Quadrilaterals

 HEGP104 — Quadrilaterals

Complete Answers to the Mathematical Problems

This guide answers the mathematical problems, construction tasks, angle questions, activities, and “Figure it Out” exercises in the attached Class 8 chapter.

 

1. Rectangle and square investigations

Problem 1. A rectangle has one diagonal of length 8 cm. What is the length of the other diagonal?

The diagonals of a rectangle are equal. Therefore, the other diagonal is 8 cm.

 

Problem 2. Where do the diagonals of a rectangle intersect?

The diagonals of a rectangle bisect each other. Therefore, they intersect at their common midpoint. Since each diagonal is 8 cm, each half-diagonal is 4 cm.

 

Problem 3. What angle must the diagonals of a rectangle make with each other?

There is no single fixed angle. The diagonals of a rectangle are equal and bisect each other, but the angle between them depends on the rectangle’s length and breadth. They are perpendicular only when the rectangle is a square.

 

Problem 4. If equal diagonals bisect each other at any angle, what quadrilateral is formed?

A quadrilateral whose diagonals are equal and bisect each other is a rectangle. If the angle between the diagonals is 90°, it is a square.

 

Problem 5. If one angle between the diagonals is (x), what are the four angles at their intersection?

The four angles are

 

[ x,\quad x,\quad 180^\circ-x,\quad180^\circ-x. ]

 

For example, if one angle is 60°, the four angles are 60°, 120°, 60°, and 120°.

 

Problem 6. What is the value of each base angle in the isosceles triangle formed by two half-diagonals?

If the vertex angle is (x), the two equal base angles are

 

[ \frac{180^\circ-x}{2}=90^\circ-\frac{x}{2}. ]

 

These angles show that the resulting quadrilateral has four right angles and is therefore a rectangle.

 

Problem 7. What additional properties do the diagonals of a square have?

The diagonals of a square are equal, bisect each other, and intersect at right angles. They also bisect the angles of the square. Thus, if one diagonal is 8 cm, the other is also 8 cm, their intersection is the midpoint of both, and the angle between them is 90°.

 

2. Figure it Out: rectangles and diagonals

Question 1. Find the remaining angles inside the rectangles.

(i) Rectangle ABCD

Using the given 30° angle and the fact that a rectangle has right angles:

 

[ \angle ABD=30^\circ, ] [ \angle CAD=60^\circ, ] [ \angle ADB=60^\circ, ] [ \angle BDC=30^\circ, ] [ \angle ACD=30^\circ, ] [ \angle ACB=60^\circ. ]

 

(ii) Rectangle PQRS

Using the given 110° angle, vertically opposite angles, supplementary adjacent angles, and the equal-diagonal property:

 

[ \angle POS=110^\circ, ] [ \angle QOP=70^\circ, ] [ \angle ROS=70^\circ, ] [ \angle OQR=35^\circ, ] [ \angle ORQ=35^\circ, ] [ \angle OQP=55^\circ, ] [ \angle OPQ=55^\circ, ] [ \angle ORS=55^\circ, ] [ \angle OSR=55^\circ. ]

 

Question 2. Construct a quadrilateral whose diagonals are equal, 8 cm long, bisect each other, and intersect at 30°, 40°, 90°, or 140°.

The construction is the same in every case:

 

1      Draw (AB=8) cm.

2      Mark its midpoint (O), so (AO=OB=4) cm.

3      At (O), construct the required angle with one ray along (OB): 30°, 40°, 90°, or 140°.

4      On the new ray and its opposite ray, mark (OC=OD=4) cm.

5      Join (A-C), (C-B), (B-D), and (D-A).

 

The resulting quadrilateral is a rectangle in all four cases. For the 90° case, it is a square.

 

Question 3. A circle has two perpendicular diameters (PL) and (AM). What is quadrilateral APML?

All four vertices lie on the circle, the diagonals are diameters of equal length, they bisect each other at the centre, and they are perpendicular. Therefore, APML is a square.

 

Question 4. How can two equal sticks and a thread be used to make an exact right angle?

Place the two equal sticks so that their midpoints coincide. Treat them as the diagonals of a quadrilateral. Join their four endpoints with a thread. The diagonals are equal and bisect each other, so the quadrilateral is a rectangle. Every rectangle has right angles; therefore, the angle at each vertex is exactly 90°.

 

Question 5. Can opposite sides being parallel and equal be used as the definition of a rectangle?

No. A quadrilateral with both pairs of opposite sides parallel and equal is a parallelogram, but it need not have right angles. A slanted parallelogram is not a rectangle. A rectangle is a parallelogram with four right angles.

 

3. Parallelograms and rhombuses

Problem 6. What are the main properties of a parallelogram?

In a parallelogram:

 

[ AB\parallel CD,\qquad AD\parallel BC, ]

 

opposite sides are equal, opposite angles are equal, adjacent angles are supplementary, and the diagonals bisect each other.

 

Problem 7. How can a parallelogram with diagonals 7 cm and 5 cm intersecting at 140° be constructed?

6      Draw (AB=7) cm.

7      Mark its midpoint (O), so (AO=OB=3.5) cm.

8      At (O), construct an angle of 140° with (OB).

9      On the two rays of the new line, mark (OC=OD=2.5) cm.

10   Join (A-C), (C-B), (B-D), and (D-A).

 

The resulting quadrilateral is the required parallelogram.

 

Problem 8. What are the main properties of a rhombus?

A rhombus has four equal sides. Its opposite sides are parallel, opposite angles are equal, adjacent angles add to 180°, and its diagonals bisect one another at right angles. The diagonals also bisect the angles of the rhombus.

 

Problem 9. Construct a rhombus whose diagonals are 4 cm and 5 cm.

11   Draw the longer diagonal (AB=5) cm.

12   Mark its midpoint (O), so (AO=OB=2.5) cm.

13   Through (O), draw a perpendicular line.

14   Mark (OC=OD=2) cm on this perpendicular.

15   Join (A-C), (C-B), (B-D), and (D-A).

 

The resulting quadrilateral is the required rhombus.

 

4. Quadrilateral angle sum

Problem 10. What is the sum of the interior angles of any quadrilateral?

Draw one diagonal. It divides the quadrilateral into two triangles. Since each triangle has angle sum 180°,

 

[ 180^\circ+180^\circ=360^\circ. ]

 

Therefore, the sum of the interior angles of every simple quadrilateral, including a concave quadrilateral, is 360°.

 

5. Kite and trapezium

Problem 11. What is a kite?

A kite is a quadrilateral with two distinct pairs of equal adjacent sides. If (AB=BC) and (CD=DA), then ABCD is a kite.

 

Its symmetry diagonal bisects the other diagonal at right angles and bisects the angles between the equal sides.

 

Problem 12. What is a trapezium?

A trapezium is a quadrilateral with at least one pair of parallel opposite sides.

 

If (PQ\parallel SR), then the interior angles on each non-parallel side are supplementary:

 

[ \angle P+\angle S=180^\circ, ] [ \angle Q+\angle R=180^\circ. ]

 

Problem 13. What additional property does an isosceles trapezium have?

An isosceles trapezium has equal non-parallel sides. Its base angles are equal:

 

[ \angle U=\angle V, ]

 

and the other pair of base angles is also equal. Adjacent angles on a leg remain supplementary.

 

6. Playing with quadrilaterals

Activity 1. Join two equilateral triangles of side 8 cm.

When two equilateral triangles are joined along a side, the outside boundary is a rhombus. All four outer sides are 8 cm because each comes from a side of one triangle. The interior angles are 60° and 120° in alternating order.

 

Activity 2. Join two isosceles triangles with sides 8 cm, 8 cm, and 6 cm.

Different joins produce different quadrilaterals. If the equal 8 cm sides form the outer boundary, the result has four equal sides and is a rhombus. If the 6 cm sides or unequal sides form the boundary, the result may be a kite or another quadrilateral depending on the joining orientation. The side lengths and diagonal arrangement should be checked from the actual join.

 

Activity 3. Join two scalene triangles with sides 6 cm, 9 cm, and 12 cm.

Joining two congruent scalene triangles along a corresponding side can produce different quadrilaterals. If the equal corresponding sides form pairs of opposite sides, a parallelogram may result. If equal sides form adjacent pairs, a kite may result. In other arrangements, a general quadrilateral is obtained. Each classification must be justified by checking parallelism, equal sides, or diagonal properties.

 

7. Figure it Out: kite, trapezium, and quadrilateral relationships

Question 1. Find all sides and angles of the quadrilateral obtained by joining two equilateral triangles of side 4 cm.

The quadrilateral is a rhombus. All four sides are 4 cm:

 

[ AB=BC=CD=DA=4\text{ cm}. ]

 

Its angles alternate between 60° and 120°:

 

[ \angle A=\angle C=60^\circ, ] [ \angle B=\angle D=120^\circ, ]

 

or the reverse arrangement, depending on which sides of the triangles are joined.

 

Question 2. Construct a kite whose diagonals are 6 cm and 8 cm.

16   Draw one diagonal, say (PQ=6) cm.

17   Construct the perpendicular bisector of (PQ), meeting it at (T).

18   On the perpendicular bisector, choose points (R) and (S) so that (RS=8) cm and (T) is the midpoint of (RS). Thus (RT=ST=4) cm.

19   Join (P-R), (R-Q), (Q-S), and (S-P).

 

The resulting quadrilateral is the required kite. Its diagonals are perpendicular and have lengths 6 cm and 8 cm.

 

Question 3. Find the remaining angles in the trapeziums.

(i) First trapezium

Using supplementary interior angles on parallel lines:

 

[ \angle R=180^\circ-105^\circ=75^\circ, ]

 

and

 

[ \angle S=180^\circ-135^\circ=45^\circ. ]

 

Thus, the remaining angles are 75° and 45°.

 

(ii) Isosceles trapezium

The given top angle is 100°. The other angle on the same leg is

 

[ 180^\circ-100^\circ=80^\circ. ]

 

Since the trapezium is isosceles, the corresponding base angle is also 80°. Thus, the remaining angles are 80° and 80°.

 

Question 4. Describe the Venn-diagram relationships among parallelograms, kites, rhombuses, rectangles, and squares.

The set relationships are:

 

•        Every square is a rectangle, a rhombus, a parallelogram, and a kite.

•        Every rectangle is a parallelogram.

•        Every rhombus is a parallelogram and a kite.

•        Not every parallelogram is a rectangle or rhombus.

•        Not every kite is a rhombus or parallelogram.

 

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

A rhombus is both a kite and a parallelogram. A square is a special case of this intersection.

 

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

Yes. A square is both a kite and a rectangle because it has equal adjacent sides and four right angles.

 

(iii) Is every kite a rhombus?

No. A rhombus is always a kite under the chapter’s definition, but a kite need not have all four sides equal. Therefore, every rhombus is a kite, but not every kite is a rhombus.

 

Question 5. If PAIR and RODS are rectangles, find (\angle IOD).

Since the marked angle between the diagonal direction and the side is 30°, the corresponding angle at the intersection of the relevant rectangle sides is also

 

[ \boxed{\angle IOD=30^\circ}. ]

 

8. Constructions and reasoning

Question 6. Construct a square with diagonal 6 cm without using a protractor.

20   Draw (AB=6) cm.

21   Construct the perpendicular bisector of (AB), meeting it at (O).

22   Since (AO=OB=3) cm, mark points (C) and (D) on the perpendicular line so that (OC=OD=3) cm.

23   Join (A-C), (C-B), (B-D), and (D-A).

 

The resulting quadrilateral is a square because its diagonals are equal, bisect each other, and are perpendicular.

 

Question 7. CASE is a square and U, V, W, X are the midpoints of its sides. What type of quadrilateral is UVWX?

UVWX is a square. If the side of the outer square is (x), then each side of the inner square is

 

[\sqrt{\left(\frac{x}{2}\right)^2+\left(\frac{x}{2}\right)^2}
=\frac{x}{\sqrt2}.]

 

All four sides are equal. Each angle is 90° because adjacent inner sides have equal slopes with respect to the outer square’s sides. Hence, UVWX is a square.

 

Question 8. If a quadrilateral has four equal sides and one angle of 90°, is it a square?

Yes. Four equal sides make the quadrilateral a rhombus. In a rhombus, opposite angles are equal and adjacent angles are supplementary. If one angle is 90°, every adjacent and opposite angle is also 90°. Therefore, the quadrilateral is a square.

 

Question 9. What type of quadrilateral has both pairs of opposite sides equal?

It is a parallelogram. Draw a diagonal. The two triangles formed have three corresponding equal sides, so they are congruent by SSS. Corresponding alternate angles are equal, which proves both pairs of opposite sides are parallel. Hence the quadrilateral is a parallelogram.

 

Question 10. Is the sum of the angles of a concave quadrilateral also 360°?

Yes. Draw a diagonal that divides the concave quadrilateral into two triangles. Each triangle has an angle sum of 180°:

 

[ 180^\circ+180^\circ=360^\circ. ]

 

Thus, the angle sum is 360°.

 

9. True-or-false questions

Question 11(i). A quadrilateral whose diagonals are equal and bisect each other must be a square.

False. A non-square rectangle also has equal diagonals that bisect each other. The diagonals must additionally be perpendicular for the rectangle to be a square.

 

Question 11(ii). A quadrilateral having three right angles must be a rectangle.

True. The fourth angle is

 

[ 360^\circ-(90^\circ+90^\circ+90^\circ)=90^\circ. ]

 

Thus, all four angles are right angles, so the quadrilateral is a rectangle.

 

Question 11(iii). A quadrilateral whose diagonals bisect each other must be a parallelogram.

True. This is a converse property of parallelograms. If the diagonals of a quadrilateral bisect each other, both pairs of opposite sides are parallel, so the quadrilateral is a parallelogram.

 

Question 11(iv). A quadrilateral whose diagonals are perpendicular to each other must be a rhombus.

False. A kite can have perpendicular diagonals without all four sides being equal. A rhombus is one possibility, but perpendicular diagonals alone are not sufficient.

 

Question 11(v). A quadrilateral in which the opposite angles are equal must be a parallelogram.

True. Let the angles be (A,B,A,B). Since their sum is 360°,

 

[ 2A+2B=360^\circ\Rightarrow A+B=180^\circ. ]

 

The consecutive interior angles are supplementary, so both pairs of opposite sides are parallel. Hence it is a parallelogram.

 

Question 11(vi). A quadrilateral in which all the angles are equal is a rectangle.

True. Since the angle sum is 360°, each angle is

 

[ 360^\circ\div4=90^\circ. ]

 

Therefore, the quadrilateral is a rectangle.

 

Question 11(vii). Isosceles trapeziums are parallelograms.

False. An isosceles trapezium has only one pair of parallel sides in general. Its non-parallel sides are equal, but they need not be parallel. Only special cases, such as a rectangle, are both isosceles trapeziums and parallelograms.

 

10. Folding and construction activities

Activity 1. What shape is formed by the first set of paper creases?

The exact result depends on the crease pattern shown, but folding a sheet into halves and quarters and making the indicated triangular crease produces a quadrilateral whose sides and angles can be identified from the symmetry of the creases. If the creases are symmetric, the resulting figure is generally a square or rectangle; if the diagonal crease is included, it may form an isosceles triangle within the quadrilateral.

 

Activity 2. How can the shown crease pattern be made?

Fold the quarter-sheet along its midpoint lines, then fold the indicated corner to the required points so that the creases are symmetric. Open the paper and trace the crease intersections. The desired quadrilateral is obtained from the boundary and the diagonal crease.

 

Activity 3. How can a square be formed from the quarter-sheet?

Fold the quarter-sheet so that the two adjacent sides are equal and the crease lines meet at right angles. Align the opposite edges carefully and press the fold. On opening the paper, the four crease segments form a square.

 

11. Summary of essential properties

Quadrilateral	Defining property	Important diagonal/angle properties
Rectangle	Four right angles	Equal diagonals that bisect each other
Square	Four equal sides and four right angles	Equal perpendicular diagonals that bisect angles and each other
Parallelogram	Both pairs of opposite sides parallel	Opposite sides/angles equal; diagonals bisect each other
Rhombus	Four equal sides	Perpendicular diagonals; diagonals bisect angles
Kite	Two pairs of equal adjacent sides	One diagonal bisects the other at right angles
Trapezium	At least one pair of opposite sides parallel	Same-side interior angles are supplementary
Isosceles trapezium	Non-parallel sides equal	Base angles are equal