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# RD Sharma solutions for Class 9 Mathematics chapter 10 - Congruent Triangles

## Chapter 10 : Congruent Triangles

#### Pages 12 - 13

Q 1 | Page 12

In Fig. 10.22, the sides BA and CA have been produced such that: BA = AD and CA = AE.
Prove that segment DE || BC.

Q 2 | Page 12

In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP
respectively. Prove that: LN = MN.

Q 3 | Page 12

In Fig. 10.23, PQRS is a square and SRT is an equilateral triangle. Prove that
(i) PT = QT (ii) ∠TQR = 15°

Q 4 | Page 13

Prove that the medians of an equilateral triangle are equal.

Q 5 | Page 13

In a ΔABC, if ∠A=l20° and AB = AC. Find ∠B and ∠C.

Q 6 | Page 13

In a ΔABC, if AB = AC and ∠B = 70°, find ∠A.

Q 7 | Page 13

The vertical angle of an isosceles triangle is 100°. Find its base angles.

Q 8 | Page 13

In Figure AB = AC and ∠ACD =105°, find ∠BAC.

Q 9 | Page 13

Find the measure of each exterior angle of an equilateral triangle.

Q 11 | Page 13

In figure, AB = AC and DB = DC, find the ratio ∠ABD : ∠ACD

Q 11 | Page 13

If the base of an isosceles triangle is produced on both sides, prove that the exterior angles so formed are equal to each other.

Q 12 | Page 13

Determine the measure of each of the equal angles of a right-angled isosceles triangle.

Q 13 | Page 13

AB is a line seg P and Q are points on opposite sides of AB such that each of them is equidistant from the points A and B (See Fig. 10.26). Show that the line PQ is perpendicular bisector of AB.

#### Page 21

Q 1 | Page 21

In Fig. 10.40, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ≅ ΔSAT

Q 2 | Page 21

Two lines AB and CD intersect at O such that BC is equal and parallel to A Prove that the lines AB and CD bisect at O.

Q 3 | Page 21

BD and CE are bisectors of ∠B and ∠C of an isosceles ΔABC with AB = AC. Prove that BD = CE.

#### Pages 38 - 39

Q 1 | Page 38

In two right triangles one side an acute angle of one are equal to the corresponding side and angle of the othe Prove that the triangles are congruent.

Q 2 | Page 38

If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosce

Q 3 | Page 39

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.

Q 4 | Page 39

PQR is a triangle in which PQ = PR and S is any point on the side PQ. Through S, a line is drawn parallel to QR and intersecting PR at T. Prove that PS = PT.

Q 5 | Page 39

In a ΔABC, it is given that AB = AC and the bisectors of ∠B and ∠C intersect at O. If M is a point on BO produced, prove that ∠MOC = ∠ABC.

Q 6 | Page 39

P is a point on the bisector of an angle ∠ABC. If the line through P parallel to AB meets BC at Q, prove that triangle BPQ is isosceles.

Q 7 | Page 39

Prove that each angle of an equilateral triangle is 60°.

Q 8 | Page 39

Angles A, B, C of a triangle ABC are equal to each other. Prove that ΔABC is equilateral.

Q 9 | Page 39

ABC is a triangle in which ∠B = 2 ∠C . D is a point on BC such that AD bisects ∠BAC and AB = CD. Prove that ∠BAC = 72°.

Q 10 | Page 39

ABC is a right angled triangle in which ∠A = 90° and AB = AC. Find ∠B and ∠C.

#### Page 47

Q 1 | Page 47

In Fig. 10.92, it is given that AB = CD and AD = BC. Prove that ΔADC ≅ ΔCBA.

Q 2 | Page 47

In a ΔPQR, if PQ = QR and L, M and N are the mid-points of the sides PQ, QR and RP
respectively. Prove that LN = MN.

#### Pages 51 - 52

Q 1 | Page 51

ABC is a triangle and D is the mid-point of BC. The perpendiculars from D to AB and AC are equal. Prove that the triangle is isosceles.

Q 2 | Page 51

ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that ΔABC is isosceles

Q 3 | Page 51

If perpendiculars from any point within an angle on its arms are congruent, prove that it lies on the bisector of that angle.

Q 4 | Page 51

In Fig. 10.99, AD ⊥ CD and CB ⊥. CD. If AQ = BP and DP = CQ, prove that ∠DAQ = ∠CBP.

Q 5 | Page 51

ABCD is a square, X and Yare points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and ∠BAY = ∠4BX.

Q 6.1 | Page 52

Which of the following statements are true (T) and which are false (F):

Sides opposite to equal angles of a triangle may be unequal

Q 6.2 | Page 52

Which of the following statements are true (T) and which are false (F):

Angles opposite to equal sides of a triangle are equal

Q 6.3 | Page 52

Which of the following statements are true (T) and which are false (F):

The measure of each angle of an equilateral triangle is 60°

Q 6.4 | Page 52

Which of the following statements are true (T) and which are false (F) :

If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isosceles.

Q 6.5 | Page 52

Which of the following statements are true (T) and which are false (F):

If the bisector of the vertical angle of a triangle bisects the base, then the triangle may be isosceles.

Q 6.5 | Page 52

Which of the following statements are true (T) and which are false (F):

The bisectors of two equal angles of a triangle are equal

Q 6.7 | Page 52

Which of the following statements are true (T) and which are false (F):

The two altitudes corresponding to two equal sides of a triangle need not be equal.

Q 7.1 | Page 52

Fill the blank in the following so that each of the following statements is true.

Sides opposite to equal angles of a triangle are ......

Q 7.2 | Page 52

Fill the blanks in the following so that each of the following statements is true.

Angle opposite to equal sides of a triangle are .....

Q 7.3 | Page 52

Fill the blank in the following so that each of the following statements is true.

In an equilateral triangle all angles are .....

Q 7.4 | Page 52

Fill the blank in the following so that each of the following statements is true.

In a ΔABC if ∠A = ∠C , then AB = ......

Q 7.5 | Page 52

Fill the blank in the following so that each of the following statements is true.

If altitudes CE and BF of a triangle ABC are equal, then AB = ....

Q 7.6 | Page 52

Fill the blank in the following so that each of the following statement is true

In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is …… CE.

Q 7.7 | Page 52

Fill the blank in the following so that each of the following statement is true.

In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then ΔABC ≅ Δ ……

#### Pages 66 - 67

Q 1 | Page 66

In ΔABC, if ∠A = 40° and ∠B = 60°. Determine the longest and shortest sides of the triangle.

Q 2 | Page 66

In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?

Q 3 | Page 66

In ΔABC, side AB is produced to D so that BD = BC. If ∠B = 60° and ∠A = 70°, prove that: (i) AD > CD (ii) AD > AC

Q 4 | Page 66

Is it possible to draw a triangle with sides of length 2 cm, 3 cm and 7 cm?

Q 5 | Page 66

O is any point in the interior of ΔABC. Prove that
(i) AB + AC > OB + OC
(ii) AB + BC + CA > OA + QB + OC
(iii) OA + OB + OC > 1/2(AB + BC + CA)

Q 10.1 | Page 67

Which of the following statements are true (T) and which are false (F)?

Sum of the three sides of a triangle is less than the sum of its three altitudes.

Q 10.2 | Page 67

Which of the following statements are true (T) and which are false (F)?

Sum of any two sides of a triangle is greater than twice the median drawn to the third side.

Q 10.3 | Page 67

Which of the following statements are true (T) and which are false (F)?

Sum of any two sides of a triangle is greater than the third side.

Q 10.5 | Page 67

Which of the following statements are true (T) and which are false (F)?

Of all the line segments that can be drawn from a point to a line not containing it, the perpendicular line segment is the shortest one.

Q 10.5 | Page 67

Which of the following statements are true (T) and which are false (F)?

If two angles of a triangle are unequal, then the greater angle has the larger side opposite to it.

Q 11.1 | Page 67

Fill in the blank to make the following statement true.

In a right triangle the hypotenuse is the .... side.

Q 11.2 | Page 67

Fill in the blank to make the following statement true.

The sum of three altitudes of a triangle is ..... than its perimeter.

Q 11.3 | Page 67

Fill in the blank to make the following statement true.

The sum of any two sides of a triangle is .... than the third side.

Q 11.4 | Page 67

Fill in the blank to make the following statement true.

If two angles of a triangle are unequal, then the smaller angle has the side opposite to it.

Q 11.5 | Page 67

Fill in the blank to make the following statement true.

Difference of any two sides of a triangle is than the third side.

Q 11.6 | Page 67

Fill in the blank to make the following statement true.

If two sides of a triangle are unequal, then the larger side has .... angle opposite to it.

Q 104 | Page 67

Which of the following statements are true (T) and which are false (F)?

Difference of any two sides of a triangle is equal to the third side.

## RD Sharma solutions for Class 9 Mathematics chapter 10 - Congruent Triangles

RD Sharma solutions for Class 9 Maths chapter 10 (Congruent Triangles) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Mathematics for Class 9 by R D Sharma (2018-19 Session) solutions in a manner that help students grasp basic concepts better and faster.

Further, we at shaalaa.com are providing such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 9 Mathematics chapter 10 Congruent Triangles are Inequalities in a Triangle, Some More Criteria for Congruence of Triangles, Properties of a Triangle, Criteria for Congruence of Triangles, Congruence of Triangles, Concept of Triangles.

Using RD Sharma Class 9 solutions Congruent Triangles exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 9 prefer RD Sharma Textbook Solutions to score more in exam.

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