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Triangles

Class 9th Mathematics Part Ii MHB Solution
Practice Set 3.1
  1. In figure 3.8, ∠ACD is an exterior angle of ΔABC. ∠B = 40°, ∠A = 70°. Find the measure…
  2. In ΔPQR, ∠P = 70° ∠Q = 65° then find ∠R.
  3. The measures of angles of a triangle are x°, (x - 20)°, (x - 40)°. Find the measure of…
  4. The measure of one of the angles of a triangle is twice the measure of its smallest…
  5. In figure 3.9, measures of some angles are given. Using the measures find the values of…
  6. In figure 3.10, line AB || line DE. Find the measures of ∠DRE and ∠ARE using given…
  7. In ΔABC, bisectors of ∠A and ∠B intersect at point O. If ∠C= 70^0 Find measure of ∠AOB.…
  8. In Figure 3.11, line AB || line CD and line PQ is the transversal. Ray PT and ray QT…
  9. Using the information in figure 3.12, find the measures of ∠a, ∠b and ∠c.…
  10. In figure 3.13, line DE || line GF ray EG and ray FG are bisectors of ∠DEF and ∠DFM…
Practice Set 3.2
  1. In each of the examples given below, a pair of triangles is shown. Equal parts of…
  2. In each of the examples given below, a pair of triangles is shown. Equal parts of…
  3. In each of the examples given below, a pair of triangles is shown. Equal parts of…
  4. In each of the examples given below, a pair of triangles is shown. Equal parts of…
  5. triangle From the information shown in the figure, in ΔABC and ΔPQR ∠ABC ≅ ∠PQR seg BC…
  6. a From the information shown in the figure., In ΔPTQ and ΔSTR ∠PTQ = ∠STR ………………………..…
  7. From the information shown in the figure, state the test assuring the congruence of…
  8. As shown in the following figure, in ΔLMN and ΔPMN, LM = PN, LN = PM. Write the test…
  9. In figure 3.24, seg AB ≅ seg CB and seg AD ≅ seg CD. Prove that ΔABD ≅ ΔCBD…
  10. In figure 3.25, ∠P ≅ ∠R seg PQ ≅ seg RQ Prove that, ΔPQT ≅ ΔPQS delta…
Practice Set 3.3
  1. Find the values of x and y using the information shown in figure 3.37. Find the measure…
  2. The length of hypotenuse of a right-angled triangle is 15. Find the length of median of…
  3. In ΔPQR, ∠Q =90^0 , PQ = 12, QR = 5 and QS is a median. Find t(QS).…
  4. In figure 3.38, point G is the point of concurrence of the medians of ΔPQR. If GT =…
Practice Set 3.4
  1. In figure 3.48, point A is on the bisector of ∠XYZ. If AX = 2cm then find AZ.…
  2. In figure 3.49, ∠RST = 56°, seg PT ⊥ ray ST, seg PT ⊥ ray ST, seg PR ⊥ ray SR and PR ≅…
  3. In ΔPQR, PQ = 10 cm, QR = 12 cm, PR = 8 cm. Find out the greatest and the smallest…
  4. In ΔFAN, ∠F = 80°, ∠A = 40°. Find out the greatest and the smallest side of the…
  5. Prove that an equilateral triangle is equiangular
  6. Prove that, if the bisector of ∠BAC of ΔABC is perpendicular to side BC, then ΔABC is…
  7. In figure 3.50, if seg PR ≅ seg PQ, show that seg PS seg PQ.
  8. In figure 3.51, in ΔABC, seg AD and seg BE are altitudes and AE = BD. Prove that seg AD…
Practice Set 3.5
  1. If ΔXYZ ∼ ΔLMN write the corresponding angles of the two triangles and also write the…
  2. In In ΔXYZ, XY = 4 cm, YZ = 6 cm, XZ = 5 cm, If ΔXYZ ∼ ΔPQR and PQ = 8 cm then find the…
  3. Draw a sketch of a pair of similar triangles. Label them. Show their corresponding…
Problem Set 3
  1. If two sides of a triangle are 5 cm and 1.5 cm, the length of its third side cannot be…
  2. In ΔPQR, If ∠R ∠Q then …………. Choose the correct alternative answer for the following…
  3. In ΔTPQ, ∠T = 65^0 , ∠P = 95^0 which of the following is a true statement? Choose the…
  4. ΔABC is isosceles in which AB = AC. seg BD and seg CE are medians. Show that BD = CE.…
  5. In ΔPQR, If PQPR and bisectors of ∠Q and ∠R intersect at S. Show that SQSR.…
  6. In figure 3.59, point D and E are on side BC of ΔABD, such that BD = CE and AD = AE.…
  7. In figure 3.60, point S is any point on side QR of ΔPQR. Prove that: PQ + QR + RP 2PS…
  8. In figure 3.61, bisector of ∠BAC intersects side BC at point D. Prove that AB BD left…
  9. In figure 3.62, seg PT is the bisector of ∠QPR. A line through R intersects ray QP at…
  10. In figure 3.63, seg AD ⊥ seg BC. seg AE is the bisector of ∠CAB and C - E - D. Prove…
Practice Set 3.1
Question 1.

In figure 3.8, ∠ACD is an exterior angle of ΔABC. ∠B = 40°, ∠A = 70°. Find the measure of ∠ACD.
Answer:

Given, ∠A = 70° and ∠B = 40°


In a triangle ABC,


The measure of an exterior angle of a triangle is equal to the sum of its remote interior angles


∠ACD is an exterior angle of triangle ABC


So, from theorem of remote interior angles,


∠ACD = ∠BAC + ∠ABC


⇒ ∠ACD = ∠A + ∠B


⇒ ∠ACD = 70° + 40° = 110°
Question 2.

In ΔPQR, ∠P = 70° ∠Q = 65° then find ∠R.
Answer:

Given, ∠P = 70° ∠Q = 65°


In a triangle we know sum of interior angles is 180°


∴ in ΔPQR


∠P + ∠Q + ∠R = 180°


70° + 65° + ∠R = 180°


∠R = 180° - 135° = 45°
Question 3.

The measures of angles of a triangle are x°, (x – 20)°, (x – 40)°. Find the measure of each angle.
Answer:

Angles of a triangle are


In a triangle we know sum of interior angles is 180°


∴ x° + (x – 20)° + (x – 40)° = 180°


x° + x° - 20° + x° - 40° = 180°


3x° = 180° + 60°


x° = 240°/3


∴ x° = 80°


Angles of the triangle are x° = 80°


(x – 20)° = 80° - 20° = 60°


(x – 40)° = 80° - 40° = 40°
Question 4.

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.
Answer:

Let the measure of the smallest angle be x


Measure of second angle = 2x


Measure of third angle = 3x


In a triangle we know sum of interior angles is 180°


∴ x + 2x + 3x = 180°


⇒ 6x = 180°


⇒ x = 180°/6


⇒ x = 30°


Measure of smallest angle = x = 30°


Measure of second angle = 2x = 2 × 30° = 60°


Measure of third angle = 3x = 3 × 30° = 90°
Question 5.

In figure 3.9, measures of some angles are given. Using the measures find the values of x, y, z.
Answer:

Given ∠TEN = 100°, ∠EMR = 140°


∠NEM = y, ∠ENM = x, ∠NME = z


In a triangle ENM


The measure of an exterior angle of a triangle is equal to the sum of its remote interior angles


∠TEN and ∠EMR is an exterior angle of triangle ENM


So from theorem of remote interior angles,


∠TEN = ∠ NME + ∠ENM


⇒ 100° = z + x ……. (1)


∠EMR = ∠NEM + ∠ENM


⇒ 140° = x + y


⇒ x = 140° - y …(2)


In a triangle we know sum of interior angles is 180°


∴ x + y + z = 180 ……….(3)


Putting (1) in (3)


⇒ y + 100° = 180°


⇒ y = 180° - 100° = 80°


Putting y in (2)


∴ x = 140° - 80°


⇒ x = 60°


Putting x in (1)


∴ 60° + z = 100°


⇒ z = 100° - 60°


⇒ z = 40°


Measure of all the angles are


x = 60°, y = 80°, z = 40°
Question 6.

In figure 3.10, line AB || line DE. Find the measures of ∠DRE and ∠ARE using given measures of some angles.
Answer:

Given ∠DAB = 70° and ∠DER = 40°


In the given figure ∠DAB = ∠ADE [Alternate Interior angles are equal]


∴ ∠ADE = ∠RDE = 70°


In ΔDER,


∠DER + ∠DRE + ∠RDE = 180°


⇒ 40° + ∠DRE + 70° = 180°


⇒ ∠DRE = 180° - 110°


⇒ ∠DRE = 70°


∵ ∠ARE is an exterior angle of triangle DER


∠ARE = ∠RDE + ∠DER = 70° + 40°


⇒ ∠ARE = 110°
Question 7.

In ΔABC, bisectors of ∠A and ∠B intersect at point O. If ∠C= 700 Find measure of ∠AOB.
Answer:

The figure is attached below:


BN and AM are the angle bisectors of angle B and A respectively.



Given ∠C = 70°


In a triangle we know sum of interior angles is 180°


In ΔABC


∠A + ∠B + ∠C = 180°


∠A + ∠B = 180° - 70°


∠A + ∠B = 110°


Now in ΔAOB


AO is the bisector of ∠A


BO is the bisector of ∠B


∴ ∠OAB = ∠A/2 and ∠OBA = ∠B/2


∠OAB + ∠OBA + ∠AOB = 180°


∠A/2 + ∠B/2 + ∠AOB = 180°


⇒ ∠AOB = 180° - (∠A + ∠B)/2


⇒ ∠AOB = 180° - 110°/2 = 180° - 55°


⇒ ∠AOB = 125°
Question 8.

In Figure 3.11, line AB || line CD and line PQ is the transversal. Ray PT and ray QT are bisectors of ∠BPQ and ∠PQD respectively.

Prove that ∠PTQ = 90°.
Answer:

Given: AB || CD, line PQ is the tranversal


Ray PT and Ray QT are bisectors of ∠BPQ and ∠PQD


To prove: ∠PTQ = 90°


Proof: Since, Ray PT and Ray QT are bisectors of ∠BPQ and ∠PQD


∠TPQ = ∠BPQ/2 ……..(1)


∠PQT = ∠PQD/2 ………(2)


Since, two parallel lines are intersected by a transversal, the interior angles on either side of the transversal are supplementary.


So, ∠BPQ + ∠PQD = 180°


Dividing both sides by 2, we get


⇒ (∠BPQ + ∠PQD)/2 = 180°/2


⇒ ∠BPQ/2 + ∠PQD/2 = 90°


In ΔPQT,


∠TPQ + ∠PQT + ∠PTQ = 180°


Substituting ∠TPQ and ∠PQT from (1) and (2) respectively


⇒ ∠BPQ/2 + ∠PQD/2 + ∠PQT = 180°


⇒ 90° + ∠PQT = 180°


⇒ ∠PQT = 180° - 90°


⇒ ∠PQT = 90°


Hence, proved.
Question 9.

Using the information in figure 3.12, find the measures of ∠a, ∠b and ∠c.
Answer:

In the given triangle


a + b + c = 180° …………(1)


c + 100° = 180° ……….(2) [angles in linear pair]


⇒ c = 180° - 100°


⇒ c = 80°


b = 70° ……………..(3) [opposite angles are equal]


Putting value of b and c in (1)


⇒ a + 70° + 80° = 180°


⇒ a = 180° - 150°


⇒ a = 30°
Question 10.

In figure 3.13, line DE || line GF ray EG and ray FG are bisectors of ∠DEF and ∠DFM respectively.

Prove that,

i. ∠DEG = 1/2∠EDF

ii. EF =FG.
Answer:

Given: line DE || line GF


Ray EG and ray FG are bisectors of and respectively


To Prove: i.


ii.


Proof: Ray EG and ray FG are bisectors of and respectively.


So, ∠DEG = ∠GEF = 1/2 ∠DEF ……………..(1)


∠DFG = ∠GFM = 1/2 ∠DFM ………..(2)


Also, ∠EDF = ∠DFG …..(3) [Alternate interior angles]


In ΔDEF


∠DFM = ∠DEF + ∠EDF


From (2) and (3)


2∠EDF = ∠DEF + ∠EDF


⇒ ∠EDF = ∠DEF


From (1)


⇒ ∠EDF = 2∠DEG


⇒ ∠DEG = 1/2 ∠EDF


Hence, (i) is proved.


Line DE || line GF


From alternate interior angles


∠DEG = ∠EGF …….(4)


From (1)


∠GEF = ∠EGF


Since, in the ΔEGF sides opposite to equal angles are equal.


∴ EF = FG


Hence, (ii) is proved.
Practice Set 3.2
Question 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.
Answer:

: By SSS congruency test

ΔABC ≅ ΔPQR


Explanation:


Given, AB = PQ


BC = QR


CA = RP


∴ By SSS congruency test


ΔABC ≅ ΔPQR


SSS : Side Side Side
Question 2.

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.
Answer:

By SAS congruency test

ΔXYZ ≅ ΔLMN


Explanation:


Given: XY = LM


∠XYZ = ∠LMN


YZ = MN


Therefore, By SAS congruency test


ΔXYZ ≅ ΔLMN


SAS: Side Angle Side
Question 3.

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.
Answer:

By ASA congruency test

ΔPRQ ≅ ΔSTU


Explanation:


Given: ∠QPR = ∠UST


PR = ST


∠PRQ = ∠STU


Therefore, By ASA congruency test


ΔPRQ ≅ ΔSTU


ASA: Angle Side Angle
Question 4.

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.
Answer:

By RHS congruency test

ΔLMN ≅ ΔPTR


Explanation:


Given: LM = PT


∠LMN = ∠PTR


LN = PR


Therefore, By RHS congruency test


ΔLMN ≅ ΔPTR


RHS: Right Hypotenuse Side
Question 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.



From the information shown in the figure, in ΔABC and ΔPQR

∠ABC ≅ ∠PQR

seg BC ≅ seg QR

∠ABC = ∠PRQ

∴ ΔABC ≅ ΔPQR ……………………____ test

∴ ∠BAC = _____……………………corresponding angles of congruent triangles.

seg AB ≅ ____………………….. corresponding sides of congruent triangles.

____ = seg PR …………………corresponding side of congruent triangles.
Answer:

Given: ∠ABC = ∠PQR


BC = QR


∠ABC = ∠PRQ


∴ ΔABC ≅ ΔPQR …………………………ASA test


ASA: angle side angle


∴ ∠BAC = ∠QPR …………………….corresponding angles of congruent triangles.


seg AB = seg PQ ………………….. corresponding sides of congruent triangles.


seg AC = seg PR ………………….. corresponding angles of congruent triangles.
Question 6.

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.



From the information shown in the figure.,

In ΔPTQ and ΔSTR

∠PTQ = ∠STR ……………………….. vertically opposite angles

seg TQ ≅ seg TR

∴ ΔPTQ ≅ ΔSTR ………………._____ test

∠TPQ ≅ ____ …………………… corresponding angles of congruent triangles.

_____ ≅ ∠TRS ………………….. corresponding angles of congruent triangles.

seg PQ ≅ _____ …………….. corresponding sides of congruent triangles.
Answer:

In ΔPTQ and ΔSTR


Given: ∠PTQ = ∠STR ……………………….. vertically opposite angles


seg TQ = seg TR


seg TP = seg TS


∴ ΔPTQ = ΔSTR ……………….SAS test


SAS: side angle side


∠TPQ = ∠TSR …………………… corresponding angles of congruent triangles.


∠TQP = ∠TRS ………………….. corresponding angles of congruent triangles.


seg PQ = seg SR …………….. corresponding sides of congruent triangles.
Question 7.

From the information shown in the figure, state the test assuring the congruence of ΔABC and ΔPQR Write the remaining congruent parts of the triangles.
Answer:

In ΔABC and ΔPQR


AB = QP


BC = PR


∠CAB = ∠RQP


∴ By RHS congruency test


ΔABC ≅ ΔQPR


∴ AC = QR ……………….. corresponding sides of congruent triangles.


∠ABC = ∠ QPR ………………… corresponding angles of congruent triangles.


∠BCA = ∠PRQ ………………… corresponding angles of congruent triangles.
Question 8.

As shown in the following figure, in ΔLMN and ΔPMN, LM = PN, LN = PM. Write the test which assures the congruence of the two triangles. Write their remaining congruent parts
Answer:

Given, In ΔLMN and ΔPNM


LM = PN


LN = PN


MN = MN


∴ By SSS congruency test


ΔLMN ≅ ΔPNM


∠LMN = ∠ PNM ………………… corresponding angles of congruent triangles.


∠LNM = ∠PMN ………………… corresponding angles of congruent triangles.


∠NLM = ∠MPN ……………….. corresponding angles of congruent triangles.
Question 9.

In figure 3.24, seg AB ≅ seg CB and seg AD ≅ seg CD.

Prove that ΔABD ≅ ΔCBD
Answer:

Given, In ΔABD and ΔCBD


AB = CB


AD = CD


BD = BD ……….[Common]


∴ By SSS congruency test


ΔABD ≅ ΔCBD
Question 10.

In figure 3.25, ∠P ≅ ∠R seg PQ ≅ seg RQ

Prove that, ΔPQT ≅ ΔPQS
Answer:

In ΔPQT and ΔRQS


∠P = ∠R …………[Given]


∠QPT = ∠QRS


PQ = RQ ………….[Given]


∠PQT = ∠RQS ………….[common]


∴ By ASA congruency


ΔPQT ≅ ΔRQS
Practice Set 3.3
Question 1.

Find the values of x and y using the information shown in figure 3.37. Find the measure of ∠ABC and ∠ACB.
Answer:

In ΔABC


Given, AB = AC


Sides of a triangle are Equal then the angles opposite to them are equal.


∠ABC = ∠ACB


∴ x = 50°


So, ∠ABD = 50° + 60° = 110°


In ΔDBC


Given, DB = DC


Sides of a triangle are Equal then the angles opposite to them are equal.


∠DBC = ∠DCB


∴ y = 60°


∠ACD = ∠ACB + ∠BCD


= 50° + 60°


∴ ∠ACD = 110°
Question 2.

The length of hypotenuse of a right-angled triangle is 15. Find the length of median of its hypotenuse.
Answer:

Length of hypotenuse of right-angled triangle = 15


We know, the length of the median of the hypotenuse is half the length


of the hypotenuse.


i.e.


Length of median of its hypotenuse = 1/2 × length of hypotenuse


Length of median of its hypotenuse = 1/2 × 15


= 7.5


∴ Length of median of its hypotenuse is 7.5
Question 3.

In ΔPQR, ∠Q =900, PQ = 12, QR = 5 and QS is a median. Find t(QS).
Answer:

ΔPQR is a right-angled triangle


So, PQ and QR are the sides and PR is the hypotenuse of ΔPQR.


∴ By Pythagoras theorem


PQ2 + QR2 = PR2


⇒ PR2 = 122 + 52 = 144 + 25 = 169


⇒ PR = 13


Length of hypotenuse of right-angled triangle = 13


We know, the length of the median of the hypotenuse is half the length


of the hypotenuse.


i.e.


Length of median of its hypotenuse = 1/2 × length of hypotenuse


Length of median of its hypotenuse = 1/2 × 13


= 6.5


∴ Length of median of its hypotenuse is 6.5
Question 4.

In figure 3.38, point G is the point of concurrence of the medians of ΔPQR. If GT = 2.5, find the lengths of PG and PT.
Answer:

Given, in ΔPQR


GT = 2.5


The point of concurrence of medians of a triangle divides each median in


the ratio 2 : 1.


Since, PT is the median.


∴ PG: GT = 2: 1




⇒ PG = 2 × 2.5 = 5


Therefore, length of PG = 5


Length of PT = PG + GT


= 5 + 2.5


Length of PT = 7.5
Practice Set 3.4
Question 1.

In figure 3.48, point A is on the bisector of ∠XYZ. If AX = 2cm then find AZ.
Answer:

Given, Point A is on the bisector of ∠XYZ


AX = 2cm


Every point on the bisector of an angle is equidistant from the sides of the


angle.


Therefore, from figure


AX = AZ


∴ AZ = 2 cm
Question 2.

In figure 3.49, ∠RST = 56°, seg PT ⊥ ray ST, seg PT ⊥ ray ST, seg PR ⊥ ray SR and PR ≅ seg PT Find the measure of ∠RSP. State the reason for your answer.
Answer:

Given, ∠RST = 56°


PT perpendicular to ST


PR perpendicular to SR


PR ≅ PT


Since, PR ≅ PT


∴ Any point equidistant from sides of an angle is on the bisector of the


angle.


Therefore, Ray SP is the bisector of ∠TSR.


That is ∠RSP = ∠TSP


Now, ∠RST = ∠RSP + ∠TSP


= 2 ∠RSP


∠RSP = 1/2 ∠RST


∠RSP = 1/2 × 56°


Therefore, ∠RSP = 28°
Question 3.

In ΔPQR, PQ = 10 cm, QR = 12 cm, PR = 8 cm. Find out the greatest and the smallest angle of the triangle.
Answer:

Given, in ΔPQR, PQ = 10 cm, QR = 12 cm, PR = 8 cm



We know, If two sides of a triangle are unequal, then the angle opposite to


the greater side is greater than angle opposite to the smaller side.


Here greater side is PQ and the smallest side is PR


∴ Angle opposite to QR = ∠QPR


Angle opposite to PR = ∠PQR


Greatest angle of triangle = ∠QPR


Smallest angle of triangle = ∠PQR
Question 4.

In ΔFAN, ∠F = 80°, ∠A = 40°. Find out the greatest and the smallest side of the triangle. State the reason.
Answer:

Given In ΔFAN,


∠F = 80°, ∠A = 40°


In a triangle sum of interior angles of the triangle is 180°


∴ ∠F + ∠A + ∠N = 180°


⇒ 80° + 40° + ∠N = 180°


⇒ ∠N = 180° - 120°


⇒ ∠N = 60°


So, ∠F = 80°, ∠N = 60°, ∠A = 40°



If two angles of a triangle are unequal then the side opposite to the greater.


Angle is greater than the side opposite to smaller angle.


Here greatest angle is ∠F and the smallest angle is ∠A


Side opposite to ∠F = NA


Side opposite to ∠A = FN


Greatest side of triangle = NA


Smallest side of triangle = FN
Question 5.

Prove that an equilateral triangle is equiangular
Answer:

Given: Equilateral triangle PQR


To Prove: ∠P ≅ ∠Q ≅∠R


Proof: PQ ≅ PR ……….[all sides of an equilateral triangle are congruent.]


∠Q ≅ ∠R [the angles opposite to the two congruent sides of a triangle are congruent (Isosceles Triangle Theorem)]


PQ ≅ QR [since all sides of an equilateral triangle are congruent.]


∠R ≅ ∠P, again, by the Isosceles Triangle Theorem


Now, since ∠Q ≅ ∠R and ∠R ≅ ∠P ,


So, ∠Q ≅ ∠P


Therefore, ∠P ≅ ∠Q.


So, equilateral triangles are equiangular.
Question 6.

Prove that, if the bisector of ∠BAC of ΔABC is perpendicular to side BC, then ΔABC is an isosceles triangle.
Answer:

Given: Bisector of ∠BAC of ΔABC is perpendicular to side BC



To Prove: ΔABC is an isosceles triangle.


Proof:


In ΔABD and ΔACD


Since, AD is the angle Bisector of ΔABC


∴ ∠BAD = ∠CAD


AD = AD ……….[Common Side]


∠ADB = ∠ADC ……[Both equal to 90°]


So, by ASA congruency test


ΔABD ≅ ΔACD


Therefore,


AB = AC ………………. corresponding sides of congruent triangles.


∠ABD = ∠ACD ……………… corresponding angles of congruent triangles.


∴ ∠ABC = ∠ACB


Since, AB = AC and ∠ABC = ∠ACB so, ΔABC is an isosceles triangle.
Question 7.

In figure 3.50, if seg PR ≅ seg PQ, show that seg PS > seg PQ.
Answer:

Given:


To prove:


Proof:


In ΔPRQ


PQ = PR …………….[given]


∠R = ∠PQ ....(i) [Angles opposite to equal sides are equal]


∠PQR > ∠S …(ii) [exterior angle of a triangle is greater than each of the opposite interior angles]


From (i) and (ii)


∠R > ∠S


PS > PR [side opposite to greater angle is longer]


⇒ PS > PQ [∵ PQ = PR]
Question 8.

In figure 3.51, in ΔABC, seg AD and seg BE are altitudes and AE = BD.

Prove that seg AD ≅ seg BE
Answer:

Given: AD and BE are altitudes


AE = BD


To prove: AD ≅ BE


Proof: AD and BE are altitudes


∠ADB = ∠BEA = 90° [Given]


In ΔADB and ΔBEA


BD = AE [Given]


∠ADB = ∠BEA = 90° [Given]


AB = BA [Common side of both the triangles]


∴ By RHS congruency


ΔADB ≅ ΔBEA


So, AD ≅ BE [corresponding sides of congruent triangles]
Practice Set 3.5
Question 1.

If ΔXYZ ∼ ΔLMN write the corresponding angles of the two triangles and also write the ratios of corresponding sides.
Answer:

Given, ΔXYZ ∼ ΔLMN


Corresponding angles of the two triangles are


∠X = ∠L


∠Y = ∠M


∠Z = ∠N


Ratios of corresponding sides.
Question 2.

In In ΔXYZ, XY = 4 cm, YZ = 6 cm, XZ = 5 cm, If ΔXYZ ∼ ΔPQR and PQ = 8 cm then find the lengths of remaining sides of ΔPQR.
Answer:

Given,


In ΔXYZ, XY = 4 cm, YZ = 6 cm, XZ = 5 cm


ΔPQR, PQ = 8 cm


ΔXYZ ∼ ΔPQR


So, Ratios of corresponding sides.



⇒


⇒


⇒ QR = 6 × 2 cm and PR = 5 × 2 cm


⇒ QR = 12 cm and PR = 10 cm
Question 3.

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.
Answer:


ΔABC ∼ ΔXYZ


Corresponding Angles


∠A = ∠X


∠B = ∠Y


∠C = ∠Z


Corresponding Sides in proportion
Problem Set 3
Question 1.

Choose the correct alternative answer for the following questions.

If two sides of a triangle are 5 cm and 1.5 cm, the length of its third side cannot be ………..
A. 3.7 cm

B. 4.1 cm

C. 3.8 cm

D. 3.4 cm
Answer:

The difference between two sides is less than third side
5 – 1.5 = 3.5

So, the third side cannot be 3.4 cm
Question 2.

Choose the correct alternative answer for the following questions.

In ΔPQR, If ∠R > ∠Q then ………….
A. QR>PR

B. PQ>PR

C. PQ<PR

D. QR<PR
Answer:


∠R > ∠Q


∴ PQ > PR
Question 3.

Choose the correct alternative answer for the following questions.

In ΔTPQ, ∠T = 650, ∠P = 950 which of the following is a true statement?
A. PQ<TP

B. PQ<TQ

C. TQ<TP<PQ

D. PQ<TP<TQ
Answer:

Sum of interior angles of a triangle = 180°

∠T + ∠P + ∠Q = 180°


⇒ 65° + 95° + ∠Q = 180°


⇒ ∠Q = 180° - 160° = 20°


Since, side opposite to greater angle is greater


∴ TP < PQ < TQ
Question 4.

ΔABC is isosceles in which AB = AC. seg BD and seg CE are medians. Show that BD = CE.
Answer:


Given: ΔABC is an isosceles triangle.


BD and CE are medians.


AB = AC


1/2 AB = 1/2 AC


Since, 1/2 AB = BE = AE and 1/2 AC = AD = CD


So, BE = CD ………….(1)


Also, ∠ABC = ∠ACB


⇒ ∠EBC = ∠DCB ……….(2)


In ΔEBC and ΔDCB


BE = CD [from (1)]


∠EBC = ∠DCB [from (2)]


BC = CB [common side]


∴ By SAS congruency


ΔEBC ≅ ΔDCB


So,


CE = BD …………..corresponding sides of congruent triangles.


∴ BD = CE
Question 5.

In ΔPQR, If PQ>PR and bisectors of ∠Q and ∠R intersect at S. Show that SQ>SR.
Answer:

Given:


SQ and SR are bisectors of ∠Q and ∠R which meet at S


PQ > PR


To Prove: SQ > SR


Proof:


PQ > PR


∠PRQ > ∠PQR [angle opposite to longer side is larger] …………(1)


SQ and SR are bisectors of ∠Q and ∠R


∴ ∠SQR = 1/2 ∠PQR and ∠SRQ = 1/2 ∠PRQ


Dividing (1) by 1/2 we get


1/2 ∠PRQ > 1/2 ∠PQR


⇒ ∠SRQ > ∠SQR


⇒ SQ > SR [sides opposite to greater angle is longer]
Question 6.

In figure 3.59, point D and E are on side BC of ΔABD, such that BD = CE and AD = AE. Show that ΔABD ≅ ΔACE.
Answer:

Given: BD = CE


AD = AE


To Prove: ΔABD ≅ ΔACE


Proof:


In ΔADE


AD = AE [given]


⇒ ∠ADE = ∠AED [angles opposite to equal sides are equal] ……(1)


Subtracting 180° from (1)


⇒ 180° - ∠ADE = 180° - ∠AED


⇒ ∠ADB = ∠AEC (2)


In ΔABD and ΔACE


BD = CE [Given]


AD = AE [Given]


∠ADB = ∠AEC [from (2)]


∴ By SAS congruency test


ΔABD ≅ ΔACE
Question 7.

In figure 3.60, point S is any point on side QR of ΔPQR.

Prove that: PQ + QR + RP > 2PS
Answer:

Given: S is any point on side QR of ΔPQR.


To Prove: PQ + QR + RP > 2PS


Proof:


We know, sum of two sides of triangle is greater than the third side


∴ In ΔPQS


PQ + QS > PS …………(1)


In Δ PSR


PR + SR > PS ……..(2)


Adding (1) and (2)


PQ + QS + PR + SR > PS + PS


⇒ PQ + QS + SR + PR > 2PS


⇒ PQ + QR + PR > 2PS [QR = QS + SR]


Hence, proved.
Question 8.

In figure 3.61, bisector of ∠BAC intersects side BC at point D.

Prove that AB > BD
Answer:

Given: AD is bisector of ∠BAC


To Prove: AB > BD


Proof: AD is bisector of ∠BAC


⇒ ∠BAD = ∠DAC …..(1)


Now, In ΔADC, ∠ ADB is the exterior angle


∠ADB > ∠DAC ..(2) [exterior angle of a triangle is greater than each


of the opposite interior angles]


Substituting ∠DAC = ∠BAD in (2)


⇒ ∠ADB > ∠BAD


⇒ AB > BD [side opposite to larger angle is larger]
Question 9.

In figure 3.62, seg PT is the bisector of ∠QPR. A line through R intersects ray QP at point S. Prove that PS = PR
Answer:

Given: PT is angle bisector of ∠QPR


⇒ ∠QPT = ∠RPT


A line through R parallel to PT intersects ray QP at S


RS || PT


To Prove: PS = PR


Proof:


PT is angle bisector of ∠QPR


⇒ ∠QPT = ∠RPT


∠QPR = ∠QPT + ∠RPT


∠QPR = 2∠RPT (1)


RS || PT, PR is the transversal


So, ∠RPT = ∠PRS [Alternate interior angles] (2)


For ΔPRS ∠RPQ is the remote exterior angle.


∠PSR + ∠PRS = ∠QPR


Substituting (1) and (2) in the above equation


∠RPT + ∠PSR = 2∠RPT


⇒ ∠PSR = ∠RPT (3)


From (2) and (3)


∠PRS = ∠PSR


⇒ PS = PR [Sides opposite to equal angles are equal]
Question 10.

In figure 3.63, seg AD ⊥ seg BC. seg AE is the bisector of ∠CAB and C - E - D. Prove that ∠DAE = 1/2 (∠B - ∠C)
Answer:

Given: AE is bisector of ∠CAB.


AD is perpendicular to CB


To Prove: ∠DAE = 1/2 (∠B - ∠C)


Proof:


We know that ∠BAE = 1/2 ∠A (1)


∠B + ∠BAD = 90°


∠BAD = 90° - ∠B ……………..(2)


On putting equations (1) and (2)


∠DAE = ∠BAE - ∠BAD


= 1/2 ∠A - (90° - ∠B)


= 1/2 ∠A – 90° + ∠B


= 1/2 ∠A - 1/2 (∠C + ∠A + ∠B) + ∠B


= 1/2 ∠A - 1/2 ∠A - 1/2 ∠B – 1/2 ∠C + ∠B


= 1/2 ∠B – 1/2 ∠C


∴ ∠DAE = 1/2 (∠B - ∠C)