RD Sharma solutions for Mathematics [English] Class 9 chapter 12 - Congruent Triangles [Latest edition]

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. 

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. 

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

In figure, 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.

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

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. 

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

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 AD. Prove that the lines AB and CD bisect at O.

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. 

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

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.  

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°. 

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

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. 

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): 

The bisectors of two equal angles of a triangle are equal 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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? 

In Δ ABC, ∠B = 35°, ∠C = 65° and the bisector of ∠BAC meets BC in P. Arrange AP, BP and CP in descending order.

In the given figure, prove that:

CD + DA + AB + BC > 2AC

In the given figure, prove that:

CD + DA + AB > BC

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)? 

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

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. 

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. 

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. 

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) 

Prove that in a quadrilateral the sum of all the sides is greater than the sum of its diagonals.

In triangles ABC and CDE, if AC = CE, BC = CD, ∠A = 60°, ∠C = 30° and ∠D = 90°. Are two triangles congruent?

In the given figure, if AB = AC and ∠B = ∠C. Prove that BQ = CP.

Mark the correct alternative in each of the following:

If ABC ≅  ΔLKM, then side of ΔLKM equal to side AC of ΔABC is

If ΔABC ≅  ΔABC is isosceles with

ΔPQR and ΔABC is not congruent to ΔRPQ, then which of the following is not true:

In triangles ABC and PQR three equality relations between some parts are as follows:
AB = QP, ∠B = ∠P and BC = PR

State which of the congruence conditions applies:

In triangles ABC and PQR, if ∠A = ∠R, ∠B = ∠P and AB = RP, then which one of the following congruence conditions applies:

In ΔPQR ≅ ΔEFD then ED =

If ΔPQR≅ ΔEFD, then ∠E =

In a ΔABC, if AB = AC and BC is produced to D such that ∠ACD = 100°, then ∠A =

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

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

In the given figure, the measure of ∠B'A'C' is

If ABC and DEF are two triangles such that ΔABC \[\cong\] ΔFDE and AB = 5cm, ∠B = 40°

In the given figure, AB ⊥ BE and FE ⊥ BE. If BC = DE and AB = EF, then ΔABD is congruent to

In the given figure, if AE || DC and AB = AC, the value of ∠ABD is

In the given 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

In the given figure, if AC is bisector of ∠BAD such that AB = 3 cm and AC =  5 cm, then CD =

D, E, F are the mid-point of the sides BC, CA and AB respectively of ΔABC. Then ΔDEF is congruent to triangle

ABC is an isosceles triangle such that AB = AC and AD is the median to base BC. Then, ∠BAD =

In the given 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 =

In the given figure, ABC is a triangle in which ∠B = 2∠C. D is a point on side BC such that ADbisects ∠BAC and AB = CD. BE is the bisector of ∠B. The measure of ∠BAC is