Mathematics · Textbook solutions

Triangles

Every solved example, exercise, and miscellaneous question — in the order the textbook teaches them. · 50 questions

Congruence of Triangles

4 q

Solved Examples

Worked · 4
  1. 3.1 SolvedEx.1
    The measures of angles of a triangle are in the ratio 5:6:75 : 6 : 7. Find the measures.
  2. 3.1 SolvedEx.2
    Observe figure 3.4 and find the measures of ∠PRS\angle PRS and ∠RTS\angle RTS. In figure 3.4, the points Q, R, S are collinear with R between Q and S; the point T lies on seg PR and seg TS is drawn. ∠PQR=40∘\angle PQR = 40^\circ, ∠QPR=30∘\angle QPR = 30^\circ and ∠TSR=20∘\angle TSR = 20^\circ.
  3. 3.1 SolvedEx.3
    Prove that the sum of exterior angles of a triangle, obtained by extending its sides in the same direction is 360∘360^\circ.
  4. 3.1 SolvedEx.4
    In figure 3.7, bisectors of ∠B\angle B and ∠C\angle C of △ABC\triangle ABC intersect at point P. Prove that ∠BPC=90+12∠BAC\angle BPC = 90 + \frac{1}{2} \angle BAC. Complete the proof filling in the blanks.

Practice Set 3.1

10 q
  1. Ex 3.1 Q1
    In figure 3.8, ∠ACD\angle ACD is an exterior angle of △ABC\triangle ABC. ∠B=40∘\angle B = 40^\circ, ∠A=70∘\angle A = 70^\circ. Find the measure of ∠ACD\angle ACD. [In figure 3.8, △ABC\triangle ABC is drawn with the side BC produced beyond C up to the point D, so that ∠ACD\angle ACD is the exterior angle of the triangle at C. No measures are marked on the figure itself.]
  2. Ex 3.1 Q2
    In △PQR\triangle PQR, ∠P=70∘\angle P = 70^\circ, ∠Q=65∘\angle Q = 65^\circ then find ∠R\angle R.
  3. Ex 3.1 Q3
    The measures of angles of a triangle are x∘x^\circ, (x−20)∘(x-20)^\circ, (x−40)∘(x-40)^\circ. Find the measure of each angle.
  4. Ex 3.1 Q4
    The measure of one of the angles of a triangle is twice the measure of its smallest angle and the measure of the other is thrice the measure of the smallest angle. Find the measures of the three angles.
  5. Ex 3.1 Q5
    In figure 3.9, measures of some angles are given. Using the measures find the values of xx, yy, zz. [In figure 3.9 the points N, M, R lie on one straight line in that order and the ray runs on beyond R; the points T, E, M lie on a second straight line in that order and the ray runs on beyond T. The segments NE, EM and MN therefore form △NEM\triangle NEM. The measures marked on the figure are ∠TEN=100∘\angle TEN = 100^\circ at E and ∠EMR=140∘\angle EMR = 140^\circ at M. The unknowns marked are x=∠ENMx = \angle ENM at N, y=∠NEMy = \angle NEM at E and z=∠EMNz = \angle EMN at M.]
  6. Ex 3.1 Q6
    In figure 3.10, line AB ∥\parallel line DE. Find the measures of ∠DRE\angle DRE and ∠ARE\angle ARE using given measures of some angles. [In figure 3.10 the ray AB starts at A and runs upward to the left. The points A, R, D lie on one straight line with R between A and D. The segments RE and DE are drawn and line DE runs on beyond E. The measures marked on the figure are ∠BAD=70∘\angle BAD = 70^\circ at A, between ray AB and ray AD, and ∠DER=40∘\angle DER = 40^\circ at E, between ray ED and ray ER.]
  7. Ex 3.1 Q7
    In △ABC\triangle ABC, bisectors of ∠A\angle A and ∠B\angle B intersect at point O. If ∠C=70∘\angle C = 70^\circ. Find measure of ∠AOB\angle AOB.
  8. Ex 3.1 Q8
    In Figure 3.11, line AB ∥\parallel line CD and line PQ is the transversal. Ray PT and ray QT are bisectors of ∠BPQ\angle BPQ and ∠PQD\angle PQD respectively. Prove that m∠PTQ=90∘m\angle PTQ = 90^\circ. [In figure 3.11 the points A, P, B lie on line AB with P between A and B, and the points C, Q, D lie on line CD with Q between C and D. The transversal PQ cuts line AB at P and line CD at Q, and the two bisectors ray PT and ray QT meet at the point T lying between the two parallel lines. No measures are marked on the figure.]
  9. Ex 3.1 Q9
    Using the information in figure 3.12, find the measures of ∠a\angle a, ∠b\angle b and ∠c\angle c. [In figure 3.12 a triangle is drawn with its base lying along a straight line that runs on beyond both base vertices, and each of the other two sides is also produced beyond the base vertex it meets. ∠a\angle a is the angle of the triangle at its top vertex. ∠b\angle b is the angle of the triangle at the left base vertex, and the angle vertically opposite to ∠b\angle b — the one between the leftward part of the base line and the downward extension of the left side — measures 70∘70^\circ. ∠c\angle c is the angle of the triangle at the right base vertex, and the angle next to ∠c\angle c on the base line — the one between the rightward part of the base line and the side that rises to the top vertex — measures 100∘100^\circ. The vertices of the triangle are not named in the figure.]
  10. Ex 3.1 Q10
    In figure 3.13, line DE ∥\parallel line GF ray EG and ray FG are bisectors of ∠DEF\angle DEF and ∠DFM\angle DFM respectively. Prove that, (i) ∠DEG=12∠EDF\angle DEG = \frac{1}{2} \angle EDF (ii) EF = FG. [In figure 3.13 the points E, F, M lie on one straight line in that order and the ray runs on beyond M, so ∠DFM\angle DFM is the exterior angle of △DEF\triangle DEF at F. The bisector ray EG of ∠DEF\angle DEF and the bisector ray FG of ∠DFM\angle DFM meet at the point G. No measures are marked on the figure; the two equal halves of ∠DEF\angle DEF are shown by small circle marks at E and the two equal halves of ∠DFM\angle DFM by cross marks at F.]

Practice Set 3.2

10 q
  1. In each of the examples given below, a pair of triangles is shown. Equal parts of triangles in each pair are marked with the same signs. Observe the figures and state the test by which the triangles in each pair are congruent.
    Ex 3.2 Q1(i)
    In the first pair of figure 3.19, △ABC\triangle ABC and △PQR\triangle PQR are drawn. Side AB and side PQ each carry one tick, side BC and side QR each carry two ticks, and side AC and side PR each carry three ticks. Complete : By ......... test, △ABC≅△PQR\triangle ABC \cong \triangle PQR.
  2. Ex 3.2 Q1(ii)
    In the second pair of figure 3.19, △XYZ\triangle XYZ and △LMN\triangle LMN are drawn. Side XY and side LM each carry one tick, ∠Y\angle Y and ∠M\angle M each carry the same arc mark, and side YZ and side MN each carry two ticks. Complete : By ......... test, △XYZ≅△LMN\triangle XYZ \cong \triangle LMN.
  3. Ex 3.2 Q1(iii)
    In the third pair of figure 3.19, △PRQ\triangle PRQ and △STU\triangle STU are drawn. ∠P\angle P and ∠S\angle S each carry the same mark, side PR and side ST each carry two ticks, and ∠R\angle R and ∠T\angle T each carry the same mark. Complete : By ......... test, △PRQ≅△STU\triangle PRQ \cong \triangle STU.
  4. Ex 3.2 Q1(iv)
    In the fourth pair of figure 3.19, △LMN\triangle LMN and △PTR\triangle PTR are drawn. ∠M\angle M and ∠T\angle T are each marked as right angles, side LM and side PT each carry one tick, and side LN and side PR — the sides opposite the right angles — each carry two ticks. Complete : By ......... test, △LMN≅△PTR\triangle LMN \cong \triangle PTR.
  5. Observe the information shown in pairs of triangles given below. State the test by which the two triangles are congruent. Write the remaining congruent parts of the triangles.
    Ex 3.2 Q2(i)
    From the information shown in figure 3.20, in △ABC\triangle ABC and △PQR\triangle PQR : ∠ABC≅∠PQR\angle ABC \cong \angle PQR; seg BC ≅\cong seg QR; ∠ACB≅∠PRQ\angle ACB \cong \angle PRQ. Complete the following : ∴△ABC≅△PQR\therefore \triangle ABC \cong \triangle PQR ....... ......... test. ∴∠BAC≅\therefore \angle BAC \cong ......... (corresponding angles of congruent triangles). seg AB ≅\cong ......... and ......... ≅\cong seg PR (corresponding sides of congruent triangles).
  6. Ex 3.2 Q2(ii)
    From the information shown in figure 3.21, in which seg PS and seg QR intersect at the point T, in △PTQ\triangle PTQ and △STR\triangle STR : seg PT ≅\cong seg ST; ∠PTQ≅∠STR\angle PTQ \cong \angle STR .... vertically opposite angles; seg TQ ≅\cong seg TR. Complete the following : ∴△PTQ≅△STR\therefore \triangle PTQ \cong \triangle STR ....... ......... test. ∴∠TPQ≅\therefore \angle TPQ \cong ......... and ......... ≅∠TRS\cong \angle TRS (corresponding angles of congruent triangles). seg PQ ≅\cong ......... (corresponding sides of congruent triangles).
  7. Ex 3.2 Q3
    From the information shown in figure 3.22, state the test assuring the congruence of △ABC\triangle ABC and △PQR\triangle PQR. Write the remaining congruent parts of the triangles. In the figure, ∠A\angle A and ∠Q\angle Q are each marked as right angles, side AB and side PQ each carry one tick, and side BC and side PR — the sides opposite those right angles — each carry two ticks.
  8. Ex 3.2 Q4
    As shown in the following figure 3.23, in △LMN\triangle LMN and △PNM\triangle PNM, LM=PNLM = PN and LN=PMLN = PM. Write the test which assures the congruence of the two triangles. Write their remaining congruent parts.
  9. Ex 3.2 Q5
    In figure 3.24, seg AB ≅\cong seg CB and seg AD ≅\cong seg CD. Prove that △ABD≅△CBD\triangle ABD \cong \triangle CBD.
  10. Ex 3.2 Q6
    In figure 3.25, ∠P≅∠R\angle P \cong \angle R and seg PQ ≅\cong seg RQ. Prove that △PQT≅△RQS\triangle PQT \cong \triangle RQS.

Practice Set 3.3

4 q
  1. Ex 3.3 Q1
    Find the values of xx and yy using the information shown in figure 3.37. Find the measure of ∠ABD\angle ABD and m∠ACDm\angle ACD. [In figure 3.37, point A lies above seg BC and point D lies below it, and seg BC is drawn. Seg AB and seg AC each carry a double tick mark, so AB≅ACAB \cong AC; seg BD and seg CD each carry a single tick mark, so BD≅CDBD \cong CD. At vertex B, the angle above seg BC is ∠ABC=x\angle ABC = x and the angle below seg BC is ∠DBC=60∘\angle DBC = 60^\circ. At vertex C, the angle above seg BC is ∠ACB=50∘\angle ACB = 50^\circ and the angle below seg BC is ∠DCB=y\angle DCB = y.]
  2. Ex 3.3 Q2
    The length of hypotenuse of a right angled triangle is 15. Find the length of median of its hypotenuse.
  3. Ex 3.3 Q3
    In △PQR\triangle PQR, ∠Q=90∘\angle Q = 90^\circ, PQ=12PQ = 12, QR=5QR = 5 and QS is a median. Find l(QS)l(QS).
  4. Ex 3.3 Q4
    In figure 3.38, point G is the point of concurrence of the medians of △PQR\triangle PQR. If GT=2.5GT = 2.5, find the lengths of PG and PT. [In figure 3.38, all three medians of △PQR\triangle PQR are drawn and they meet at the point G. The point T lies on side QR, so seg PT is the median drawn from P and it passes through G. No lengths or angle measures are printed in the figure.]

Practice Set 3.4

8 q
  1. Ex 3.4 Q1
    In figure 3.48, point A is on the bisector of ∠XYZ\angle XYZ. If AX = 2 cm then find AZ. In the figure, the two arcs marked at the vertex Y show that ∠XYA≅∠AYZ\angle XYA \cong \angle AYZ; the point X lies on one arm of ∠XYZ\angle XYZ with seg AX ⊥\perp ray YX at X, and the point Z lies on the other arm with seg AZ ⊥\perp ray YZ at Z. Both right angles are marked in the figure.
  2. Ex 3.4 Q2
    In figure 3.49, ∠RST=56∘\angle RST = 56^\circ, seg PT ⊥\perp ray ST, seg PR ⊥\perp ray SR and seg PR ≅\cong seg PT. Find the measure of ∠RSP\angle RSP. State the reason for your answer. In the figure, T lies on ray ST and R lies on ray SR, and the point P lies in the interior of ∠RST\angle RST; the right angles are marked at T and at R.
  3. Ex 3.4 Q3
    In △PQR\triangle PQR, PQ = 10 cm, QR = 12 cm, PR = 8 cm. Find out the greatest and the smallest angle of the triangle.
  4. Ex 3.4 Q4
    In △FAN\triangle FAN, ∠F=80∘\angle F = 80^\circ, ∠A=40∘\angle A = 40^\circ. Find out the greatest and the smallest side of the triangle. State the reason.
  5. Ex 3.4 Q5
    Prove that an equilateral triangle is equiangular.
  6. Ex 3.4 Q6
    Prove that, if the bisector of ∠BAC\angle BAC of △ABC\triangle ABC is perpendicular to side BC, then △ABC\triangle ABC is an isosceles triangle.
  7. Ex 3.4 Q7
    In figure 3.50, if seg PR ≅\cong seg PQ, show that seg PS >> seg PQ. In the figure, the points Q, R and S are collinear with R lying between Q and S, and seg PQ, seg PR and seg PS are drawn from the point P which is not on line QS. The double tick marks on seg PQ and on seg PR show that these two segments are congruent.
  8. Ex 3.4 Q8
    In figure 3.51, in △ABC\triangle ABC, seg AD and seg BE are altitudes and AE = BD. Prove that seg AD ≅\cong seg BE. In the figure, D lies on side BC with seg AD ⊥\perp side BC, and E lies on side AC with seg BE ⊥\perp side AC; both right angles are marked in the figure.

Similar Triangles and Proportionality

1 q

Solved Examples

Worked · 1
  1. 3.5 SolvedEx.1
    Some information is shown in △ABC\triangle ABC and △PQR\triangle PQR in figure 3.57. Observe it. Hence find the lengths of side AC and PQ. [From Fig. 3.57 — in △ABC\triangle ABC: ∠B=90∘\angle B = 90^\circ, AB=3AB = 3, CB=4CB = 4; in △PQR\triangle PQR: ∠Q=90∘\angle Q = 90^\circ, PR=7.5PR = 7.5, RQ=6RQ = 6. ∠A\angle A and ∠P\angle P carry the same mark, so ∠A=∠P\angle A = \angle P.]

Practice Set 3.5

3 q
  1. Ex 3.5 Q1
    If △XYZ∼△LMN\triangle XYZ \sim \triangle LMN, write the corresponding angles of the two triangles and also write the ratios of corresponding sides.
  2. Ex 3.5 Q2
    In △XYZ\triangle XYZ, XY=4XY = 4 cm, YZ=6YZ = 6 cm, XZ=5XZ = 5 cm, If △XYZ∼△PQR\triangle XYZ \sim \triangle PQR and PQ=8PQ = 8 cm then find the lengths of remaining sides of △PQR\triangle PQR.
  3. Ex 3.5 Q3
    Draw a sketch of a pair of similar triangles. Label them. Show their corresponding angles by the same signs. Show the lengths of corresponding sides by numbers in proportion.

Problem Set 3

10 q
  1. Choose the correct alternative answer for the following questions.
    Prob Q1(i)
    If two sides of a triangle are 5 cm and 1.5 cm, the lenght of its third side cannot be . . . . . . . .
    1. A.
      3.7 cm
    2. B.
      4.1 cm
    3. C.
      3.8 cm
    4. D.
      3.4 cm
  2. Prob Q1(ii)
    In △PQR\triangle PQR, If ∠R>∠Q\angle R > \angle Q then . . . . . . . . .
    1. A.
      QR>PRQR > PR
    2. B.
      PQ>PRPQ > PR
    3. C.
      PQ<PRPQ < PR
    4. D.
      QR<PRQR < PR
  3. Prob Q1(iii)
    In △TPQ\triangle TPQ, ∠T=65∘\angle T = 65^\circ, ∠P=95∘\angle P = 95^\circ which of the following is a true statement ?
    1. A.
      PQ<TPPQ < TP
    2. B.
      PQ<TQPQ < TQ
    3. C.
      TQ<TP<PQTQ < TP < PQ
    4. D.
      PQ<TP<TQPQ < TP < TQ
  4. Prob Q2
    △ABC\triangle ABC is isosceles in which AB=ACAB = AC. Seg BD and seg CE are medians. Show that BD=CEBD = CE.
  5. Prob Q3
    In △PQR\triangle PQR, If PQ>PRPQ > PR and bisectors of ∠Q\angle Q and ∠R\angle R intersect at S. Show that SQ>SRSQ > SR.
  6. Prob Q4
    In figure 3.59, point D and E are on side BC of △ABC\triangle ABC, such that BD=CEBD = CE and AD=AEAD = AE. Show that △ABD≅△ACE\triangle ABD \cong \triangle ACE. (In the figure the points lie in the order B - D - E - C.)
  7. Prob Q5
    In figure 3.60, point S is any point on side QR of △PQR\triangle PQR. Prove that : PQ+QR+RP>2PSPQ + QR + RP > 2PS
  8. Prob Q6
    In figure 3.61, bisector of ∠BAC\angle BAC intersects side BC at point D. Prove that AB>BDAB > BD
  9. Prob Q7
    In figure 3.62, seg PT is the bisector of ∠QPR\angle QPR. A line parallel to seg PT and passing through R intersects ray QP at point S. Prove that PS=PRPS = PR.
  10. Prob Q8
    In figure 3.63, seg AD ⊥\perp seg BC. seg AE is the bisector of ∠CAB\angle CAB and C - E - D. Prove that ∠DAE=12(∠C−∠B)\angle DAE = \frac{1}{2} (\angle C - \angle B)