What is the difference between a rhombus and a square?
No, see solution.
Practice makes perfect
We will investigate whether a rhombus that is not a square can be inscribed in a circle. Let's first remember the definitions of a rhombus and a square.
Square: A square is a parallelogram with for congruent sides and four right angles.
As we can see, the only difference between a rhombus and a square is their interior angles. Since both of them are special types of parallelograms, we can continue by determining the type of parallelogram that be inscribed in a circle.
Let's visualize the theorem. To see how the angles change, we can move vertices A, B, C, and D. Let's try!
As we move the vertices of the quadrilateral, we can see that the opposite angles add up to 180^(∘). This is a result of Corollary 3.
m∠ A + m∠ C =180^(∘)
m∠ B + m∠ D =180^(∘)
Opposite Angles of Inscribed Parallelograms
Since we want ABCD to be a parallelogram, we will also consider Theorem 6-5.
Theorem 6-5
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
We can examine a visual expression of this theorem for a better understanding. Let ABCD be a parallelogram.
∠ A≅ ∠ C ⇔ m∠ A = m∠ C
∠ B≅ ∠ D ⇔ m∠ B = m∠ D
If we combine the corollary and the theorem, we can conclude that each interior angle of the parallelogram must be a right angle.
c|c
m∠ A+m∠ C=180 m∠ A= m∠ C & m∠ B+m∠ D=180 m∠ B= m∠ D
⇓ & ⇓
m∠ A = m∠ C= 90 & m∠ B = m∠ D=90
Conclusion
By the definition of a rectangle, a parallelogram inscribed in a circle must be a rectangle. Note that it could also be a square because a square is a special type of rectangle. As a result, we cannot inscribe a rhombus that is not a square in a circle.
