CBSE Class 8 Maths Understanding Quadrilaterals HOTs

HOTS

1. The diagonal of a rectangle is thrice its smaller side. Find the ratio of its sides.
Answer: 2 √2 : 1

2. In the given design, eight isosceles trapeziums surround a regular octagon. What is the measure of ∠B in trapezium ABCD? 

Answer: 45°

3. In the adjoining figure, ABCD is a kite. If ∠BCD=52° and ∠ADB = 42°, find the values of x, y and z.

Answer: x=42°, y=96°, z=64°

4. In the adjoining figure, ABCD is a rhombus and EDC is an equilateral triangle. If ∠DAB = 48°, find  

a. ∠BEC
b. ∠DEB
c. ∠ BFC
Answer: a. 36° b. 24° c. 96°

5. PQRS is a parallelogram. PS is produced to M so that SM=SR and MR is produced to meet PQ produced at N. Prove that QN=QR.

CHALLENGES

1. In an equiangular polygon, the measure of each exterior angle is 25% of the measure of each interior angle. Find the number of sides it has.

2. In a septagon, two of the angles are 130° each and the remaining angles are equal. Find the equal angles.

3. A polygon has n sides. Two of its angles are right angles and each of the remaining angles is 144°. Find the value of n.

4. Is there a polygon which has only two types of interior angles 120° and 60°? If so, how many sides does such a polygon have?
[Hint: If n is the number of sides and if k angles are of 60°, then n–k angles are of 120° and (n–2)180=60k+120(n–k)].

5. In the adjacent figure, the pentagon is such that AB=BC=CD and AE=ED. Moreover, ∠ABC=∠BCD=∠AED=130°. Find the measures of ∠BAE and ∠EDC. 

6. Prove that the perpendicular drawn from the vertex of a regular pentagon to the opposite side bisects that side.

7. For what values of m and n, is it possible for the external angle of a regular m-gon to be equal to the internal angle of a regular n-gon. Check your answer.
[Hint: (n–2) π/n = 2 π/m.]

8. In a convex polygon, sum of all the angles except one is 2280°. How many sides does the polygon have? What is the measure of this exceptional angle?
[Hint: If x is the exceptional angle, then (n–2)180=2280+x, and 0<x<180.]

9. Let ABCD be a parallelogram. The diagonals bisect in E.
a. If AB=6 cm and AD=4 cm, find CD and BC.
b. If DE=5 cm and AE=7 cm, find BD and AC.
c. If ∠DAB=72°, find the measure of ∠C BA.
d. If AD=(x+2y), BC=(2x+3), DC=(x+7) and AB=(3y+2), find AB and BC.

10. In the figure, ∠CAB=60°. ACFG and ABDE are squares. The diagonal FA meets the diagonal BE at H. Find the measure of ∠EAH. 

11. AB and BC are two adjacent sides of an ‘n’ sided regular polygon (see Fig). If ∠ACB= 1/4 ∠ABC , what is n? 

12. If an angle of a parallelogram is two-third of its adjacent angle, what is the smallest angle of the parallelogram?

13. In a parallelogram ABCD, M is the midpoint of BD and BM bisects ∠B. Find ∠AMB.

14. On the sides of a square ABCD, equilateral triangles ABP, BCQ, CDR and DAS are constructed, externally. Show that PQRS is a square.

15. On the sides of a square ABCD, isosceles triangles ABP, BCQ, CDR and DAS, all with apex angle 20°, are constructed, externally. Show that PQRS is a square. Can you generalise this?

16. On the sides of a rectangle ABCD, equilateral triangles ABP, BCQ, CDR and DAS are constructed, externally. Show that PQRS is a rhombus. Show that the result is still true if congruent isosceles triangles are constructed.

17. Suppose ABCD is a square. Let E, F, G, H be the midpoints of AB, BC, CD, DA respectively. Prove that EFGH is a square. What happens if you start with a rectangle ABCD?

18. In a right angled triangle ABC with ∠B=90°, points D, E, F are respectively on AB, BC, CA such that AF=AD and CE=CF. Find ∠EFD.

19. The angle bisectors of ∠A and ∠B of an isosceles trapezium ABCD meet at P and that of ∠C and ∠D meet at Q. Prove that ∠ P+∠Q=180°. What if trapezium ABCD is not isosceles? 

SUMMARY

1. A curve is a set of connected points.
2. A simple curve is a curve that does not intersect itself at any point other than possibly at the endpoints.
3. A closed curve is a curve that begins and ends at the same point.
4. A polygon is a simple closed curve composed of a finite number of line segments.
5. The line segment joining two vertices of a polygon is called an edge or side.
6. A polygon in which all the sides are equal and all the angles are equal is called a regular polygon.
7. Polygons which are not regular are called irregular polygons. That is, their sides and angles are not equal.
8. A polygon is said to be convex if all its diagonals lie inside the polygon.
9. A polygon is said to be concave if at least one of the diagonals lie outside it.
10. Sum of the interior angles of a polygon = Number of triangles in the polygon × 180°
11. Sum of the exterior angles of a polygon is 360°.
12. A quadrilateral in which one pair of opposite sides is parallel is called a trapezium.
13. A quadrilateral in which both the pairs of opposite sides are parallel is called a parallelogram.
14. A parallelogram in which all the angles are right angles is called a rectangle.
15. A parallelogram in which all the sides are equal is called a rhombus.
16. A rectangle in which all the sides are equal is called a square.
17. In a parallelogram
• both pairs of opposite sides are parallel
• both pairs of opposite sides are equal
• diagonals bisect each other
• both pairs of opposite angles are equal
18. In a rectangle the diagonals are equal and they bisect each other.
19. In a rhombus diagonals bisect each other at right angles.
20. In a square diagonals are equal and they bisect each other at right angles.

ERROR ANALYSIS

1. Given, to prove, figure part are not written by the students.
2. Steps of proof with reasoning, must be written.
3. Students do not label the figures properly and also do not draw neat diagrams.
4. Application of the results of geometry must be given sufficient practice.

ACTIVITY I

To verify by paper cutting and pasting, that the sum of the exterior angles drawn in order of any polygon is 360°.
Learning Objective : To understand the exterior angle property of a polygon.
Pre-requisite knowledge: Familiarity with exterior angles of a polygon and a complete angle.
Material Required : Coloured and white sheets of paper, a ruler, a pencil, a pair of scissors and a pair of compasses.
Procedure : a. For triangle

Step 1 : Draw a triangle on a coloured sheet and name it ABC. Make exterior angles in an order at each vertex of this triangle and name them as X, Y and Z. Figure (a).

Step 2 : Cut out all the three exterior angles. Paste them on a white sheet of paper at a point P so that there is no gap between them as shown in Figure (b).

Observations.
1. All angles together form a .................. angle (straight, reflex, complete)
2. The sum of the exterior angles of a triangle taken in order is............. 

Procedure : b. For polygons
Step 3 : Draw a quadrilateral, a pentagon and a hexagon on a coloured sheet of paper. Mark their exterior
angles taken in order at each vertex.
Step 4 : Repeat Step 2 for each of these polygons. [See Figure (f), Figure (g) and Figure (h)]

Observations: 

1. The sum of the exterior angles of a quadrilateral taken in an order is ............
2. The sum of the exterior angles of a pentagon taken in an order is............
3. The sum of the exterior angles of a hexagon taken in an order is............
4. The sum of the exterior angles in each polygon taken in an order is ............

ACTIVITY II

Solve the following crossword puzzle, hints are given below : 

Across 

2. A polygon in which all the interior angles are 1 A polygon in which at least one interior less than 180°.
Answer: Convex Polygon

5. A quadrilateral DEFG in which DE=EF and FG=GD
Answer: Kite

6. A quadrilateral LMNO with MN || OL.
Answer: Trapezium

8. A simple closed rectilinear figure.
Answer: Polygon

9. Quadrilateral WX YZ in which ZW || XY and WX || ZY.
Answer: Parallelogram

10. Rectangle ABCD in which BC=CD
Answer: Square

Down

1. A polygon in which at least one interior angle is more than 180°
Answer: Concave Polygon

3. An equilateral triangle is a _________ polygon.
Answer: Regular

4. Line segment AC in a quadrilateral ABCD
Answer: Diagonal

7. Parallelogram ABCD in which ∠B=90°
Answer: Rectangle

11. Parallelogram PQRS with RS=SP.
Answer: Rhombus