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R.D. Sharma solutions Mathematics for Class 9 by R D Sharma (2018-19 Session) chapter 10 Congruent Triangles

Chapters

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

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

Chapter 10 - Congruent Triangles

Page 0

In a ΔABC, if ∠A = 55°, ∠B = 40°, find ∠C. 

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. 

If the angles of a triangle are in the ratio 1: 2 : 3, determine three angles. 

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

 

The angles of a triangle are (x − 40)°, (x − 20)° and `(1/2x-10)^@.` find the value of x 

The angles of a triangle are arranged in ascending order of magnitude. If the difference
between two consecutive angles is 10°, find the three angles. 

Prove that the medians of an equilateral triangle are equal. 

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

 

Two angles of a triangle are equal and the third angle is greater than each of those angles
by 30°. Determine all the angles of the triangle. 

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

If one angle of a triangle is equal to the sum of the other two, show that the triangle is a
right triangle. 

ABC is a triangle in which ∠A — 72°, the internal bisectors of angles B and C meet in O.
Find the magnitude of ∠ROC.

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

The bisectors of base angles of a triangle cannot enclose a right angle in any case. 

 

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

If the bisectors of the base angles of a triangle enclose an angle of 135°, prove that the triangle is a right triangle.  

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

In a ΔABC, ∠ABC = ∠ACB and the bisectors of ∠ABC and ∠ACB intersect at O such that ∠BOC = 120°. Show that ∠A = ∠B = ∠C = 60°. 

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 

 

Can a triangle have two right angles? Justify your answer in case.

Can a triangle have two obtuse angles?  Justify your answer in  case.

Can a triangle have two acute angles?Justify your answer in case. 

Can a triangle have  All angles more than 60°? Justify your answer in case.

 

Can a triangle have All angles less than 60° Justify your answer in case.

Can a triangle have All angles equal to 60°? Justify your answer in case.

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

If each angle of a triangle is less than the sum of the other two, show that the triangle is acute angled. 

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 0

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. 

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. 

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

 

Page 0

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

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 isosceles.  

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. 

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

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.  

 

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

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

In Figure 10.24, AB = AC and ∠ACD =105°, find ∠BAC. 

 

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

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

 

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. 

 

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

Page 0

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. 

 

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. 

 

Page 0

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): 

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. 

Prove that the perimeter of a triangle is greater than the sum of its altitudes. 

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 ……

Page 0

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) 

In Fig. 10.131, 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)? 

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. 

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

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