Quadrilaterals | FIO

Question 13

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

Solution

We will use the properties of rectangles and the sum of angles in a triangle to find the required angle.

Step 1 — Understand the rectangle PAIR

We are told that PAIR is a rectangle. This means all its interior angles are right angles, which are $90^\circ$ . So, the angle at vertex I, $\angle AIR$ , is $90^\circ$ . Since point O lies on the line segment AI, the angle $\angle RIO$ is also $90^\circ$ . This tells us that triangle RIO is a right-angled triangle (a triangle with one $90^\circ$ angle).

Diagram 1

Step 2 — Find angle ROI

In any triangle, the sum of all three interior angles is always $180^\circ$ . For our triangle RIO, we know two angles: $\angle RIO = 90^\circ$ and $\angle ORI = 30^\circ$ . Let us find the third angle, $\angle ROI$ .

$\angle RIO + \angle ORI + \angle ROI = 180^\circ$

$90^\circ + 30^\circ + \angle ROI = 180^\circ$

$120^\circ + \angle ROI = 180^\circ$

$\angle ROI = 180^\circ - 120^\circ$

$\boxed{60^\circ}$

Step 3 — Understand the rectangle RODS

We are also told that RODS is a rectangle. Just like PAIR, all interior angles of rectangle RODS are $90^\circ$ . So, the angle at vertex O, $\angle ROD$ , is $90^\circ$ .

Diagram 2

Step 4 — Find angle IOD

From the diagram, we can see that the angle $\angle ROD$ is made up of two smaller angles: $\angle ROI$ and $\angle IOD$ . So, we can write:

$\angle ROD = \angle ROI + \angle IOD$

We know that $\angle ROD = 90^\circ$ (from Step 3) and we found $\angle ROI = 60^\circ$ (from Step 2). Let us substitute these values into the equation.

$90^\circ = 60^\circ + \angle IOD$

To find $\angle IOD$ , we subtract $60^\circ$ from both sides.

$\angle IOD = 90^\circ - 60^\circ$

$\boxed{30^\circ}$

Answer

(i) $\angle\text{IOD} = \mathbf{30^\circ}$

More questions in FIO

Find all the other angles inside the following rectangles.

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

(i) 30° (ii) 40° (iii) 90° (iv) 140°

Consider a circle with centre O. Line segments PL and AM are two perpendicular diameters of the circle. What is the figure APML? Reason and/or experiment to figure this out.

We have seen how to get 90° 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° using these?

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?

Find the remaining angles in the following quadrilaterals.

Using the diagonal properties, construct a parallelogram whose diagonals are of lengths 7 cm and 5 cm, and intersect at an angle of 140°.

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

Find all the sides and the angles of the quadrilateral obtained by joining two equilateral triangles with sides 4 cm.

Q10

Construct a kite whose diagonals are of lengths 6 cm and 8 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 $\angle\text{IOD}$ .

Q14

Construct a square with diagonal 6 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°, 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°? 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.