True or false?
- (i) A parallelogram with a right angle is a rectangle.
- (ii) A rhombus with perpendicular diagonals is a square.
- (iii) If the diagonals of a parallelogram are equal, then it is a rectangle.
Step-by-step solution
Idea: Use what a parallelogram already gives: opposite angles equal and adjacent angles adding to 180°. One right angle then spreads to all four. For (ii), remember that perpendicular diagonals are a property of every rhombus, so they add nothing new.
(i) A parallelogram with a right angle is a rectangle.
- Let ∠A = 90° in parallelogram ABCD. Adjacent angles add to 180°, so ∠B = ∠D = 90°, and opposite angles are equal, so ∠C = ∠A = 90°.½ mark
- A parallelogram with all angles 90° is a rectangle. True.½ mark
(ii) A rhombus with perpendicular diagonals is a square.
- The diagonals of every rhombus are perpendicular. So the statement would make every rhombus a square.½ mark
- A rhombus with angles 60° and 120° (picture) has perpendicular diagonals but no right angle, so it is not a square. False. (A rhombus with equal diagonals is a square.)½ mark
(iii) If the diagonals of a parallelogram are equal, then it is a rectangle.
- In parallelogram ABCD with AC = BD: in ∆ABC and ∆BAD, AB is common, BC = AD (opposite sides), AC = BD (given). So ∆ABC ≅ ∆BAD (SSS) and ∠ABC = ∠BAD.½ mark
- But ∠ABC + ∠BAD = 180° (adjacent angles), so each is 90°. A parallelogram with a right angle is a rectangle. True.½ mark
Answer to write in the exam
(i)
ABCD parallelogram, ∠A = 90°
∠A + ∠B = 180° ⇒ ∠B = 90°; ∠C = ∠A = 90°, ∠D = ∠B = 90°
∴ All angles 90° ⇒ rectangle. True.
(ii)
Diagonals of every rhombus are perpendicular
A rhombus with angles 60°, 120° has perpendicular diagonals but is not a square
∴ False.
(iii)
In ∆ABC and ∆BAD: AB common, BC = AD, AC = BD ⇒ ∆ABC ≅ ∆BAD (SSS)
⇒ ∠B = ∠A; ∠A + ∠B = 180° ⇒ ∠A = ∠B = 90°
∴ Rectangle. True.
Common mistakes that cost marks
- Answering (ii) “true” because a square has perpendicular diagonals. The question is about the converse direction, and every rhombus has perpendicular diagonals.
- In (iii), dropping the word “parallelogram”: a quadrilateral with equal diagonals need not be a rectangle (an isosceles trapezium).
- In (i), checking only that ∠C = 90° and stopping. Show all four angles are right angles.
How this can come in the exam
Which of these is enough to make a rhombus a square?
- Its diagonals bisect each other
- Its diagonals are perpendicular
- Its diagonals are equal
- Its opposite angles are equal
Show answer
(C) Its diagonals are equal
A rhombus is a parallelogram, and a parallelogram with equal diagonals is a rectangle; a rhombus that is a rectangle is a square. The other three hold for every rhombus.
Try one yourself
True or false: a parallelogram whose diagonals are equal and perpendicular is a square.
Show answer
True. Equal diagonals make it a rectangle; perpendicular diagonals that bisect each other make all sides equal (a rhombus); a rectangle that is a rhombus is a square.
More questions like this
- The diagonal AC of a parallelogram ABCD bisects ∠A. Show that it also bisects ∠C and that ABCD is a rhombus.
- The following questions examine converses of true properties. Answer them with Yes or No. If your answer is No, what extra condition can you add so that the answer becomes Yes?
- Let ABCD be a parallelogram with AB ≠ BC. Show that the pairwise intersection points of the four angle bisectors form the vertices of a rectangle (see the figure). Why did we assume AB ≠ BC?
- On a piece of paper, draw and then cut out two copies of the same triangle. Keep one copy aside. On the other, mark the midpoint of each side. (You can do this by folding the paper to join two vertices at a time.) Draw the triangle made by the three midpoints and cut the paper along each of these lines. Now you have four smaller triangles. Compare them to each other and to the intact copy of the original triangle. What do you notice?
- Try to prove directly that the four smaller triangles are congruent. You will see that no congruence test applies, because many quantities are unknown.
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