# RD SHARMA Solutions for Class 9 Maths Chapter 12 - Congruent Triangles

## Chapter 12 - Congruent Triangles Exercise Ex. 12.1

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

Prove that the medians of an equilateral triangle are equal.

The vertical angle of an isosceles triangle is 100^{o}. Find its base angles.

In fig., AB = Ac and ∠ACD = 105°, find ∠BAC.

In fig., AB =AC and DB = DC, find the ratio ∠ABD = ∠ACD.

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

ABC is a right-angled triangle in which A = 90^{o} and AB = AC. Find B and C.

AB is a line segment. 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.). Show that the line PQ is perpendicular bisector of AB.

## Chapter 12 - Congruent Triangles Exercise Ex. 12.2

In fig., it is given RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3 prove that ΔRBT ≅ ΔSAT.

## Chapter 12 - Congruent Triangles Exercise Ex. 12.3

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

Let ABC and DEF be two right triangles.

^{o}each.

So, AB = BC = AC

Now, AB = AC

We also have

AC = BC

⇒ ∠B = ∠A (angles opposite to equal sides of a triangle are equal)

So, we have

∠A = ∠B = ∠C

Now, in ΔABC

∠A + ∠B + ∠C = 180

^{o}

⇒ ∠A + ∠A + ∠A = 180

^{o}

⇒ 3∠A = 180

^{o}

⇒ ∠A = 60

^{o}

⇒ ∠A = ∠B = ∠C = 60

^{o}

Hence, in an equilateral triangle all interior angles are of 60

^{o}.

## Chapter 12 - Congruent Triangles Exercise Ex. 12.4

In fig., it is given that Ab = CD and AD = BC. prove that ΔADC ≅ ΔCBA

## Chapter 12 - Congruent Triangles Exercise Ex. 12.5

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

Which of the following statements are True (T) and which are False (f):

(i) Sides opposite to equal angles of a triangle may be unequal.

(ii) Angles opposite to equal sides of a triangle are equal.

(iii) The measure of each angle of an equilaterial triangle is 60^{o}.

(iv) If the altitude from one vertex of a triangle bisects the opposite side, then the triangle may be isoscles.

(v) The bisectors of two equal angles of a traingle are equal.

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

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

(viii) If any two sides of a right triangle are respectively equal to two sides of other right triagnle, then the two triangles are congruent.

(ix) Two right triangles are congruent if hypotenuse and a side of one triangle are respectively equal to the hypotenuse and a side of the other triangle.

(i) False

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

(vii) False

(viii) False

(ix) True

(i) equal

(ii) equal

(iii) equal

(iv) BC

(v) AC

(vi) equal to

(vii) EFD

## Chapter 12 - Congruent Triangles Exercise Ex. 12.6

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

Here, 2 + 3 < 7

Hence, it is not possible because triangle can be drawn only if the sum of any two sides is greater than third side.

In fig., prove that:

i. CD + DA + AB + BC > 2AC

ii. CD + DA + AB > BC

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

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

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

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

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

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

(vi) 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.

(i) False

(ii) True

(iii) True

(iv) False

(v) True

(vi) True

(i) largest

(ii) less

(iii) greater

(iv) smaller

(v) less

(vi) greater

## Chapter 12 - Congruent Triangles Exercise 12.85

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, then the measure of vertex angle of the triangle is

(a) 100°

(b) 120°

(c) 110°

(d) 130°

Which of the following is not a criterion for congruence of triangles?

(a) SAS

(b) SSA

(c) ASA

(d) SSS

## Chapter 12 - Congruent Triangles Exercise 12.86

## Chapter 12 - Congruent Triangles Exercise 12.87

In the figure, ABC is an isosceles triangle whose side AC is produced to E. Through C, CD is drawn parallel to BA. The value of x is

(a) 52°

(b) 76°

(c) 156°

(d) 104°

In figure, X is a point in the interior of square ABCD. AXYZ is also a square. If DY = 3 cm and AZ = 2 cm, then BY =

(a) 5 cm

(b) 6 cm

(c) 7 cm

(d) 8 cm

## Chapter 12 - Congruent Triangles Exercise 12.88

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