Q1. How many sides does a quadrilateral have?

Ans: 4 sides.

Q2. How many diagonals does a quadrilateral have?

Ans: 2 diagonals.

Q3. What is a rectangle?

Ans: A quadrilateral with all angles \(90^\circ\).

Q4. What is a square?

Ans: A rectangle with all sides equal (or all sides equal and all angles \(90^\circ\)).

Q5. Property: Opposite sides of a parallelogram are — ?

Ans: Congruent (equal in length).

Q6. In a rectangle, diagonals are — ?

Ans: Equal (congruent) and bisect each other.

Q7. In a rhombus, diagonals are — ?

Ans: Perpendicular bisectors of each other and they bisect opposite angles.

Q8. What defines a trapezium?

Ans: Exactly one pair of opposite sides is parallel.

Q9. How many elements (sides, angles, diagonals) in a quadrilateral?

Ans: 10 elements.

Q10. If a square has side 6 cm, its diagonal equals?

Ans: \(6\sqrt{2}\) cm (≈ 8.485 cm).

Q11. If a rhombus has diagonals 16 cm and 12 cm, its side equals?

Ans: side \(= \sqrt{8^2 + 6^2} = \sqrt{100} = 10\) cm.

Q12. If diagonals of rectangle are 26 cm and one side is 24 cm, the other side equals?

Ans: other side \(= \sqrt{26^2 - 24^2} = \sqrt{100}=10\) cm.

Q13. In a parallelogram, diagonals — ?

Ans: Bisect each other.

Q14. In a kite, what is special about diagonals?

Ans: One diagonal is perpendicular bisector of the other.

Q15. If a square has side 8 cm, diagonal equals?

Ans: \(8\sqrt{2}\) cm (≈ 11.314 cm).

Q16. Are opposite angles of a rhombus equal?

Ans: Yes, opposite angles are congruent.

Q17. Number of right angles in a rectangle?

Ans: Four \(90^\circ\) angles.

Q18. What is required at minimum (count) to construct a general quadrilateral?

Ans: 5 independent elements (sides/angles/diagonals) appropriately placed.

Q19. True or False: In a rectangle diagonals are perpendicular.

Ans: False — diagonals are equal but not generally perpendicular (they are perpendicular only in a square).

Q20. If diagonals of rhombus are 24 and 70, half-diagonals are?

Ans: \(12\) and \(35\) respectively (half of 24 and 70).

Q1. Steps to construct a quadrilateral when four sides and one diagonal are given (example PQRS with \(PQ,QR,RS,SP,QS\) given)?

Ans: Construct triangles \(SPQ\) and \(SRQ\) using the given side lengths and diagonal \(QS\). Place them so they share side \(SQ\); join \(P\) to \(R\) to complete quadrilateral.

Q2. Steps to construct quadrilateral when three sides and two diagonals are given (example \(WX YZ\))?

Ans: Construct triangles that use the known diagonals and sides (e.g. \(WXZ\) and \(WZY\)), draw both triangles sharing appropriate diagonal; join remaining vertices to form quadrilateral.

Q3. Show how to construct a quadrilateral if two adjacent sides and three angles are given (example \(LEFT\) with \(EL,EF\) and \(\angle L,\angle E,\angle F\) given).

Ans: Draw side \(EL\). At \(E\) draw angle \(\angle E\) and mark EF length along it. From \(L\) draw ray at \(\angle L\) and from \(F\) draw ray at \(\angle F\). Their intersection is \(T\). Join vertices to complete.

Q4. Construct a quadrilateral when three sides and two included angles are given (ex: \(QR,RS,SP\) and \(\angle R,\angle S\)).

Ans: Draw \(QR\). At \(R\) draw ray making \(\angle R\) and mark \(RS\) on it. At \(S\) draw ray making \(\angle S\) and mark \(SP\) on it. Join \(P\) to \(Q\).

Q5. If diagonals of a rhombus are 16 cm and 12 cm, find perimeter.

Ans: side \(= \sqrt{8^2+6^2}=10\) cm; perimeter = \(4\times10=40\) cm.

Q6. In rectangle ABCD, if \(AB=8\) cm, \(BP=8.5\) cm where \(P\) is midpoint of diagonal, find diagonal BD and side BC.

Ans: \(BD=2\times BP=17\) cm. Using Pythagoras: \(BC=\sqrt{BD^2 - CD^2}=\sqrt{17^2 - 8^2}=\sqrt{289-64}= \sqrt{225}=15\) cm.

Q7. Find the diagonal of a square of side 6 cm.

Ans: diagonal \(=6\sqrt{2}\) cm (exact). Numerically \( \approx 8.485\) cm.

Q8. In rhombus BEST, if \(\angle BTS = 110^\circ\) find \(\angle TBS\).

Ans: Diagonal bisects opposite angles, so angle at B which is formed is split: \(\angle TBS = \tfrac{1}{2}(180^\circ - 110^\circ) = 35^\circ\). (Detailed reasoning in chapter text.)

Q9. If \( \angle BES = 110^\circ\) and it's a rhombus, find \(\angle TBE\) (internal angle at B part made by diagonal).

Ans: Opposite angle equal ⇒ \(\angle BES =110^\circ\). Adjacent interior angles sum to \(360^\circ\), giving the half-split angles as \(70^\circ\) and halves of those as \(35^\circ\).

Q10. If a square has side 6 cm, show using Pythagoras that diagonal \(=\sqrt{72}\).

Ans: \(d^2=6^2+6^2=36+36=72\Rightarrow d=\sqrt{72}=6\sqrt{2}.\)

Q11. Construct rectangle with \(AB=6.0\) cm and \(BC=4.5\) cm — short construction steps.

Ans: Draw \(AB=6.0\). At \(B\) draw perpendicular and mark \(BC=4.5\). From \(A\) draw perpendicular equal to \(BC\). Join endpoints to form rectangle.

Q12. Construct a rhombus with side 4 cm and \(\angle K = 75^\circ\) — steps.

Ans: Draw side \(KL=4\). At \(K\) make angle \(75^\circ\), mark \(KM=4\). With compasses from L and M draw arcs radius 4 to locate N. Join vertices to make rhombus.

Q13. If diagonals of rhombus are 16 and 12, find side (show formula used).

Ans: side \(= \sqrt{(16/2)^2+(12/2)^2}=\sqrt{8^2+6^2}=\sqrt{100}=10\) cm.

Q14. If diagonal of rectangle is 26 and one side 24, other side equals?

Ans: \( \sqrt{26^2 - 24^2} = \sqrt{676-576} = \sqrt{100} = 10\) cm.

Q15. Square side 8 cm → diagonal length?

Ans: \(8\sqrt{2}\) cm.

Q16. Measure of one angle of a rhombus is \(50^\circ\); remaining angles are?

Ans: Opposite angle = \(50^\circ\); adjacent angles = \(180^\circ - 50^\circ = 130^\circ\). So angles: \(50^\circ, 130^\circ, 50^\circ, 130^\circ.\)

Q17. State two properties of a kite.

Ans: (1) Two pairs of adjacent sides congruent. (2) One diagonal is perpendicular bisector of the other; one pair of opposite angles congruent.

Q18. Parallelogram PQRS: if \(PS=5.4\) cm then \(QR=\) ?

Ans: \(QR=PS=5.4\) cm (opposite sides equal).

Q19. In same parallelogram, if \(TS=3.5\) cm where \(T\) is diagonal midpoint, find \(QS\).

Ans: \(QS = 2\times TS = 7.0\) cm (diagonals bisect each other).

Q20. In parallelogram, if \(\angle QRS = 118^\circ\), find \(\angle QPS\) where \(P\) opposite vertex.

Ans: Opposite angles equal ⇒ \(\angle QPS = 118^\circ.\)

Q1. Construct quadrilateral \(PQRS\) with \(PQ=5.6\) cm, \(QR=5\) cm, \(SP=4.3\) cm, \(RS=7\) cm and diagonal \(QS=6.2\) cm — give stepwise construction.

Ans (steps):

1. Draw segment \(QS\) of length \(6.2\) cm.
2. At \(Q\) construct triangle \(Q R S\): mark \(QR=5\) on a ray from \(Q\) and draw arc of radius \(7\) cm from that point to get \(S\) on the other side so that \(RS=7\) and \(QS\) is already fixed.
3. At \(S\) construct triangle \(S P Q\): from \(S\) draw arc of radius \(4.3\) cm, from \(Q\) draw arc of radius \(5.6\) cm; their intersection gives \(P\).
4. Join \(P\) to \(Q\) and \(P\) to \(R\) to finish quadrilateral.\)

Q2. Construct quadrilateral \(WXYZ\) when \(YZ=4\) cm, \(ZX=6\) cm, \(WX=4.5\) cm, \(ZW=5\) cm, \(YW=6.5\) cm — steps.

Ans (steps):

1. Draw triangle \(WZY\) with sides \(ZW=5\), \(WY=6.5\), \(YZ=4\).
2. From \(Z\) draw ray and mark \(ZX=6\) on it.
3. From \(W\) draw arc radius \(WX=4.5\); intersection with ray from \(Z\) gives \(X\).
4. Join \(X\) to \(Y\) to complete quadrilateral.

Q3. Construct \(LEFT\) with \(EL=4.5\), \(EF=5.5\), \(\angle L=60^\circ\), \(\angle E=100^\circ\), \(\angle F=120^\circ\) — explain how rays intersect to find \(T\).

Ans: Draw \(EL\). At \(E\) draw ray making \(100^\circ\) and mark \(EF=5.5\). At \(L\) draw ray of \(60^\circ\). At \(F\) draw ray of \(120^\circ\). Intersection of rays from \(L\) and \(F\) gives \(T\). Join to get quadrilateral.

Q4. Construct \(PQRS\) with \(QR=5\), \(RS=6.2\), \(SP=4\), \(\angle R=62^\circ\), \(\angle S=75^\circ\) — steps.

Ans: Draw \(QR=5\). At \(R\) draw ray at \(62^\circ\) and mark \(RS=6.2\) on it. At \(S\) draw ray at \(75^\circ\) and mark \(SP=4\) on it. Join \(P\) to \(Q\) to complete.

Q5. In rectangle ABCD, diagonals intersect at P. If \(AB=8\) and \(BP=8.5\), find \(BC\) showing Pythagoras steps.

Ans: \(BD=2\times BP = 17\). Since \(BD\) is diagonal and \(AB=8\): \(BC=\sqrt{BD^2 - AB^2}=\sqrt{17^2 - 8^2}=\sqrt{289-64}=\sqrt{225}=15\) cm.

Q6. In rhombus BEST, diagonals meet at A. If \(m\angle BTS = 110^\circ\), find \(m\angle TBS\) and \(m\angle TBE\) (show reasoning).

Ans: Opposite angles equal ⇒ \(\angle BES=\angle BTS=110^\circ\). The other two angles sum to \(360-220=140^\circ\) so each is \(70^\circ\). Diagonal bisects angles, so \(\angle TBS=\tfrac{1}{2}\times70^\circ=35^\circ\).

Q7. In same rhombus: given \(TE=\tfrac{1}{2} \cdot 24 =12\) and \(BS=\tfrac{1}{2}\cdot70 =35\), find \(TS\).

Ans: With half-diagonals \(TA=12\) and \(AS=35\), \(TS = \sqrt{12^2+35^2}=\sqrt{144+1225}=\sqrt{1369}=37\) cm.

Q8. Construct square \(WXYZ\) with side \(5.2\) cm — give short steps and diagonal length.

Ans: Draw \(WX=5.2\). At \(W\) and \(X\) draw perpendiculars of length \(5.2\) to locate other vertices; join. Diagonal \(=5.2\sqrt{2}\) cm.

Q9. If ratio of angles of quadrilateral CWPR is \(7:9:3:5\), find angles and type of quadrilateral.

Ans: Sum \(=24x=360\Rightarrow x=15.\) Angles: \(105^\circ,135^\circ,45^\circ,75^\circ.\) Only one pair of opposite angles sum to \(180^\circ\) (105+75=180), so exactly one pair of opposite sides parallel ⇒ trapezium.

Q10. Given parallelogram PQRS with diagonals intersection at T. If \(PS=5.4\) cm and \(TS=3.5\) cm, find \(QR,QS\).

Ans: \(QR=PS=5.4\) cm (opposite sides). \(QS=2\times TS=7.0\) cm (diagonals bisect each other).

Q11. Solve: Adjacent angles of parallelogram are \((5x-7)^\circ\) and \((4x+25)^\circ\). Find them.

Ans: Sum = 180 ⇒ \(5x-7+4x+25=180\Rightarrow9x+18=180\Rightarrow x=18.\) Angles: \(5x-7=83^\circ,\;4x+25=97^\circ.\)

Q12. Given parallelogram with diagonal half-length \(TS=3.5\). Find full diagonal \(QS\) and explain use of midpoint property.

Ans: \(QS=2\cdot TS=7.0\) cm because diagonals bisect each other (midpoint property).

Q13. Construct parallelogram ABCD with \(BC=7\) cm, \(\angle ABC=40^\circ\), \(AB=3\) cm — steps.

Ans: Draw \(AB=3\). At \(B\) draw ray at \(40^\circ\) and mark \(BC=7\). From \(A\) draw a line parallel to \(BC\); from \(C\) draw line parallel to \(AB\); their intersection gives \(D\). Join to complete.

Q14. Ratio of consecutive angles 1:2:3:4 — find angles and quadrilateral type.

Ans: Sum \(=10x=360\Rightarrow x=36.\) Angles: \(36^\circ,72^\circ,108^\circ,144^\circ.\) No opposite angles equal or sum neatly to 180 (except 36+144=180 actually — so one pair opposite sums to 180). Since only one opposite pair sums 180, that indicates it's a trapezium (only one pair parallel). (Give reasoning in text.)

Q15. Construct quadrilateral BARC with \(BA=BC=4.2\), \(AC=6.0\), \(AR=CR=5.6\) — outline steps.

Ans: Construct triangle \(ABC\) with sides \(BA=BC=4.2\) and \(AC=6.0\). From \(A\) and \(C\) draw arcs radius \(5.6\); intersection gives \(R\). Join to complete.

Q16. Construct PQRS with \(PQ=3.5, QR=5.6, RS=3.5, \angle Q=110^\circ, \angle R=70^\circ\). If PQRS is a parallelogram which given info is unnecessary?

Ans: In parallelogram opposite sides are equal so \(RS=PQ=3.5\) is redundant. Also if \(\angle Q=110^\circ\) then \(\angle R=70^\circ\) (supplementary), so giving both angles is consistent; the side equality is the unnecessary data.

Q17. Prove in a rectangle that diagonals bisect each other (sketch of proof using congruent triangles).

Ans (sketch): In rectangle ABCD, diagonals AC and BD meet at O. Triangles AOB and COD are congruent (RHS: AO=CO, BO=DO, right angles). So AO=CO and BO=DO → diagonals bisect each other.

Q18. Show how diagonals of rhombus bisect angles (short proof idea).

Ans (idea): In rhombus all sides equal. Triangle pairs formed by diagonal are isosceles, so diagonal bisects vertex angles. (Formal: use congruence of triangles formed by equal sides.)

Q19. If parallelogram angles are \(83^\circ\) and \(97^\circ\), verify adjacency/supplement property.

Ans: Adjacent angles sum \(83 + 97 = 180^\circ\) — consistent with parallelogram property that adjacent angles are supplementary.

Q20. Using diagonal half-lengths 12 and 35, compute perimeter of rhombus.

Ans: side \(=\sqrt{12^2 + 35^2} = \sqrt{144+1225}=37.\) Perimeter = \(4\times37 = 148\) cm.

PS8.1 Q1(1). Construct quadrilateral MORE with \(MO=5.8\) cm, \(OR=4.4\) cm, \(\angle M=58^\circ, \angle O=105^\circ, \angle R=90^\circ\).

Ans (constructive steps):

1. Draw \(MO=5.8\) cm.
2. At \(O\) draw ray making \(105^\circ\) (interior angle); on it mark \(OR=4.4\) cm to locate \(R\).
3. From \(M\) draw ray making \(58^\circ\) and from \(R\) draw perpendicular (since \(\angle R=90^\circ\)); their intersection gives \(E\).
4. Join vertices to get quadrilateral MORE. (Refine with compass for accurate alignment.)

PS8.1 Q1(2). Construct DEFG with \(DE=4.5\), \(EF=6.5\), \(DG=5.5\), \(DF=7.2\), \(EG=7.8\).

Ans (steps):

1. Construct diagonal \(EG=7.8\) cm.
2. At \(E\) construct triangle \(E D F\) such that \(ED=4.5\) and \(EF=6.5\) using arcs from \(E\).
3. At \(G\) construct triangle \(G D F\) using \(GD=5.5\) and \(DF=7.2\). Intersection of arcs gives \(D\) and \(F\).
4. Join to form quadrilateral DEFG.

PS8.1 Q1(3). In ABCD, \(AB=6.4\), \(BC=4.8\), \(\angle A=70^\circ,\angle B=50^\circ,\angle C=140^\circ\) — how to construct?

Ans:

1. Draw \(AB=6.4\). At \(A\) draw ray at \(70^\circ\). At \(B\) draw ray at \(50^\circ\) and mark \(BC=4.8\) on it to locate \(C\).
2. From \(C\) draw ray making appropriate angle \(140^\circ\) (interior) and from \(A\) draw ray parallel or using remaining info to locate \(D\). Intersection gives \(D\).

PS8.1 Q1(4). Construct LMNO with \(LM=LO=6\), \(ON=NM=4.5\), \(OM=7.5\).

Ans (steps):

1. Draw segment \(OM=7.5\).
2. From \(O\) draw arc radius 6 to locate points at distance 6 (possible L); from that point draw arcs radius 4.5 to find N, etc.
3. Construct two triangles that share required sides and join carefully to satisfy all length conditions.

PS8.2 Q1. Draw rectangle ABCD with \(AB=6.0\) cm and \(BC=4.5\) cm — steps.

Ans: Draw \(AB=6.0\). At \(B\) draw perpendicular and mark \(BC=4.5\). At \(A\) draw perpendicular of length 4.5. Join endpoints to complete rectangle.

PS8.2 Q2. Draw square WXYZ with side 5.2 cm — steps.

Ans: Draw \(WX=5.2\). At both \(W\) and \(X\) draw perpendicular rays; mark length 5.2 on both to get Y and Z; join to form square. Diagonals intersect at equal halves and are perpendicular.

PS8.2 Q3. Draw rhombus KLMN with side 4 cm and \(\angle K=75^\circ\) — steps.

Ans: Draw \(KL=4\). At \(K\) draw ray of \(75^\circ\) and place \(KM=4\) on it. With compass from \(L\) and \(M\) draw arcs radius 4; their intersection gives \(N\). Join to complete rhombus.

PS8.2 Q4. If diagonal of rectangle is 26 cm and one side is 24 cm, find the other side.

Ans: Other side \(= \sqrt{26^2 - 24^2} = \sqrt{676-576} = \sqrt{100} = 10\) cm.

PS8.2 Q5. Diagonals of rhombus ABCD are 16 cm and 12 cm. Find side and perimeter.

Ans: Half-diagonals \(=8\) and \(6\). Side \(= \sqrt{8^2 + 6^2} = \sqrt{64+36}=10\) cm. Perimeter \(=4\times10=40\) cm.

PS8.2 Q6. Find diagonal of square with side 8 cm.

Ans: diagonal \(=8\sqrt{2}\) cm (≈ 11.3137 cm).

PS8.2 Q7. If one angle of rhombus is \(50^\circ\), find remaining three angles.

Ans: Opposite angle = \(50^\circ\). Adjacent angles are \(180^\circ - 50^\circ = 130^\circ\). So angles: \(50^\circ,130^\circ,50^\circ,130^\circ.\)

PS8.3 Q1. Opposite angles of a parallelogram are \((3x-2)^\circ\) and \((50-x)^\circ\). Find each angle.

Ans: Opposite angles equal ⇒ \(3x-2 = 50-x\Rightarrow 4x=52\Rightarrow x=13.\) So that angle \(=3x-2=37^\circ\). Opposite angle is also \(37^\circ\). The other pair each equals \(180-37=143^\circ\).

PS8.3 Q2. In the figure of parallelogram WXYZ with intersection O, answer: (1) If \(WZ=4.5\) find \(XY\). (2) If \(YZ=8.2\) find \(XW\). (3) If \(OX=2.5\) find \(OZ\). (4) If \(WO=3.3\) find \(WY\). (5) If \(\angle WZY=120^\circ\) find \(\angle WXY\) and \(\angle XWZ\).

Ans:

1. \(XY = WZ = 4.5\) cm (opposite sides equal).
2. \(XW = YZ = 8.2\) cm.
3. Diagonals bisect each other ⇒ \(OX = OZ = 2.5\) cm.
4. \(WY = 2\times WO = 6.6\) cm (O is midpoint of diagonal WY).
5. Opposite angles equal. If \(\angle WZY = 120^\circ\) (angle at \(Z\)), then angle at opposite vertex \(X\) (\(\angle WXY\)) = \(120^\circ\). Adjacent angles are supplementary ⇒ \(\angle XWZ = 60^\circ\).

PS8.3 Q3. Construct parallelogram ABCD with \(BC=7\) cm, \(\angle ABC=40^\circ\), \(AB=3\) cm — outline steps.

Ans: Draw \(AB=3\). At \(B\) draw ray making \(40^\circ\) and mark \(BC=7\). Draw through \(A\) a line parallel to \(BC\), and through \(C\) a line parallel to \(AB\); their intersection gives \(D\). Join vertices.

PS8.3 Q4. Ratio of consecutive angles of a quadrilateral is \(1:2:3:4\). Find each angle and name type.

Ans: Sum \(=10x=360\Rightarrow x=36.\) Angles: \(36^\circ,72^\circ,108^\circ,144^\circ.\) Here \(36^\circ + 144^\circ = 180^\circ\) so one pair of opposite angles are supplementary and the other pair are supplementary? Actually we have only one opposite pair summing to 180 — this indicates exactly one pair of opposite sides are parallel ⇒ trapezium.

PS8.3 Q5. Construct BARC with \(BA=BC=4.2\), \(AC=6.0\), \(AR=CR=5.6\) — steps.

Ans: Construct isosceles triangle \(ABC\) with AB=BC=4.2 and AC=6.0. From \(A\) and \(C\) draw arcs radius 5.6; their intersection gives \(R\). Join to complete quadrilateral.

PS8.3 Q6. Construct PQRS with given sides and angles; if PQRS is parallelogram which given info is unnecessary?

Ans: If it is a parallelogram then opposite sides equal; since \(RS=PQ\) is implied, giving both is unnecessary. So the redundant information is either \(RS\) (or PQ) if the parallelogram condition is already stated.