RD SHARMA Solutions for Class 9 Maths Chapter 12 - Congruent Triangles
Chapter 12 - Congruent Triangles Exercise 12.85
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
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°
Solution 9
Question 10
Which of the following is not a criterion for congruence of triangles?
(a) SAS
(b) SSA
(c) ASA
(d) SSS
Solution 10
Chapter 12 - Congruent Triangles Exercise 12.86
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Chapter 12 - Congruent Triangles Exercise 12.87
Question 1
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°
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
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
Solution 5
Chapter 12 - Congruent Triangles Exercise 12.88
Question 1
Solution 1
Chapter 12 - Congruent Triangles Exercise Ex. 12.1
Question 1
In fig., the sides BA and CA have been produced such that BA = AD and CA = AE.
Solution 1
Question 2
Solution 2
Question 3
Prove that the medians of an equilateral triangle are equal.
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
The vertical angle of an isosceles triangle is 100o. Find its base angles.
Solution 6
Question 7
In fig., AB = Ac and ∠ACD = 105°, find ∠BAC.
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
In fig., AB =AC and DB = DC, find the ratio ∠ABD = ∠ACD.
Solution 10
Question 11
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 = 90o and AB = AC. Find B and C.
Solution 11
Question 12
Solution 12
Question 13
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.
Solution 13
Chapter 12 - Congruent Triangles Exercise Ex. 12.2
Question 1
Solution 1
Question 2
In fig., it is given RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3 prove that ΔRBT ≅ ΔSAT.
Solution 2
Question 3
Solution 3
Chapter 12 - Congruent Triangles Exercise Ex. 12.3
Question 1
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.
Solution 1
Let ABC and DEF be two right triangles.
Question 2
Solution 2
Question 3
Solution 3
Question 4
Show that the angles of an equilateral triangle are 60o each.
Solution 4
Let us consider that ABC is an equilateral triangle.
So, AB = BC = AC
Now, AB = AC
So, AB = BC = AC
Now, AB = AC
⇒ ∠C = ∠B (angles opposite to equal sides of a triangle are equal)
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 = 180o
⇒ ∠A + ∠A + ∠A = 180o
⇒ 3∠A = 180o
⇒ ∠A = 60o
⇒ ∠A = ∠B = ∠C = 60o
Hence, in an equilateral triangle all interior angles are of 60o.
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 = 180o
⇒ ∠A + ∠A + ∠A = 180o
⇒ 3∠A = 180o
⇒ ∠A = 60o
⇒ ∠A = ∠B = ∠C = 60o
Hence, in an equilateral triangle all interior angles are of 60o.
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Chapter 12 - Congruent Triangles Exercise Ex. 12.4
Question 1
In fig., it is given that Ab = CD and AD = BC. prove that ΔADC ≅ ΔCBA
Solution 1
Question 2
Solution 2
Chapter 12 - Congruent Triangles Exercise Ex. 12.5
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
In fig., AD ⊥ CD and CB ⊥ CD. If AQ = BP an DP = CQ, prove that ∠DAQ = ∠CBP.
Solution 4
Question 5
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 60o.
(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.
Solution 5
(i) False
(ii) True
(iii) True
(iv) False
(v) True
(vi) False
(vii) False
(viii) False
(ix) True
Question 6
Solution 6
(i) equal
(ii) equal
(iii) equal
(iv) BC
(v) AC
(vi) equal to
(vii) EFD
Question 7
Solution 7
Chapter 12 - Congruent Triangles Exercise Ex. 12.6
Question 1
Solution 1
Question 2
Solution 2
Question 3
Solution 3
Question 4
Is it possible to draw a triangle with sides of length 2cm, 3cm and 7 cm?
Solution 4
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.
Question 5
Solution 5
Question 6
Solution 6
Question 7
In fig., prove that:
i. CD + DA + AB + BC > 2AC
ii. CD + DA + AB > BC
Solution 7
Question 8
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.
Solution 8
(i) False
(ii) True
(iii) True
(iv) False
(v) True
(vi) True
Question 9
Solution 9
(i) largest
(ii) less
(iii) greater
(iv) smaller
(v) less
(vi) greater
Question 10
Solution 10
Question 11
Solution 11
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