1-Mark Questions (20) — with Answers
Q1. Define a point, line, and plane (as basic terms).
Ans. They are undefined basic concepts; a line/plane are sets of points (line in the sense of straight line).
Q2. What is the coordinate of a point on a number line?
Ans. The real number associated with the point; e.g., point at 3 has coordinate \(3\).
Q3. Distance between two points \(A(x_1)\) and \(B(x_2)\) on a number line?
Ans. \(d(A,B)=|x_2-x_1|\).
Q4. If \(d(P,Q)+d(Q,R)=d(P,R)\), what is true?
Ans. Points \(P,Q,R\) are collinear and \(Q\) lies between \(P\) and \(R\) (\(P-Q-R\)).
Q5. What is a segment?
Ans. The union of endpoints \(A,B\) and all points between them; written as \(\overline{AB}\).
Q6. Define a ray \(\overrightarrow{AB}\).
Ans. Union of seg \(AB\) and all points \(P\) with \(A-B-P\); endpoint is \(A\).
Q7. When are two segments congruent?
Ans. When their lengths are equal: if \(AB=CD\) then \(\overline{AB}\cong\overline{CD}\).
Q8. State the midpoint condition.
Ans. If \(A-M-B\) and \(AM\cong MB\), then \(M\) is the unique midpoint of \(\overline{AB}\).
Q9. Define perpendicularity of segments/rays.
Ans. If their containing lines are perpendicular (\(90^\circ\)), the segments/rays are said perpendicular.
Q10. Distance of a point \(C\) from a line \(\ell\)?
Ans. Length of perpendicular from \(C\) to \(\ell\); foot is the perpendicular foot.
Q11. Write reflexive, symmetric, transitive properties of segment congruence.
Ans. \(\overline{AB}\cong\overline{AB}\); if \(\overline{AB}\cong\overline{CD}\) then \(\overline{CD}\cong\overline{AB}\); if \(\overline{AB}\cong\overline{CD}\) and \(\overline{CD}\cong\overline{EF}\) then \(\overline{AB}\cong\overline{EF}\).
Q12. What is a conditional statement?
Ans. A statement in “If–then” form: antecedent (if-part) ⇒ consequent (then-part).
Q13. Define converse of a conditional.
Ans. Statement obtained by interchanging antecedent and consequent.
Q14. What is a theorem and a proof?
Ans. A theorem is a statement proved logically; the logical argument is the proof.
Q15. State Euclid’s postulate: through two points…
Ans. There is one and only one line passing through two points.
Q16. Opposite (vertical) angles formed by intersecting lines are…
Ans. Congruent (equal in measure).
Q17. If \(A-B-C\) and \(AB=6.5\), \(BC=3.2\), find \(AC\).
Ans. \(AC=AB+BC=9.7\).
Q18. If coordinate of \(A\) is \(-5\) and \(B\) is \(3\), \(d(A,B)=?\)
Ans. \(|3-(-5)|=8\).
Q19. If a number \(p>2\) is prime, then it is…
Ans. Odd.
Q20. What figure is formed by three non‑collinear points?
Ans. A triangle.
2-Mark Questions (20) — with Solutions
Q1. On a number line, find \(d(A,B)\) for (i) \(x_A=1,x_B=7\) (ii) \(x_A=6,x_B=-2\).
Ans. (i) \(|7-1|=6\). (ii) \(|-2-6|=8\).
Q2. If \(A(\,-3\,),B(5)\), compute \(d(A,B)\).
Ans. \(|5-(-3)|=8\).
Q3. If \(A(-4),B(-5)\), find \(d(A,B)\).
Ans. \(|-5-(-4)|=1\).
Q4. If \(A-B-C\), \(AC=11\), \(BC=6.5\), find \(AB\).
Ans. \(AB=AC-BC=4.5\).
Q5. If \(R-S-T\), \(ST=3.7\), \(RS=2.5\), find \(RT\).
Ans. \(RT=RS+ST=6.2\).
Q6. Coordinates of a point \(P\) are \(-7\). Find points 8 units away.
Ans. \(-7\pm8\Rightarrow -15,\;1\).
Q7. If \(\overline{AB}\cong\overline{CD}\) and \(\overline{CD}\cong\overline{EF}\), prove \(\overline{AB}\cong\overline{EF}\).
Ans. By transitivity of congruence (equal lengths).
Q8. Define and give an example of opposite rays.
Ans. Rays with common endpoint going in opposite directions on a line, e.g., \(\overrightarrow{RP}\) and \(\overrightarrow{RQ}\) if \(P-R-Q\).
Q9. Write conditional and converse: “Diagonals of a rectangle are congruent.”
Ans. If a quadrilateral is a rectangle, then its diagonals are congruent. Converse: If a quadrilateral’s diagonals are congruent, then it is a rectangle (not always true).
Q10. State Euclid’s 5th postulate informally.
Ans. If interior angles on the same side of a transversal sum to less than two right angles, the lines meet on that side.
Q11. Show that distance on a number line is non‑negative.
Ans. \(d=|x_2-x_1|\ge0\). Equality 0 only when \(x_1=x_2\).
Q12. If \(x\) and \(y\) are coordinates of \(A,B\), write \(d(A,B)\).
Ans. \(|y-x|\).
Q13. Prove that opposite (vertical) angles are equal (outline).
Ans. From linear pairs: \(\angle AOC+\angle BOC=180^\circ\) and \(\angle BOC+\angle BOD=180^\circ\Rightarrow \angle AOC=\angle BOD\).
Q14. If \(AM=MB\) and \(A-M-B\), what is \(M\)? Unique?
Ans. Midpoint; yes, every segment has a unique midpoint.
Q15. If \(AB=5\) cm, \(BP=2\) cm, \(AP=3.4\) cm, compare segments.
Ans. \(AP>BP\); also \(AP+BP=5.4>AB\) so \(P\) is not on segment AB.
Q16. Find coordinates at distance 13 from \(A(5)\).
Ans. \(5\pm13\Rightarrow -8,\;18\).
Q17. Write “Angles in a linear pair are supplementary” in if–then form.
Ans. If two angles form a linear pair, then their sum is \(180^\circ\).
Q18. Write converse of: “If corresponding angles are congruent then lines are parallel.”
Ans. If two lines are parallel, then the corresponding angles (by a transversal) are congruent.
Q19. Identify antecedent and consequent: “If all sides of a triangle are congruent then all angles are congruent.”
Ans. Antecedent: triangle has all sides congruent. Consequent: all its angles are congruent.
Q20. State when three points are collinear using distances.
Ans. For distinct \(P,Q,R\): collinear iff one of \(d(P,Q)+d(Q,R)=d(P,R)\), etc., holds.
3-Mark Questions (20) — with Solutions
Q1. Explain “betweenness” with an example using coordinates.
Ans. Let \(P(-5),Q(-2),R(4)\). \(d(P,Q)=3\), \(d(Q,R)=6\), \(d(P,R)=9\). Since \(3+6=9\), \(P-Q-R\).
Q2. City problem: \(d(U,A)=215\), \(d(V,A)=140\), \(d(U,V)=75\). Who lies between whom?
Ans. \(75+140=215\Rightarrow V\) is between \(U\) and \(A\).
Q3. If \(d(P,R)=7\), \(d(P,Q)=10\), \(d(Q,R)=3\), identify collinearity & betweenness.
Ans. \(10\neq7+3\) and \(7\neq10+3\), \(3\neq10+7\) ⇒ not collinear.
Q4. For \(d(R,S)=8\), \(d(S,T)=6\), \(d(R,T)=4\), decide.
Ans. \(8\neq6+4\), \(6\neq8+4\), \(4\neq8+6\) ⇒ not collinear.
Q5. For \(d(A,B)=16\), \(d(C,A)=9\), \(d(B,C)=7\).
Ans. \(9+7=16\Rightarrow C\) is between \(A\) and \(B\).
Q6. If \(L,M,N\) with \(d(L,M)=11\), \(d(M,N)=12\), \(d(N,L)=8\).
Ans. No equality like \(a+b=c\) holds ⇒ not collinear.
Q7. If \(X,Y,Z\) collinear with \(d(X,Y)=17\), \(d(Y,Z)=8\), find \(d(X,Z)\).
Ans. \(17+8=25\Rightarrow d(X,Z)=25\).
Q8. Show a ray as a set using union.
Ans. \(\overrightarrow{AB}=\overline{AB}\cup\{P\mid A\!-
B\!-
P\}\).
Q9. Give a direct proof that vertical angles are equal.
Ans. As in text: use two linear‑pair sums equal to \(180^\circ\) to eliminate common angle.
Q10. Give an indirect proof that a prime \(p>2\) is odd.
Ans. Assume \(p\) even ⇒ divisible by 2 contradicts definition of prime with only divisors 1 and \(p\).
Q11. Write converse for: “Interior angles supplementary ⇒ lines parallel.” Is the converse true?
Ans. Converse: If lines are parallel, the interior angles on same side are supplementary (true by parallel‑line properties).
Q12. If \(AB=5\), \(BP=2\), \(AP=3.4\), can \(P\) be between \(A\) and \(B\)?
Ans. If between, \(AP+PB=AB\). But \(3.4+2=5.4\ne5\). So \(P\) is not between.
Q13. Write two pairs of opposite rays with common endpoint \(R\) on line \(PQ\).
Ans. \(\overrightarrow{RP},\overrightarrow{RQ}\) and any extension in opposite directions on the same line.
Q14. Distance from a point to a line: define and illustrate.
Ans. The perpendicular length; draw from the point to the line; foot gives shortest distance.
Q15. Find points 13 units from \(A(5)\) and verify using absolute value.
Ans. \(x=5\pm13\Rightarrow-8,18\). Check: \(|-8-5|=13\), \(|18-5|=13\).
Q16. If three points are collinear, how many lines pass through them?
Ans. Exactly one line.
Q17. If three points are non‑collinear, how many lines are determined?
Ans. Three lines (each pair determines one).
Q18. Write the “if–then” form: “The diagonals of a rectangle are congruent.”
Ans. If a quadrilateral is a rectangle, then its diagonals are congruent.
Q19. State and use the midpoint property to find \(AM\) if \(AB=8\) and \(M\) is midpoint.
Ans. \(AM=MB=AB/2=4\).
Q20. If \(x\) and \(y\) are numbers with \(|y-x|=0\), what can you say about points?
Ans. \(x=y\); the two points coincide (distance zero).
All Textbook Exercises — Perfect Solutions
Practice Set 1.1
Q1. Using the given number line (Fig. 1.5), find distances like \(d(B,E)\), \(d(J,A)\), …
Ans. Method: Read each point’s coordinate from the diagram, then compute \(|x_2-x_1|\). Example: If \(x_B=-4\) and \(x_E=3\), then \(d(B,E)=|3-(-4)|=7\). Repeat similarly for all pairs.
Q2. If coordinates of \(A\) and \(B\) are \(x\) and \(y\), find \(d(A,B)\) for the given values.
Ans. Use \(|y-x|\): (i) \(|7-1|=6\); (ii) \(|-2-6|=8\); (iii) \(|7-(-3)|=10\); (iv) \(|-5-(-4)|=1\); (v) \(|-6-(-3)|=3\); (vi) \(|-8-4|=12\).
Q3. From triples of distances, decide betweenness or non‑collinearity.
Ans. Check if one equals sum of other two: (i) 10 ?= 7+3 ⇒ true ⇒ \(P-Q-R\) with \(Q\) between \(P,R\). (ii) 6 ?= 8+4 (no); 8 ?= 6+4 (no); 4 ?= 8+6 (no) ⇒ not collinear. (iii) 16 = 9+7 ⇒ \(A-C-B\). (iv) none match ⇒ not collinear. (v) 15 ?= 7+8 ⇒ yes ⇒ \(Y\) between \(X,Z\). (vi) 8 ?= 5+6 ⇒ no; 6 ?= 5+8 ⇒ no; 5 ?= 8+6 ⇒ no ⇒ not collinear.
Q4. On a number line, \(d(A,C)=10\), \(d(C,B)=8\). Find \(d(A,B)\) in all possibilities.
Ans. If \(A-C-B\): \(AB=AC+CB=18\). If \(C\) lies outside segment \(AB\): \(AB=|AC-CB|=2\).
Q5. Collinear \(X,Y,Z\) with \(d(X,Y)=17\), \(d(Y,Z)=8\). Find \(d(X,Z)\).
Ans. \(d(X,Z)=17+8=25\).
Q6. (i) \(A-B-C\), \(AC=11\), \(BC=6.5\). (ii) \(R-S-T\), \(ST=3.7\), \(RS=2.5\). (iii) \(X-Y-Z\), \(XZ=\tfrac{37}{?}\) (as per text), \(XY=7\). Find the remaining lengths.
Ans. (i) \(AB=AC-BC=4.5\). (ii) \(RT=RS+ST=6.2\). (iii) If \(XZ=\tfrac{37}{?}\) is \(37/ ?\) (text typo). In standard version: if \(XZ=37\), \(XY=7\), then \(YZ=30\).
Q7. Figure formed by three non‑collinear points?
Ans. Triangle.
Practice Set 1.2
Q1. Using coordinates in table (A:−3, B:5, C:2, D:−7, E:9), decide congruent pairs among: \(\overline{DE},\overline{AB}\); \(\overline{BC},\overline{AD}\); \(\overline{BE},\overline{AD}\).
Ans. Lengths: \(DE=|9-(-7)|=16\); \(AB=|5-(-3)|=8\) ⇒ not congruent. \(BC=|2-5|=3\); \(AD=|-7-(-3)|=4\) ⇒ not congruent. \(BE=|9-5|=4\); \(AD=4\) ⇒ congruent.
Q2. If \(M\) is midpoint of \(AB\) and \(AB=8\), find \(AM\).
Ans. \(AM=MB=4\).
Q3. If \(P\) is midpoint of \(CD\) and \(CP=2.5\), find \(CD\).
Ans. \(CP=PD\Rightarrow CD=5\).
Q4. If \(AB=5\) cm, \(BP=2\) cm, \(AP=3.4\) cm, compare segments and decide if \(P\) lies on segment \(AB\).
Ans. \(AP>BP\). Since \(AP+BP=5.4>AB\), \(P\) is not on segment \(AB\).
Q5. Refer Fig. 1.13. Answer (opposite ray of \(\overrightarrow{RP}\), intersections/unions of rays, etc.).
Ans. Typical: Opposite ray of \(\overrightarrow{RP}\) is \(\overrightarrow{RQ}\). Intersections: \(\overrightarrow{PQ}\cap\overrightarrow{RP}=\{P\}\) if \(P\) common; union \(\overrightarrow{PQ}\cup\overrightarrow{QR}\) is a path from \(P\) through \(Q\) to \(R\); seg \(QR\subset\overrightarrow{Q R}\) and \(\overrightarrow{R Q}\); pair of opposite rays with endpoint \(R\): \(\overrightarrow{RS},\overrightarrow{RQ}\); two rays with endpoint \(S\): \(\overrightarrow{SP},\overrightarrow{ST}\); intersection \(\overrightarrow{SP}\cap\overrightarrow{ST}=\{S\}\).
Q6. Using Fig. 1.14 number line, find equidistant points and distances asked.
Ans. Method: Read coordinates and use \(|x_2-x_1|\). Points equidistant from B are those symmetric about B; similarly for Q. Compute: \(d(U,V)=|x_V-x_U|\), etc.
Practice Set 1.3 (Conditionals & Converses)
Q1. Write in “if–then” form.
Ans. (i) If a quadrilateral is a parallelogram, then opposite angles are congruent. (ii) If a quadrilateral is a rectangle, then its diagonals are congruent. (iii) If a triangle is isosceles, then the segment joining the vertex and midpoint of base is perpendicular to the base.
Q2. Write converses.
Ans. (i) If alternate angles (by a transversal) are congruent, then the lines are parallel. (ii) If a pair of interior angles on same side are supplementary, then the lines are parallel. (iii) If diagonals of a quadrilateral are congruent, then it is a rectangle (not necessarily always true without extra conditions).
Problem Set 1
Q1. MCQs
Ans. (i) A (only one) (ii) C (one) (iii) C (one or three) (iv) C (\(|5-(-2)|=7\)) (v) B (\(10-2=8\)).
Q2. P(3), Q(−5), R(6). Decide T/F.
Ans. \(d(P,Q)=8\), \(d(Q,R)=11\), \(d(P,R)=3\). (i) \(8+11\neq3\) ⇒ F. (ii) \(3+11\neq8\) ⇒ F. (iii) \(3+8=11\) ⇒ T. (iv) \(8-3\neq11\) ⇒ F.
Q3. Find distances for pairs of coordinates.
Ans. (i) \(|6-3|=3\) (ii) \(|-1-(-9)|=8\) (iii) \(|5-(-4)|=9\) (iv) \(|-2-x|=|x+2|\) (v) \(|(x-3)-(x+3)|=6\) (vi) \(|-47-(-25)|=22\) (vii) \(|-85-80|=165\).
Q4. Point P is at −7. Find points 8 units away.
Ans. \(-7\pm8\Rightarrow -15,\;1\).
Q5. (i) If \(A-B-C\), \(AC=17\), \(BC=6.5\), find \(AB\). (ii) If \(P-Q-R\), \(PQ=3.4\), \(QR=5.7\), find \(PR\).
Ans. (i) \(AB=17-6.5=10.5\). (ii) \(PR=3.4+5.7=9.1\).
Q6. A(1). Find coordinates 7 units away.
Ans. \(1\pm7\Rightarrow -6,\;8\).
Q7. Write in conditional form.
Ans. (i) If a quadrilateral is a rhombus, then it is a square. (Note: generally false; textbook likely expects: “Every square is a rhombus.”) (ii) If two angles form a linear pair, then their sum is \(180^\circ\). (iii) If three segments pairwise meet at endpoints to form a closed figure, then the figure is a triangle. (iv) If a number has only two divisors, then it is prime.
Q8. Write converses.
Ans. (i) If a figure is a triangle, then sum of interior angles is \(180^\circ\). (Converse of the given is: If sum is \(180^\circ\), then the figure is a triangle.) (ii) If two angles are complementary, then their sum is \(90^\circ\). (Converse: If sum is \(90^\circ\), then they are complementary.) (iii) If two lines are parallel, then corresponding angles are congruent. (iv) If a number is divisible by 3, then the sum of its digits is divisible by 3. (Converse is generally true as a criterion.)
Q9. Identify antecedent and consequent.
Ans. (i) Antecedent: all sides of triangle are congruent. Consequent: all angles are congruent. (ii) Antecedent: quadrilateral is a parallelogram. Consequent: diagonals bisect each other.
Q10*. Draw labelled figures and write given/To Prove (ideas).
Ans. (i) Two equilateral triangles: Given: \(\triangle ABC,\triangle PQR\) with all sides equal in each; To prove: \(\triangle ABC\sim\triangle PQR\). (ii) Linear pair congruent ⇒ each right angle: Given: \(\angle AOB\) and \(\angle BOC\) linear pair and equal; Prove: each \(90^\circ\). (iii) Altitudes on two sides congruent ⇒ those sides congruent: Given: \(h_b=h_c\); Prove: \(b=c\).
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