Quadrilaterals | FIO

Question 16

If a quadrilateral has four equal sides and one angle of 90∘90^\circ, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.

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Solution
Understand the Question
  • A square is defined as a quadrilateral with four equal sides and four right angles (90∘90^\circ).
  • A quadrilateral with four equal sides is a rhombus.
  • To determine if having just one 90∘90^\circ angle makes it a square, we need to prove geometrically whether the remaining three angles must also equal 90∘90^\circ, and then confirm this by construction and measurement.

If a quadrilateral has four equal sides and one angle of 90∘90^\circ, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.

Step 1 · Analyze △ABD\triangle \text{ABD} using Angle Sum Property

Let ABCD\text{ABCD} be a quadrilateral with all equal sides: AB=BC=CD=DA\text{AB} = \text{BC} = \text{CD} = \text{DA}

Given that one angle is 90∘90^\circ, let ∠DAB=90∘\angle \text{DAB} = 90^\circ. Draw diagonal BD\text{BD}.Diagram 1

In △ABD\triangle \text{ABD}, since AB=AD\text{AB} = \text{AD}, △ABD\triangle \text{ABD} is an isosceles right-angled triangle. Therefore, ∠ABD=∠ADB=x\angle \text{ABD} = \angle \text{ADB} = x.

∠ABD+∠ADB+∠DAB=180∘∠ABD+∠ADB+90∘=180∘∠ABD+∠ADB=180∘−90∘∠ABD+∠ADB=90∘\begin{aligned} \angle \text{ABD} + \angle \text{ADB} + \angle \text{DAB} &= 180^\circ \\[0.6em] \angle \text{ABD} + \angle \text{ADB} + 90^\circ &= 180^\circ \\[0.6em] \angle \text{ABD} + \angle \text{ADB} &= 180^\circ - 90^\circ \\[0.6em] \angle \text{ABD} + \angle \text{ADB} &= 90^\circ \end{aligned} x+x=90∘2x=90∘x=90∘2=45∘\begin{aligned} x + x &= 90^\circ \\[0.6em] 2x &= 90^\circ \\[0.6em] x &= \dfrac{90^\circ}{2} = 45^\circ \end{aligned}

∠ABD=45∘,∠ADB=45∘\angle \text{ABD} = 45^\circ, \quad \angle \text{ADB} = 45^\circ

Step 2 · Prove Triangle Congruence and Determine All Angles

Comparing △ABD\triangle \text{ABD} and △CBD\triangle \text{CBD}:

  • AB=CB\text{AB} = \text{CB} (equal sides)
  • AD=CD\text{AD} = \text{CD} (equal sides)
  • BD=BD\text{BD} = \text{BD} (common side)

By SSS congruence criterion: △ABD≅△CBD\triangle \text{ABD} \cong \triangle \text{CBD}

Corresponding angles of congruent triangles are equal (CPCT): ∠BCD=∠BAD=90∘\angle \text{BCD} = \angle \text{BAD} = 90^\circ ∠CBD=∠ABD=45∘\angle \text{CBD} = \angle \text{ABD} = 45^\circ ∠CDB=∠ADB=45∘\angle \text{CDB} = \angle \text{ADB} = 45^\circ

Now, calculate the remaining vertex angles:

∠ABC=∠ABD+∠CBD=45∘+45∘=90∘\begin{aligned} \angle \text{ABC} &= \angle \text{ABD} + \angle \text{CBD} \\[0.6em] &= 45^\circ + 45^\circ \\[0.6em] &= 90^\circ \end{aligned} ∠ADC=∠ADB+∠CDB=45∘+45∘=90∘\begin{aligned} \angle \text{ADC} &= \angle \text{ADB} + \angle \text{CDB} \\[0.6em] &= 45^\circ + 45^\circ \\[0.6em] &= 90^\circ \end{aligned}

Since all four sides are equal and all four interior angles are 90∘90^\circ, quadrilateral ABCD\text{ABCD} is a square.

Step 3 · Verify by Construction and Measurement

Construction Steps:

  1. Draw a line segment AB=5 cm\text{AB} = 5\text{ cm}.
  2. At point A\text{A}, construct a ray making an angle of 90∘90^\circ with AB\text{AB}.
  3. Mark point D\text{D} on this ray such that AD=5 cm\text{AD} = 5\text{ cm}.
  4. From point D\text{D}, draw an arc of radius 5 cm5\text{ cm}.
  5. From point B\text{B}, draw an arc of radius 5 cm5\text{ cm} intersecting the previous arc at point C\text{C}.
  6. Join BC\text{BC} and CD\text{CD} to complete quadrilateral ABCD\text{ABCD}.Diagram 2

Measurement:

  • Sides: AB=BC=CD=DA=5 cm\text{AB} = \text{BC} = \text{CD} = \text{DA} = 5\text{ cm}
  • Angles: ∠DAB=∠ABC=∠BCD=∠CDA=90∘\angle \text{DAB} = \angle \text{ABC} = \angle \text{BCD} = \angle \text{CDA} = 90^\circ

Measurement confirms that the resulting figure is a square.

Answer

Yes, a quadrilateral with four equal sides and one angle of 90∘90^\circ will always be a square.

Common Mistakes
  • Assuming 4 Equal Sides Imply a Square: Four equal sides define a rhombus, which may have non-right angles. However, if any one angle of a rhombus is 90∘90^\circ, all angles become 90∘90^\circ, making it a square.
  • Overlooking Construction Rules: When constructing vertex C\text{C}, ensure both arcs are drawn with the exact side length (5 cm5\text{ cm}) from B\text{B} and D\text{D}.

More questions in FIO

Q1

Find all the other angles inside the following rectangles.

Q2

Draw a quadrilateral whose diagonals have equal lengths of 8 cm8\text{ cm} that bisect each other, and intersect at an angle of

(i) 30∘30^\circ

(ii) 40∘40^\circ

(iii) 90∘90^\circ

(iv) 140∘140^\circ

Q3

Consider a circle with centre OO. Line segments PLPL and AMAM are two perpendicular diameters of the circle. What is the figure APMLAPML? Reason and/or experiment to figure this out.

Q4

We have seen how to get 90∘90^\circ using paper folding. Now, suppose we do not have any paper but two sticks of equal length, and a thread. How do we make an exact 90∘90^\circ using these?

Q5

We saw that one of the properties of a rectangle is that its opposite sides are parallel. Can this be chosen as a definition of a rectangle? In other words, is every quadrilateral that has opposite sides parallel and equal, a rectangle?

Q6

Find the remaining angles in the following quadrilaterals.

Q7

Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm7\text{ cm} and 5 cm5\text{ cm}, and intersect at an angle of 140∘140^\circ.

Q8

Using the diagonal properties, construct a rhombus whose diagonals are of lengths 4 cm and 5 cm.

Q9

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm4\text{ cm}.

Q10

Construct a kite whose diagonals are of lengths 6 cm6\text{ cm} and 8 cm8\text{ cm}.

Q11

Find the remaining angles in the following trapeziums—

Q12

Draw a Venn diagram showing the set of parallelograms, kites, rhombuses, rectangles, and squares. Then, answer the following questions—

(i) What is the quadrilateral that is both a kite and a parallelogram?

(ii) Can there be a quadrilateral that is both a kite and a rectangle?

(iii) Is every kite a rhombus? If not, what is the correct relationship between these two types of quadrilaterals?

Q13

If PAIR and RODS are two rectangles, find ∠IOD\angle \text{IOD}.

Q14

Construct a square with diagonal 6 cm6\text{ cm} without using a protractor.

Q15

CASE is a square. The points U, V, W and X are the midpoints of the sides of the square. What type of quadrilateral is UVWX? Find this by using geometric reasoning, as well as by construction and measurement. Find other ways of constructing a square within a square such that the vertices of the inner square lie on the sides of the outer square, as shown in Figure (b).

Q16

If a quadrilateral has four equal sides and one angle of 90∘90^\circ, will it be a square? Find the answer using geometric reasoning as well as by construction and measurement.

Q17

What type of a quadrilateral is one in which the opposite sides are equal? Justify your answer.

Hint: Draw a diagonal and check for congruent triangles.

Q18

Will the sum of the angles in a quadrilateral such as the following one also be 360∘360^\circ? Find the answer using geometric reasoning as well as by constructing this figure and measuring.

Q19

State whether the following statements are true or false. Justify your answers.

(i) A quadrilateral whose diagonals are equal and bisect each other must be a square.

(ii) A quadrilateral having three right angles must be a rectangle.

(iii) A quadrilateral whose diagonals bisect each other must be a parallelogram.

(iv) A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.

(v) A quadrilateral in which the opposite angles are equal must be a parallelogram.

(vi) A quadrilateral in which all the angles are equal is a rectangle.

(vii) Isosceles trapeziums are parallelograms.

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