expansion of irrational numbers is non-terminating, non-recurring, and 17\sqrt{17} is irrational, what can we conclude about the decimal expansion of 17\sqrt{17}?

Q4

Given that y=x2+6y = x^2 + 6 and x=−1x = -1, what can we conclude about the value of yy?

Q5

Given that ABCD is a parallelogram and ∠B=80∘\angle B = 80^\circ. What can you conclude about the other angles of the parallelogram?

Q6

Given that PQRSPQRS is a cyclic quadrilateral and also its diagonals bisect each other. What can you conclude about the quadrilateral?

Q7

Given that p\sqrt{p} is irrational for all primes pp and also suppose that 3721 is a prime. Can you conclude that 3721\sqrt{3721} is an irrational number? Is your conclusion correct? Why or why not?

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