Quadrilaterals | FIO

Question 18

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.

Question diagram 1
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Solution
Understand the Question
  • Any quadrilateral can be divided into two triangles by drawing a single diagonal.
  • Since the sum of interior angles in any triangle is 180∘180^\circ, the sum of all interior angles in the quadrilateral is the sum of angles of both triangles: 180∘+180∘=360∘180^\circ + 180^\circ = 360^\circ.
  • This geometric property holds true for all quadrilaterals (both convex and concave), which can also be verified empirically by drawing the figure and measuring each angle with a protractor.

Step 1 · Divide the Quadrilateral into Two Triangles

Draw a diagonal BD\text{BD} connecting vertices B\text{B} and D\text{D} in quadrilateral ABCD\text{ABCD}. This divides ABCD\text{ABCD} into two triangles: △ABD\triangle \text{ABD} and △CBD\triangle \text{CBD}.Diagram 1

Using the angle sum property of a triangle (sum of interior angles is 180∘180^\circ):

In △ABD\triangle \text{ABD}: ∠A+∠ABD+∠ADB=180∘…(1)\angle A + \angle ABD + \angle ADB = 180^\circ \quad \dots (1)

In △CBD\triangle \text{CBD}: ∠C+∠CBD+∠CDB=180∘…(2)\angle C + \angle CBD + \angle CDB = 180^\circ \quad \dots (2)

Step 2 · Sum the Angles of the Quadrilateral

Adding equations (1)(1) and (2)(2): (∠A+∠ABD+∠ADB)+(∠C+∠CBD+∠CDB)=180∘+180∘(\angle A + \angle ABD + \angle ADB) + (\angle C + \angle CBD + \angle CDB) = 180^\circ + 180^\circ

Rearranging the terms: ∠A+(∠ABD+∠CBD)+∠C+(∠ADB+∠CDB)=360∘\angle A + (\angle ABD + \angle CBD) + \angle C + (\angle ADB + \angle CDB) = 360^\circ

From the figure:

  • ∠B=∠ABD+∠CBD\angle B = \angle ABD + \angle CBD
  • ∠D=∠ADB+∠CDB\angle D = \angle ADB + \angle CDB

Substituting these into the equation: ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^\circ

Step 3 · Verification by Construction and Measurement

  1. Construct the quadrilateral on paper according to the given shape.
  2. Measure each interior angle (∠A,∠B,∠C,∠D\angle A, \angle B, \angle C, \angle D) using a protractor.
  3. Adding the four measured angles yields ∠A+∠B+∠C+∠D≈360∘\angle A + \angle B + \angle C + \angle D \approx 360^\circ (any slight variance is due to minor measurement error).
Answer

Yes, the sum of the angles in the quadrilateral is 360∘360^\circ.

Common Mistakes
  • Convex vs. Non-Convex Confusion: Assuming the angle sum property of 360∘360^\circ only applies to convex quadrilaterals; it applies to all simple quadrilaterals.
  • Diagonal Selection in Non-Convex Shapes: In a non-convex (concave) quadrilateral, ensure the diagonal is chosen so it lies completely inside the figure to cleanly split it into two triangles.
  • Measurement Errors: Misaligning the baseline of the protractor when measuring reflex or obtuse interior angles.

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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