Opposite sides of a parallelogram
The strips slide and fit exactly.
Equal opposite sides?
In a rectangle the opposite sides are equal. A leaning parallelogram looks as if its opposite sides are equal too. But looks can fool us.
Are the opposite sides of a leaning parallelogram equal?
What this lesson covers
The idea
The opposite sides of a parallelogram are equal, as shown by congruent triangles (AAS) formed by a diagonal.
Equal opposite sides?
In a rectangle the opposite sides are equal. A leaning parallelogram looks as if its opposite sides are equal too. But looks can fool us.
Are the opposite sides of a leaning parallelogram equal?
- Yes, always
- Only when it is a rectangle
- No, never
Slide the strips
The orange strip is a copy of AB. Slide it along the blue rails. Where does it land? Then slide the blue strip, a copy of AD, along the orange rails. Change the shape and try again.
Equal, by congruence
The opposite sides of a parallelogram are equal, as shown by congruent triangles (AAS) formed by a diagonal.
Both strips fitted exactly, in every shape. Why? Join the diagonal BD and compare ∆ABD and ∆CDB:
∠A = ∠C (opposite angles of a parallelogram), ∠ADB = ∠CBD (alternate angles, AD ∥ BC), and BD is common. So ∆ABD ≅ ∆CDB by AAS.
Corresponding sides are equal: AB = CD and AD = CB.
Notes
The opposite sides of a parallelogram are equal, as shown by congruent triangles (AAS) formed by a diagonal.
Check yourself
In parallelogram ABCD, AB = 7 cm and BC = 4 cm. How long is CD?
Answer: 7 cm
CD is opposite AB, and opposite sides of a parallelogram are equal. So CD = AB = 7 cm.
A parallelogram has neighbouring sides of 8 cm and 5 cm. What is its perimeter?
Answer: 26 cm
The sides are 8, 5, 8 and 5, because opposite sides are equal. 8 + 5 + 8 + 5 = 26 cm.
In parallelogram PQRS, PQ = 2x + 3 and SR = 15. What is x?
Answer: 6
Opposite sides are equal, so 2x + 3 = 15. Then 2x = 12 and x = 6.
In the proof that AB = CD, why are ∆ABD and ∆CDB congruent?
- ∠A = ∠C, ∠ADB = ∠CBD and BD is common: AAS — correct. Yes! Two pairs of equal angles and a common side make the triangles congruent by AAS.
- Because AB = CD. That is what we want to prove. We cannot use it to show the triangles congruent.
- Because the triangles look the same. Looking the same is not a proof. We need two angles and a side: AAS.