Class 8 Maths Part 1 Chapter 4 Questions with Solutions

Concept: A rectangle has four right angles. Its diagonals are equal and bisect each other. Use triangle angle sum, vertically opposite angles and equal base angles in isosceles triangles.

1. For figure (i), the given angle is BAC = 30 degrees. Since angle DAB is 90 degrees, angle CAD = 60 degrees.
2. The diagonal creates congruent right triangles, so matching acute angles repeat at the opposite corners.
3. Thus angle ABD = 30 degrees, angle DBC = 60 degrees, angle ADB = 60 degrees, angle CDB = 30 degrees, angle ACD = 30 degrees and angle ACB = 60 degrees.
4. At the intersection O, the acute angles AOD and BOC are 60 degrees, and the obtuse angles AOB and COD are 120 degrees.
5. For figure (ii), vertically opposite angles give angle POS = 110 degrees and angle QOP = angle ROS = 70 degrees.
6. Each half-triangle formed by equal half-diagonals is isosceles, so the base angles are equal.
7. So in triangle QOR, each base angle is (180 - 110) divided by 2 = 35 degrees.
8. In triangles QOP and ROS, each base angle is (180 - 70) divided by 2 = 55 degrees.

Final Answer: Figure (i): BAC = ABD = CDB = ACD = 30 degrees; CAD = DBC = ADB = ACB = 60 degrees; AOD = BOC = 60 degrees and AOB = COD = 120 degrees. Figure (ii): POS = 110 degrees, QOP = ROS = 70 degrees, OQR = ORQ = OPS = OSP = 35 degrees, and OQP = OPQ = ORS = OSR = 55 degrees.

Concept: A quadrilateral whose diagonals are equal and bisect each other is a rectangle. The chosen angle between the diagonals fixes the shape, but each diagonal is still split into two 4 cm halves.

1. Draw AB = 8 cm and mark its midpoint O.
2. Through O, draw a second line making the required angle with AB.
3. On this second line, mark points C and D so that OC = OD = 4 cm.
4. Join A to C, C to B, B to D and D to A.
5. Repeat the same construction for 30 degrees, 40 degrees, 90 degrees and 140 degrees.

Final Answer: The construction is obtained by taking two equal 8 cm diagonals that share midpoint O and meet at the required angle.

Concept: All radii of the same circle are equal. Perpendicular diameters create four right angles at the centre, and the endpoints are evenly spaced around the circle.

1. OP, OL, OA and OM are all radii, so they are equal.
2. The diameters are perpendicular, so each central angle is 90 degrees.
3. The chords AP, PM, ML and LA subtend equal central angles, so these four sides are equal.
4. Each angle of APML stands on a diameter of the circle. An angle in a semicircle is a right angle, so all four angles are 90 degrees.

Final Answer: APML is a square.

Concept: If two equal segments are used as the diagonals of a quadrilateral and they bisect each other, the quadrilateral formed by their endpoints is a rectangle. A rectangle has right angles.

1. Let the equal sticks be AB and CD.
2. Place them so that their midpoints meet at O.
3. Tie or pass a thread around the four endpoints A, C, B and D.
4. The two sticks now act as equal diagonals that bisect each other.
5. The thread forms a rectangle, so each corner angle is 90 degrees.

Final Answer: Put the equal sticks through the same midpoint and pass the thread around their endpoints. The thread forms a rectangle, giving exact right angles.

Concept: Opposite sides parallel and equal describe a parallelogram. A rectangle needs the extra condition that all angles are right angles.

1. Every rectangle has opposite sides parallel and equal.
2. But a slanted parallelogram also has opposite sides parallel and equal.
3. A slanted parallelogram need not have any right angle.
4. Therefore the condition is not enough to define a rectangle.

Final Answer: No. That condition defines a parallelogram, not necessarily a rectangle.

Concept: Use special quadrilateral properties and the angle-sum rule. Adjacent angles in a parallelogram add to 180 degrees, opposite angles in a parallelogram are equal, and equal sides often create equal base angles.

1. In (i), the given angle is 40 degrees, so the adjacent angle is 140 degrees. Opposite angles match.
2. In (ii), the adjacent angle to 110 degrees is 70 degrees, and the opposite angles match.
3. In (iii), the source diagram is solved using the rhombus angle property. A rhombus diagonal bisects the angles at its endpoints, so the 30 degree split repeats and gives 60 degree angles at V and X.
4. Adjacent angles in a rhombus are supplementary, so the remaining angles U and W are 120 degrees.
5. In (iv), use the same rhombus angle property for AOIE. Diagonal OE bisects angle E, and opposite angles in a rhombus are equal.
6. Since one part at E is 20 degrees, the other part at E is 20 degrees; angle E is 40 degrees, so opposite angle O is also 40 degrees and its two parts are 20 degrees each.

Final Answer: (i) E = 140 degrees, R = 140 degrees, A = P = 40 degrees. (ii) Q = 70 degrees, S = 70 degrees, R = 110 degrees. (iii) XVU = XVW = 30 degrees, WXU = UVW = 60 degrees, UXV = WXV = 30 degrees, U = W = 120 degrees. (iv) OEA = IEO = AOE = EOI = 20 degrees, O = E = 40 degrees, and A = I = 140 degrees.

Concept: The diagonals of a parallelogram bisect each other. So the 7 cm diagonal is split into 3.5 cm and 3.5 cm, and the 5 cm diagonal is split into 2.5 cm and 2.5 cm.

1. Draw AB = 7 cm and mark its midpoint O.
2. At O, draw a line making 140 degrees with AB.
3. On this line, mark C and D on opposite sides of O so that OC = OD = 2.5 cm.
4. Join A to C, C to B, B to D and D to A.

Final Answer: The constructed quadrilateral ADBC is the required parallelogram.

Concept: The diagonals of a rhombus bisect each other at right angles. So draw one diagonal, then draw the perpendicular bisector for the other diagonal.

1. Draw AB = 5 cm and mark its midpoint O.
2. At O, draw a line perpendicular to AB.
3. On this perpendicular line, mark C and D so that OC = OD = 2 cm.
4. Join A to C, C to B, B to D and D to A.

Final Answer: The quadrilateral ADBC is the required rhombus.

Concept: Each angle of an equilateral triangle is 60 degrees. When two equilateral triangles share a side, the outer boundary has four equal 4 cm sides.

1. Each equilateral triangle has all sides equal to 4 cm.
2. The outside quadrilateral uses four of those equal sides, so all its sides are 4 cm.
3. At the two outer sharp vertices, each angle is 60 degrees.
4. At the two vertices on the shared side, two 60 degree angles combine to 120 degrees.

Final Answer: All four sides are 4 cm. The angles are 60 degrees, 120 degrees, 60 degrees and 120 degrees.

Concept: In a kite, one diagonal is the perpendicular bisector of the other diagonal. This creates two pairs of adjacent equal sides.

1. Draw PQ = 6 cm.
2. Draw the perpendicular bisector of PQ and let it meet PQ at T.
3. On the perpendicular line, choose points R and S on opposite sides of T so that RS = 8 cm.
4. Join P to R, R to Q, Q to S and S to P.

Final Answer: PRQS is a kite with diagonals 6 cm and 8 cm.

Concept: Interior angles on the same side of a transversal between parallel lines add to 180 degrees. In an isosceles trapezium, base angles are equal.

1. In (i), the top and bottom bases are parallel. The angle above the 105 degree angle is 75 degrees because co-interior angles add to 180 degrees.
2. The angle above the 135 degree angle is 45 degrees by the same rule.
3. In (ii), the trapezium is isosceles, so the two top angles are equal. The other top angle is 100 degrees.
4. Each bottom angle is supplementary to a 100 degree top angle, so both bottom angles are 80 degrees.

Final Answer: (i) The remaining top angles are 45 degrees and 75 degrees. (ii) The other top angle is 100 degrees, and the two bottom angles are 80 degrees each.

Concept: Use the chapter definitions carefully: rhombuses and squares lie in the kite-parallelogram overlap, rectangles lie in parallelograms, and the square is the kite-rectangle exception.

1. Place rhombuses in the overlap of kites and parallelograms.
2. Place squares with rhombuses in that common region.
3. Place rectangles inside parallelograms, with squares in the small overlap of rectangles and rhombuses.
4. Thus a square is the quadrilateral that can be both a kite and a rectangle; the printed key's No is for a non-square rectangle.
5. A general kite is not necessarily a rhombus; every rhombus is a kite.

Final Answer: (i) Rhombus and square. (ii) Yes, a square is both a kite and a rectangle; a non-square rectangle is not. (iii) No. A rhombus is a kite, whereas a kite need not be a rhombus.

Concept: In rectangles, opposite sides are parallel and adjacent sides are perpendicular. Parallel side directions preserve the angle shown in the diagram.

1. Use the given 30 degree angle between the slanted side and the rectangle direction.
2. In rectangle PAIR, OI is perpendicular to RI. In rectangle RODS, OD is perpendicular to RO.
3. The angle between two lines is equal to the angle between their perpendiculars, so angle IOD matches the given angle between RI and RO.

Final Answer: Angle IOD = 30 degrees.

Concept: The diagonals of a square are equal, bisect each other and are perpendicular. A perpendicular bisector can be made with compass arcs.

1. Draw AB = 6 cm.
2. Construct the perpendicular bisector of AB and mark its midpoint O.
3. On the perpendicular line, mark C and D so that OC = OD = 3 cm.
4. Join A to C, C to B, B to D and D to A.

Final Answer: ACBD is the required square with diagonal 6 cm.

Concept: Midpoints on adjacent sides of a square create congruent right isosceles triangles. This makes all sides of the inner quadrilateral equal and all its angles right angles.

1. Let the side of the square be x. Then each half-side is x/2.
2. In right triangle CUV, CU = CV = x/2, so UV squared equals x squared over 2.
3. The same calculation applies to UX, XW and WV, so all four sides are equal.
4. Each corner angle of the inner quadrilateral becomes 90 degrees from the pair of 45 degree angles around it.
5. Construction and measurement also confirm this: draw the square, mark the side midpoints, join them, and measure four equal sides and four right angles in UVWX.
6. Other inner squares can be made by choosing one point on a side and then choosing points at the same fractional distance around the next three sides, as in the tilted Figure (b) pattern.

Final Answer: UVWX is a square. Measurement confirms equal sides and right angles, and tilted inner squares are obtained by taking matching fractional points on consecutive sides.

Concept: A quadrilateral with four equal sides is a rhombus. In a rhombus, opposite angles are equal and adjacent angles are supplementary.

1. Four equal sides make the quadrilateral a rhombus.
2. In a rhombus, adjacent angles add to 180 degrees.
3. If one angle is 90 degrees, each adjacent angle must also be 90 degrees.
4. The opposite angle is equal to the first angle, so it is also 90 degrees.
5. For construction and measurement, draw four equal sides with one right angle, complete the quadrilateral, then measure the other three angles. They are all 90 degrees.

Final Answer: Yes. It will be a square.

Concept: If both pairs of opposite sides are equal, a diagonal creates two congruent triangles by SSS. Equal alternate angles then show that both pairs of opposite sides are parallel.

1. Draw diagonal AC in quadrilateral ABCD.
2. In triangles ABC and CDA, AB = CD, BC = AD and AC is common.
3. So the triangles are congruent by SSS.
4. Corresponding alternate angles are equal, giving AB parallel CD and BC parallel AD.

Final Answer: The quadrilateral is a parallelogram.

Concept: A diagonal can split the quadrilateral into two triangles. Each triangle has angle sum 180 degrees.

1. Join BD to split the quadrilateral into triangles ADB and BDC.
2. The sum of angles in triangle ADB is 180 degrees.
3. The sum of angles in triangle BDC is 180 degrees.
4. Adding both triangle sums gives the angle sum of the quadrilateral.
5. For construction and measurement, draw a similar concave quadrilateral, measure all four interior angles including the reflex angle at D, and add them. The total is 360 degrees.

Final Answer: Yes. The angle sum is 360 degrees.

Concept: Use the defining diagonal and angle properties of rectangles, squares, parallelograms, rhombuses, kites and trapeziums.

1. (i) False: equal diagonals that bisect each other give a rectangle, but it need not be a square.
2. (ii) True: three right angles force the fourth angle to be right because the angle sum is 360 degrees.
3. (iii) True: diagonals that bisect each other give a parallelogram.
4. (iv) False: perpendicular diagonals alone can also occur in a kite, so they do not force a rhombus.
5. (v) True: if both pairs of opposite angles are equal, adjacent angles add to 180 degrees and the quadrilateral is a parallelogram.
6. (vi) True: if all angles are equal, each is 360 degrees divided by 4, which is 90 degrees, so the quadrilateral is a rectangle.
7. (vii) False: an isosceles trapezium need not have two pairs of parallel opposite sides.

Final Answer: (i) False. (ii) True. (iii) True. (iv) False. (v) True. (vi) True. (vii) False.