# Understanding Quadrilaterals

**Exercise 1**

**1.** Given here are some figures:

Classify each of them on the basis of the following

(a) Simple curve (b) Simple closed curve

(c) Polygon (d) Convex polygon

(e) Concave polygon

**2.** How many diagonals does each of the following have?

(a) A convex quadrilateral (b) A regular hexagon (c) A triangle

**3. **What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try)

**4.** Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.)

What can you say about the angle sum of a convex polygon with number of sides?

**5.** What is regular polygon? State the name of a regular polygon of:

(a) 3 sides

(b) 4 sides

(c) 6 sides

**6.**Find the angles measures x in the following figures:

**7.** (a) Find x + y + z (b) Find x + y + z + w

**Exercise 2**

**8. ** Find x in the following figures:

**9. **Find the measure of each exterior angle of a regular polygon of:

(a) 9 sides (b) 15 sides

**10. ** How many sides does a regular polygon have, if the measure of an exterior angle is 24°?

**11. ** How many sides does a regular polygon have if each of its interior angles is 165°?

**12. ** (a) Is it possible to have a regular polygon with of each exterior angle as 22°?

(b) Can it be an interior angle of a regular polygon? Why?

**13. ** (a) What is the minimum interior angle possible for a regular polygon? Why?

(b) What is the maximum exterior angle possible for a regular polygon?

**Exercise 3**

**14.** Given a parallelogram ABCD. Complete each statement along with the definition or property used.

(i) AD = _________

(ii) ∠ DCB = __________

(iii) OC = _________

(iv) *m* ∠DAB + *m* ∠CDA = _______

**15. ** Consider the following parallelograms. Find the values of the unknowns x, y, z.

Note: For getting correct answer, read 3° = 30° in figure (iii)

**16.** Can a quadrilateral ABCD be a parallelogram, if:

(i) ∠ D + ∠ B = 180°?

(ii) AB = DC = 8 cm, AD = 4 cm and BC = 4.4 cm?

(iii) ∠ A = 70° and ∠ C = 65°?

**17.**Draw a rough figure of a quadrilateral that is not a parallelogram but has exactly two opposite angles of equal measures.

**18.** The measure of two adjacent angles of a parallelogram are in the ratio 3 : 2. Find the measure of each of the angles of the parallelogram.

**19.** Two adjacent angles of a parallelogram have equal measure. Find the measure of the angles of the parallelogram.

**20. ** The adjacent figure HOPW is a parallelogram. Find the angle measures x, y and z. State the properties you use to find them.

∠ HOP + 70° =180°

**21. ** The following figures GUNS and RUNS are parallelograms. Find x and y. (Lengths are in cm)

**22.** In the figure, both RISK and CLUE are parallelograms. Find the value of x.

**23.**Explain how this figure is a trapezium. Which is its two sides are parallel?

**25. ** Find the measure of ∠ P and ∠ S if in given figure.

(If you find *m* ∠ R is there more than one method to find *m* ∠ P)

**Exercise 4**

**26. **State whether true or false:

(a) All rectangles are squares.

(b) All rhombuses are parallelograms.

(c) All squares are rhombuses and also rectangles.

(d) All squares are not parallelograms.

(e) All kites are rhombuses.

(f) All rhombuses are kites.

(g) All parallelograms are trapeziums.

(h) All squares are trapeziums.

**27.** Identify all the quadrilaterals that have:

(a) four sides of equal lengths.

(b) four right angles.

**28. ** Explain how a square is:

(a) a quadrilateral (b) a parallelogram

(c) a rhombus (d) a rectangle

**29.** Name the quadrilateral whose diagonals:

(i) bisect each other.

(ii) are perpendicular bisectors of each other.

(iii) are equal.

**30. ** Explain why a rectangle is a convex quadrilateral.

**31. ** ABC is a right-angled triangle and 0 is the mid-point of the side opposite to the right angle. Explain why 0 is equidistant from A, B and C. (The dotted lines are drawn additionally to help you.)