# Class 9 RD Sharma Solution – Chapter 10 Congruent Triangles- Exercise 10.5

### Question 1. 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.

Solution:

Given: D is the mid-point of BC so, BD = DC, ED = FD, and ED âŠ¥ AB, FD âŠ¥ AC, so ED = FD

Prove: Î”ABC is an isosceles triangle

In Î”BDE and Î”CDF

ED = FD [Given]

BD = DC [D is mid-point]

âˆ BED = âˆ CFD = 90Â°

By RHS congruence criterion

Î”BDE â‰… Î”CDF

So, now by C.P.C.T

BE = CF â€¦ (i)

Now, in â–³AED and â–³AFD

ED = FD [Given]

âˆ AED = âˆ AFD = 90Â°

By RHS congruence criterion

â–³AED â‰… â–³AFD

So, now by C.P.C.T

So, EA = FA â€¦ (ii)

Now by adding equation (i) and (ii), we get

BE + EA = CF + FA

AB = AC

So, Î”ABC is an isosceles triangle because two sides of the triangle are equal.

Hence proved

### Question 2. ABC is a triangle in which BE and CF are, respectively, the perpendiculars to the sides AC and AB. If BE = CF, prove that Î”ABC is isosceles.

Solution:

Given: BE âŠ¥ AC, CF âŠ¥ AS, BE = CF.

To prove: Î”ABC is isosceles

In Î”BCF and Î”CBE,

âˆ BFC = CEB = 90Â° [Given]

BC = CB [Common side]

And CF = BE [Given]

By RHS congruence criterion

Î”BFC â‰… Î”CEB

So, now by C.P.C.T

âˆ FBC = âˆ EBC

âˆ ABC = âˆ ACB

and AC = AB [Because opposite sides to equal angles are equal]

So, Î”ABC is isosceles

Hence proved

### Question 3. If perpendiculars from any point within an angle on its arms are congruent. Prove that it lies on the bisector of that angle.

Solution:

Let us consider âˆ ABC and BP is an arm withinâˆ ABC

So now draw perpendicular from point P on arm BA and BC, i.e., PN and PM

Prove: BP is the angular bisector of âˆ ABC.

In Î”BPM and Î”BPN

âˆ BMP = âˆ BNP = 90Â° [Given]

MP = NP [Given]

BP = BP [Common side]

So, by RHS congruence criterion

Î”BPM â‰… Î”BPN

So, by C.P.C.T

âˆ MBP = âˆ NBP

and BP is the angular bisector of âˆ ABC.

Hence proved

### Question 4. In figure, AD âŠ¥ CD and CB âŠ¥ CD. If AQ = BP and DP = CQ, prove that âˆ DAQ = âˆ CBP.

Solution:

Given that AD âŠ¥ CD, CB âŠ¥ CD, AQ = BP and DP = CQ,

Prove:âˆ DAQ = âˆ CBP

We have DP = CQ

So by adding PQ on both sides, we get

DP + PQ = CQ + PQ

DQ = CP … (i)

In Î”DAQ and Î”CBP

We have

âˆ ADQ = âˆ BCP = 90Â° [Given]

And DQ = PC [From (i)]

So, by RHS congruence criterion

Î”DAQ â‰… Î”CBP

So, by C.P.C.T

âˆ DAQ = âˆ CBP

Hence proved

### Question 5. ABCD is a square, X and Y are points on sides AD and BC respectively such that AY = BX. Prove that BY = AX and âˆ BAY = âˆ ABX.

Solution:

In ABCD square,

X and Y are points on sides AD and BC

So, AY = BX.

To prove: BY = AX and âˆ BAY = âˆ ABX

Now, join Band X, A and Y

So,

âˆ DAB = âˆ CBA = 90Â° [Given ABCD is a square]

Also, âˆ XAB = âˆ YAB = 90Â°

In Î”XAB and Î”YBA

âˆ XAB = âˆ YBA = 90Â° [given]

AB = BA [Common side]

So, by RHS congruence criterion

Î”XAB â‰… Î”YBA

So, by C.P.C.T

BY = AX

âˆ BAY = âˆ ABX

Hence proved

### Question 6. 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 equilateral triangle is 60

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

(v) The bisectors of two equal angles of a triangle 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 triangle, then the two triangles are congruent.

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

Solution:

(i) False

(ii) True

(iii) True

(iv) False

(v) True

(vi) False

(vii) False

(viii) False

(ix) True

### Question 7. Fill the blanks In the following so that each of the following statements is true.

(i) Sides opposite to equal angles of a triangle are ___

(ii) Angle opposite to equal sides of a triangle are ___

(iii) In an equilateral triangle all angles are ___

(iv) In Î”ABC, if âˆ A = âˆ C, then AB =

(v) If altitudes CE and BF of a triangle ABC are equal, then AB  ___

(vi) In an isosceles triangle ABC with AB = AC, if BD and CE are its altitudes, then BD is ___ CE.

(vii) In right triangles ABC and DEF, if hypotenuse AB = EF and side AC = DE, then Î”ABC â‰… Î” ___

Solution:

(i) Equal

(ii) Equal

(iii) Equal

(iv) AB = BC

(v) AB = AC

(vi) BD is equal to CE

(vii) Î”ABC â‰… Î”EFD.

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