#### Chapters

Chapter 2: Exponents of Real Numbers

Chapter 3: Rationalisation

Chapter 4: Algebraic Identities

Chapter 5: Factorisation of Algebraic Expressions

Chapter 6: Factorisation of Polynomials

Chapter 7: Introduction to Euclid’s Geometry

Chapter 8: Lines and Angles

Chapter 9: Triangle and its Angles

Chapter 10: Congruent Triangles

Chapter 11: Co-ordinate Geometry

Chapter 12: Heron’s Formula

Chapter 13: Linear Equations in Two Variables

Chapter 14: Quadrilaterals

Chapter 15: Areas of Parallelograms and Triangles

Chapter 16: Circles

Chapter 17: Constructions

Chapter 18: Surface Areas and Volume of a Cuboid and Cube

Chapter 19: Surface Areas and Volume of a Circular Cylinder

Chapter 20: Surface Areas and Volume of A Right Circular Cone

Chapter 21: Surface Areas and Volume of a Sphere

Chapter 22: Tabular Representation of Statistical Data

Chapter 23: Graphical Representation of Statistical Data

Chapter 24: Measures of Central Tendency

Chapter 25: Probability

#### RD Sharma Mathematics Class 9 by R D Sharma (2018-19 Session)

## Chapter 10 : Congruent Triangles

#### Pages 12 - 13

In Fig. 10.22, the sides BA and CA have been produced such that: BA = AD and CA = AE.

Prove that segment DE || BC.

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.

In Fig. 10.23, PQRS is a square and SRT is an equilateral triangle. Prove that

(i) PT = QT (ii) ∠TQR = 15°

Prove that the medians of an equilateral triangle are equal.

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

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

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

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

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

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

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

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

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

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

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.

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

#### Pages 38 - 39

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.

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

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

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.

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.

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.

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

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

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°.

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

#### Page 47

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

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

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.

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

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

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

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.

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

Sides opposite to equal angles of a triangle may be unequal

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

Angles opposite to equal sides of a triangle are equal

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

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

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.

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.

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

The bisectors of two equal angles of a triangle are equal

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.

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

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

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

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

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

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

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

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

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 = ....

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.

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

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

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

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

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

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)

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.

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.

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.

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.

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.

Fill in the blank to make the following statement true.

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

Fill in the blank to make the following statement true.

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

Fill in the blank to make the following statement true.

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

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.

Fill in the blank to make the following statement true.

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

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.

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 Mathematics Class 9 by R D Sharma (2018-19 Session)

#### Textbook solutions for Class 9

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

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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.

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