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#### Question 1:

In the given figure, the sides BA  and CA have been produced such that BA = AD and CA = AE. Prove that segment DE || BC.

It is given that

We have to prove that

Now considering the two triangles we have

In

(Given)

(Given)

We need to show to prove.

Now

(Vertically opposite angle)

So by congruence criterion we have

So and

Then

, and

Hence from above conditions .

#### Question 2:

In a ΔPQR, if PQQR and L, M and N are the mid-points of the sides PQ, OR, and RP respectively. Prove that LN = MN.

It is given that

And is the mid point of

So

And is the mid point of

So

And is the mid point of

So

We have to prove that

In we have

(Equilateral triangle)

Then

, and

, and

Similarly comparing and we have

, and

And (Since N is the mid point of )

So by congruence criterion, we have

Hence.

#### Question 3:

Prove that the medians of an equilateral triangle are equal.

We have to prove that the median of an equilateral triangle are equal.

Let be an equilateral triangle with as its medians.

Let

In we have

(Since similarly)

(In equilateral triangle, each angle)

And (common side)

So by congruence criterion we have

This implies that,

Similarly we have

Hence .

#### Question 4:

In a Δ ABC, if ∠A = 120° and AB = AC. Find ∠B and ∠C.

In, it is given that

, and

We have to find, and

Since and

Then (as AB = AC)

Now

(By property of triangle)

Thus,

, as (given)

So,

Since,, so

Hence .

#### Question 5:

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

In it is given that

, and

We have to find.

Since

Then (isosceles triangles)

Now

(As given)

Thus

(Property of triangle)

Hence .

#### Question 6:

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

Suppose in the isosceles triangle ΔABC it is given that

We have to find the base angle.

Now vertical angle (given)

And

Since then

Now

(By property of triangle)

So

Hence the base angle is.

#### Question 7:

In the given figure, AB = AC and ∠ACD = 105°, find ∠BAC.

It is given that

We have to find.

(Isosceles triangle)

Now

Since exterior angle of isosceles triangle is the sum of two internal base angles

Now

So, (By property of triangle)

Hence .

#### Question 8:

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

We have to find the measure of each exterior angle of an equilateral triangle.

It is given that the triangle is equilateral

So, and

Since triangle is equilateral

So,

Now we have to find the exterior angle.

As we know that exterior angle of the triangle is sum of two interior angles

Thus

Hence each exterior angle is.

#### Question 9:

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

It is given that the base of an isosceles triangle is produced on both sides.

We have to prove that the exterior angles so formed are equal to each other.

That is we need to show that

Let the is isosceles having base and equal sides AB and AC

Then, and

(Isosceles triangles)

Now

.........(1)

And,

.......(2)

Thus

........(3)

Now from equation (2)

.........(4)

Since

Hence from equation (3) and (4)

#### Question 10:

In the given figure, AB = AC and DB DC, find the ratio ∠ABD : ∠ACD.

It is given that

We have to find the ratio.

Since

And

So we have,

So

Hence .

#### 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 = 90° and AB = AC. Find ∠B and ∠C.

It is given that

Is right angled triangle

And

We have to find and

Since

(Isosceles triangle)

Now

(Property of triangle)

()

So

Hence

#### Question 12:

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

It is given that

Δis a square and Δ is an equilateral triangle.

We have to prove that

(1) and (2)

(1)

Since,

(Angle of square)

(Angle of equilateral triangle)

Similarly, we have

Thus in and we have

(Side of square)

And (equilateral triangle side)

So by congruence criterion we have

Hence.

(2)
Since
QR = RS ( Sides of Square)
RS = RT (Sides of Equilateral triangle)

We get
QR = RT

Thus, we get
$\angle TQR=\angle RTQ$  (Angles opposite to equal sides are equal)

Now, in the triangle TQR, we have

$\angle TQR+\angle RTQ+\angle QRT={180}^{0}\phantom{\rule{0ex}{0ex}}\angle TQR+\angle TQR+{150}^{0}={180}^{0}\phantom{\rule{0ex}{0ex}}2\angle TQR+{150}^{0}={180}^{0}\phantom{\rule{0ex}{0ex}}2\angle TQR={180}^{0}-{150}^{0}\phantom{\rule{0ex}{0ex}}2\angle TQR={30}^{0}\phantom{\rule{0ex}{0ex}}\angle TQR=\frac{{30}^{0}}{2}={15}^{0}\phantom{\rule{0ex}{0ex}}$

#### 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 Figure). Show that the line PQ is perpendicular bisector of AB.

It is given that

P and Q are equidistant from A and B that is

, and

We are asked to show that line PO is perpendicular bisector of line AB.

First of all we will show that ΔAQP and ΔQBP are congruent to each other and ultimately we get the result.

Consider the triangles AQP and QBP in which

AP=BP, AQ=BQ, PQ=PQ

So by SSS property we have

Implies that

Now consider the triangles ΔAPC and ΔPCB in which

And

So by SAS criterion we find that,

So this implies that AC=BC and

But

Hence PQ is perpendicular bisector of AB.

#### Question 14:

In a ∆ABC, D is the mid-point of AC such that BD = $\frac{1}{2}$ AC. Show that ∠ABC is a right angle.

In a ∆ABC, D is the mid-point of AC such that BD = $\frac{1}{2}$AC.

D is the mid-point of AC.

∴ AD = CD = $\frac{1}{2}$AC

⇒ AD = CD = BD           (BD = $\frac{1}{2}$AC)

In ∆ABD,

∴ ∠ABD = ∠A          .....(1)      (In a triangle, equal sides have equal angles opposite to them)

In ∆CBD,

CD = BD

∴ ∠CBD = ∠C           .....(2)     (In a triangle, equal sides have equal angles opposite to them)

Adding (1) and (2), we get

∠ABD + ∠CBD = ∠A + ∠C

⇒ ∠B = ∠A + ∠C     .....(3)

In ∆ABC,

∠A + ∠B + ∠C = 180º        (Angle sum property of triangle)

⇒ ∠B + ∠B = 180º             [Using (3)]

⇒ 2∠B = 180º

⇒ ∠B = $\frac{180°}{2}$ = 90º

Thus, ∠ABC is a right angle.

#### Question 15:

ABC is a right triangle with AB = AC. Bisector of ∠A meets BC at D. Prove that BC = 2AD.

In ∆ABC,

AB = AC     (Given)

∴ ∠C = ∠B      .....(1)       (In a triangle, angles opposite to equal sides are equal)

Also, ∠A = 90º     (Given)

Now,

∠A + ∠B + ∠C = 180º    (Angle sum property of triangle)

⇒ 90º + 2∠B = 180º        [Using (1)]

⇒ 2∠B = 180º − 90º = 90º

⇒ ∠B = $\frac{90°}{2}$ = 45º

∴ ∠C = ∠B = 45º

It is given that, AD is the bisector of ∠A.

∴ ∠CAD = ∠BAD$=\frac{\angle \mathrm{A}}{2}=\frac{90°}{2}$ = 45º

In ∆ABD,

∠B = ∠BAD      (Each measure 45º)

∴ AD = BD       .....(2)     (In a triangle, sides opposite to equal angles are equal)

In ∆ACD,

∠C = ∠CAD      (Each measure 45º)

∴ AD = CD       .....(3)     (In a triangle, sides opposite to equal angles are equal)

Adding (2) and (3), we have

Hence proved.

#### Question 16:

ABC is a right triangle right angled at B such that ∠BCA = 2∠BAC. Show that AC = 2BC.

∆ABC is a right triangle right angled at B such that ∠BCA = 2∠BAC.

Produce CB to D such that BC = BD.

In ∆ABD and ∆ABC,

BD = BC                   (Construction)

∠ABD = ∠ABC       (90º each)

AB = AB                   (Common)

∴∆ABD ≅ ∆ABC     (SAS congruence axiom)

So, AD = AC             .....(1)         (CPCT)

Now,

∠BCA = 2∠BAC

⇒ ∠BCA = ∠BAC + ∠BAC

⇒ ∠BCA = ∠BAC + ∠BAD      [Using (2)]

In ∆ACD,

⇒ AD = CD                (Sides opposite to equal angles in a triangle are equal)

⇒ AC = BC + BD      [Using (1)]

⇒ AC = BC + BC      (BC = BD)

⇒ AC = 2BC

#### Question 1:

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

It is given that

Is bisector of and is bisector of.

And is isosceles with

We have to prove that

If will be sufficient to prove to show that

Now in these two triangles

Since, so

Now as BD and CE are bisector of the respectively, so

, and

BC=BC

So by congruence criterion we have

Hence Proved.

#### Question 2:

In the given figure, it is given that RT = TS, ∠1 = 2∠2 and ∠4 = 2∠3. Prove that ΔRBT ΔSAT.

It is given that

We have to prove that

Now

In we have

(Isosceles triangle)     .......(1)

Now we have

(Vertically opposite angles)

(Since, given)

.......(2)

Subtracting equation (2) from equation (1) we have

Now in and we have

(Given)

So all the criterion for the two triangles and are satisfied to be congruent

Hence by congruence criterion we have proved.

#### Question 3:

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.

It is given that

We have to prove that the lines and bisect at.

If we prove that, then

We can prove and bisects at.

Now in and

(Given)

(Since and is transversal)

And (since and is transversal)

So by congruence criterion we have,

, so

Hence and bisect each other at.

#### Question 1:

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.

It is given that

We are asked to show that

Let us assume

, and are right angled triangle.

Thus in and, we have

And (given)

Hence by AAs congruence criterion we haveProved.

#### Question 2:

If the bisector of the exterior vertical angle of a triangle be parallel to the base. Show that the triangle is isosceles.

We have to prove that is isosceles.

Let Δ be such that the bisector of is parallel to

The base, we have

(Corresponding angles)

(Alternate angle)

(Since)

Hence is isosceles.

#### Question 3:

In an isosceles triangle, if the vertex angle is twice the sum of the base angles, calculate the angles of the triangle.

In the triangle ABC it is given that the vertex angle is twice of base angle.

We have to calculate the angles of triangle.

Now, let be an isosceles triangle such that

Then

(Given)

(

Now (property of triangle)

Hence

#### Question 4:

Prove that each angle of an equilateral triangle is 60°

We have to prove each angle of an equilateral triangle is.

Here

(Side of equilateral triangle)

...........(1)

And

(Side of equilateral triangle)

..........(2)

From equation (1) and (2) we have

Hence

Now

That is (since)

Hence Proved.

#### Question 5:

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

It is given that

We have to prove that triangle ΔABC is equilateral.

Since (Given)

So,     ..........(1)

And (given)

So       ........(2)

From equation (1) and (2) we have

Now from above equation if we have

Given condition satisfy the criteria of equilateral triangle.

Hence the given triangle is equilateral.

#### Question 6:

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

It is given that

We have to find and.

Since so,

Now (property of triangle)

(Since )

Here

Then

Hence

#### Question 7:

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.

It is given that

We have to prove

In we have

(Given)

So,

Now (Given)

Since corresponding angle are equal, so

That is,

Henceproved.

#### Question 8:

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.

It is given that

In,

We have to prove that

Now

(Given)

Thus

........(1)

In, we have

So, {from equation (1)}

Hence Proved.

#### Question 9:

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 the following figure it is given that sides AB and PQ are parallel and BP is bisector of

We have to prove that is an isosceles triangle.

(Since BP is the bisector of)    ........(1)

(Since and are parallel)     .......(2)

Now from equation (1) and (2) we have

So

Now since and is a side of.

And since two sides and are equal, so

Hence is an isosceles triangle.

#### Question 10:

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°.

It is given that in

And bisects

We have to prove that

Now let

(Given)

Sinceis a bisector of so let

Let be the bisector of

If we join we have

In

So

In triangle and we have

(Given)

(Proved above)

So by congruence criterion, we have

And

, and (since)

In we have

Since,

And,

So,

In we have

Here,

HenceProved.

#### Question 11:

Bisectors of angles B and C of an isosceles triangle ABC with AB = AC intersect each other at O. Show that external angle adjacent to ∠ABC is equal to ∠BOC.

∠ABD is the external angle adjacent to ∠ABC.

∆ABC is an isosceles triangle.

AB = AC               (Given)

∴ ∠C = ∠ABC      .....(1)        (In a triangle, equal sides have equal angles opposite to them)

Also, OB and OC are the bisectors of ∠B and ∠C, respectively.

Similarly,

In ∆BOC,

∠OBC + ∠OCB + ∠BOC = 180º               (Angle sum property of triangle)

Now,

∠ABD + ∠ABC = 180º             .....(5)        (Linear pair)

From (4) and (5), we have

∠ABD + ∠ABC = ∠ABC + ∠BOC

⇒ ∠ABD = ∠BOC

Thus, the external angle adjacent to ∠ABC is equal to ∠BOC.

#### Question 1:

In the given figure, it is given that ABCD and AD = BC. Prove that ΔADC ΔCBA.

It is given that

We have to prove that.

Now in triangles and we have

(Given)

(Given)

So (common)

Each side of is equal to .

Hence, by congruence criterion we have Proved.

#### Question 2:

In a ΔPQR, if PQQR and L, M and N are the mid-points of the sides PQ, QR and RP respectively. Prove that LN = MN.

It is given that

and L, M, N are the mid points of sides, , and respectively.

We have to prove that

Now using the mid point theorem, we have

And

Similarly we have

In triangle and we have

(Proved above)

(Proved above)

And (common)

So, by congruence criterion, we have

And

Then

HenceProved.

#### 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.

We have to prove thatis isosceles.

Let and be perpendicular from D on AB and AC respectively.

In order to prove that

We will prove that

Now in and we have

(Since D is mid point of BC)

(Given)

So by congruence criterion we have

And

Hence is isosceles.

#### 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 ΔABC is isosceles.

It is given that

, and

And.

We have to prove is isosceles.

To prove is isosceles we will prove

For this we have to prove

Now comparing and we have

(Given)

(Common side)

So, by right hand side congruence criterion we have

So (since sides opposite to equal angle are equal)

Hence is isosceles.

#### 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.

Let P be a point within such that

We have to prove that P lies on the bisector of

In and we have

(We have)

(Common)

So by right hand side congruence criterion, we have

So,

Hence P lies on the bisector of proved.

#### Question 4:

In the given figure, ADCD and CBCD. If AQ = BP and DP = CQ, prove that
DAQ = ∠CBP.

It is given that

, and

If and

We have to prove that

In triangles and we have

(Since given)

So

And (given)

So by right hand side congruence criterion we have

So

HenceProved.

#### 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 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 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.

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

#### Question 6:

Fill in 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 a ABC if ∠A = ∠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 ΔABC Δ .........

(1)

(2)

(3)

(4)

(5)

(6)

(7)

#### Question 7:

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

It is given ABCD is a square and

We have to prove that and

In right angled trianglesand Δ we have

And, and

So by right hand side congruence criterion we have

So (since triangle is congruent)

HenceProved.

#### Question 8:

ABCD is a quadrilateral such that AB = AD  and CB = CD. Prove that AC is the perpendicular bisector of BD.

In quadrilateral ABCD, AB = AD and BC = CD. Let AC and BD intersect at O.

AC = AC        (Common)

BC = CD        (Given)

∴ ∆ABC ≅ ∆ADC       (SSS congruence criterion)

⇒ ∠BAC = ∠DAC       (CPCT)

∠BAO = ∠DAO        (Proved above)

AO = AO                   (Common)

∴ ∆ABO ≅ ∆ADO       (SAS congruence axiom)

⇒ OB = OD             .....(1)      (CPCT)

∠AOB = ∠AOD       .....(2)      (CPCT)

Now,

∠AOB + ∠AOD = 180º         (Linear pair)

⇒ 2∠AOB = 180º          [Using (2)]

⇒ ∠AOB = 90º        .....(3)

From (1) and (3), we conclude that AC is the perpendicular bisector of BD.

#### Question 9:

ABC is a right triangle such that AB = AC and bisector of angle C intersects the side AB at D. Prove that AC + AD = BC.

∆ABC is a right triangle such that AB = AC. CD is the bisector of ∠C which intersects AB at D.

Let AB = AC = x units

In right ∆ABC,

${\mathrm{BC}}^{2}={\mathrm{AB}}^{2}+{\mathrm{AC}}^{2}$        (Pythagoras theorem)

$⇒{\mathrm{BC}}^{2}={x}^{2}+{x}^{2}=2{x}^{2}$

$⇒\mathrm{BC}=\sqrt{2}x$

In ∆ABC, CD is the bisector of ∠C.

$\therefore \frac{\mathrm{AD}}{\mathrm{BD}}=\frac{\mathrm{AC}}{\mathrm{BC}}$          (The bisector of an angle of a triangle divides the opposite side in the ratio of sides containing the angle)

$⇒\frac{\mathrm{AD}}{x-\mathrm{AD}}=\frac{x}{\sqrt{2}x}$

$⇒\sqrt{2}\mathrm{AD}=x-\mathrm{AD}$

$⇒\mathrm{AD}\left(\sqrt{2}+1\right)=x$

$⇒\mathrm{AD}=\frac{x}{\sqrt{2}+1}=x\left(\sqrt{2}-1\right)$

Now,

$\mathrm{AC}+\mathrm{AD}=x+\sqrt{2}\left(x-1\right)=\sqrt{2}x=\mathrm{BC}$

Hence proved.

#### Question 10:

O is a point in the interior of a square ABCD such that OAB is an equilateral triangle. Show that ∆OCD is an isosceles triangle.

O is a point in the interior of square ABCD. ∆OAB is an equilateral triangle.

Now,

∠DAB = ∠CBA        .....(1)         (Measure of each angle of a square is 90º)

∠OAB = ∠OBA        .....(2)         (Measure of each angle of an equilateral triangle is 60º)

Subtracting (2) from (1), we get

∠DAB − ∠OAB = ∠CBA − ∠OBA

OA = OB      (Sides of an equilateral triangle are equal)

AD = BC       (Sides of a square are equal)

∴ ∆OAD ≅ ∆OBC       (SAS congruence axiom)

⇒ OD = OC             (CPCT)

In ∆OCD,

OC = OD

∴ ∆OCD is an isosceles triangle.    (A triangle whose two sides are equal is an isosceles triangle)

#### Question 11:

ABCD is a quadrilateral in which AB = BC and AD = CD. Show that BD bisects both the angles ∠ABC and ∠ADC.

In ∆ABD and ∆CBD,

AB = CB        (Given)

BD = BD        (Common)

∴ ∆ABD ≅ ∆CBD       (SSS congruence criterion)

So, ∠ABD = ∠CBD       .....(1)       (CPCT)

From (1) and (2), we conclude that

BD bisects both ∠ABC and ∠ADC.

#### Question 12:

Line segment joining the mid-points M and N of parallel sides AB and DC, respectively of a trapezium ABCD is perpendicular to both the sides AB and DC. Prove that AD = BC.

ABCD is a trapezium with AB || CD. M and N are the mid-points of sides AB and AC, respectively.

Join AN and BN.

In ∆AMN and ∆BMN,

AM = BM                      (M is the mid-point of AB)

∠AMN = ∠BMN           (MN ⊥ AB)

MN = MN                      (Common)

∴ ∆AMN ≅ ∆BMN        (SAS congruence axiom)

So, AN = BN         .....(1)       (CPCT)

∠ANM = ∠BNM            (CPCT)

Now,

∠DNM = ∠CNM            (90º each)

∴ ∠DNM − ∠ANM = ∠CNM − ∠BNM

⇒ ∠AND = ∠BNC      .....(2)

In ∆AND and ∆BNC,

DN = CN                       (N is the mid-point of CD)

∠AND = ∠BNC           [From (2)]

AN = BN                      [From (1)]

∴ ∆AND ≅ ∆BNC        (SAS congruence axiom)

Hence proved.

#### Question 1:

In ΔABC, if ∠A = 40° and ∠B = 60°. Determine the longest and shortest sides of the triangle.

In the triangle ABC it is given that

We have to find the longest and shortest side.

Here

Now is the largest angle of the triangle.

So the side in front of the largest angle will be the longest side.

Hence will be the longest

Since is the shortest angle so that side in front of it will be the shortest.

And is shortest side

Hence Is longest and is shortest.

#### Question 2:

In a ΔABC, if ∠B = ∠C = 45°, which is the longest side?

In the triangle ABC it is given that

We have to find the longest side.

Here

(Since)

Now is the largest angle of the triangle.

So the side in front of the largest angle will be the longest side.

Hence BC will be the longest side.

#### Question 3:

In Δ ABC, side AB is produced to D so that BD = BC. If ∠B = 60° and ∠A = 70°, prove that :

It is given that

, and

We have to prove that

(1)

(2)

(1)

Now

And since BD=BC, so, and

That is,

Now

And, so

Hence (1) (Side in front of greater angle will be longer)

And (2) Proved.

#### Question 4:

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

As we know that a triangle can only be formed if

The sum of two sides is greater than the third side.

Here we have 2 cm, 3 cm and 7 cm as sides.

(Since 5 is less than 7)

Hence the sum of two sides is less than the third sides

So, the triangle will not exist.

#### Question 5:

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

It is given that

AP is the bisector of

We have to arrange, and in descending order.

In we have

(As AP is the bisector of)

So (Sides in front or greater angle will be greater)              ........(1)

In we have

(As AP is the bisector of)

Since,

So                   ..........(2)

Hence

From (1) & (2) we have

#### Question 6:

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

We have to prove that the perimeter of a triangle is greater than the sum of its altitude.

In

, ,

We have to prove

Since

So and

........(1)

Now consider then

, and

Again consider

, and

Adding (1), (2) and (3), we get

Hence the perimeter of a triangle is greater than the sum of all its altitude.

#### Question 7:

In the given figure, prove that:

(i) CD + DA + AB + BC > 2AC

(ii) CD + DA + AB > BC

(1) We have to prove that

In we have

(As sum of two sides of triangle is greater than third one)   ........(1)

In we have

(As sum of two sides of triangle is greater than third one)   .........(2)

Hence

Adding (1) & (2) we get

Proved.

(2) We have to prove that

In we have

(As sum of two sides of triangle is greater than third one)

Proved.

#### 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.

(1)

(2)

(3)

(4)

(5)

(6)

#### Question 9:

Fill in the blanks to make the following statements true.

(i) In a right triangle the hypotenuse is the ...... side.

(ii) The sum of three altitudes of a triangle is ....... than its perimeter.

(iii) The sum of any two sides of a triangle is ..... than the third side.

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

(v) Difference of any two sides of a triangle is ...... than the third side.

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

(1)

(2)

(3)

(4)

(5)

(6)

#### Question 10:

O is any point in the interior of Δ ABC. Prove that

(i) AB + AC > OB + OC

(ii) AB + BC + CA > OA + OB + OC

(iii) OA + OB + OC > $\frac{1}{2}$(AB + BC + CA)

It is given that, is any point in the interior of

We have to prove that

(1) Produced to meet at.

In we have

.........(1)

And in we have

.........(2)

Adding (1) & (2) we get

HenceProved.

(2) We have to prove that

From the first result we have

..........(3)

And

.........(4)

HenceProved.

(3) We have to prove that

In triangles, and we have

HenceProved.

#### Question 11:

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

We have to prove that the sum of four sides of quadrilateral is greater than sum of diagonal.

Since the sum of two sides of triangle is greater than third side.

In we have

..........(1)

In we have

..........(2)

In we have

.........(3)

In we have

.........(4)

Adding (1) & (2) & (3) and (4) we get

Hence Proved.

#### Question 12:

Prove that in a triangle, other than an equilateral triangle, angle opposite to the longest side is greater than $\frac{2}{3}$of a right angle.

Let AC be the longest side in the ∆ABC.

Now,

AC > AB

⇒ ∠B > ∠C         .....(1)             (In a triangle, greater side has greater angle opposite to it)

Also,

AC > BC

⇒ ∠B > ∠A         .....(2)             (In a triangle, greater side has greater angle opposite to it)

From (1) and (2), we have

∠B + ∠B > ∠A + ∠C

⇒ ∠B + ∠B + ∠B > ∠A + ∠B + ∠C

⇒ 3∠B > 180º                      (Using angle sum property of a triangle)

⇒ ∠B > $\frac{1}{3}$ × 180º

Or ∠B > $\frac{2}{3}$ × 90º

Thus, the angle opposite to the longest side is greater than $\frac{2}{3}$of a right angle.

Hence proved.

#### Question 13:

D is any point on side AC of a ∆ABC with AB = AC. Show that CD < BD.

It is given that, D is any point on side AC of a ∆ABC with AB = AC.

In ∆ABC,

AB = AC      (Given)

∴ ∠ACB = ∠ABC         (In a triangle, equal sides have equal angles opposite to them)

Now, ∠ABC > ∠DBC

⇒ ∠ACB > ∠DBC            (∠ACB = ∠ABC)

In ∆BCD,

∠DCB > ∠DBC

⇒ BD > CD                       (In a triangle, greater angle has greater side opposite to it)

Or CD < BD

#### Question 1:

Mark the correct alternative in each of the following:

If ABC $\cong$ ΔLKM, then side of ΔLKM equal to side AC of ΔABC is

(a) LK

(b) KM

(c) LM

(d) None of these

It is given that

As triangles are congruent, same sides will be equal.

So

Hence (c).

#### Question 2:

If ΔABC $\cong$ ΔABC is isosceles with

(a) AB = AC

(b) AB = BC

(c) AC = BC

(d) None of these

It is given that and is isosceles

Since triangles are congruent so as same side are equal

Hence (a).

#### Question 3:

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

(a) BC = PQ

(b) AC = PR

(c) AB = PQ

(d) QR = BC

If and is not congruent to

Since and compare corresponding sides you will see

(As )

Hence (a) , is not true.

#### Question 4:

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:

(a) SAS

(b) ASA

(c) SSS

(d) RHS

In and

It is given that

Since two sides and an angle are equal so it obeys

Hence (a).

#### Question 5:

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

(a) SAS

(b) ASA

(c) SSS

(d) RHS

In and

It is given that

Since given two sides and an angle are equal so it obeys

Hence (b).

#### Question 6:

In ΔPQR $\cong$ ΔEFD then ED =

(a) PQ

(b) QR

(c) PR

(d) None of these

If

We have to find

Since, as in congruent triangles equal sides are decided on the basis of “how they are named”.

Hence (c).

#### Question 7:

If ΔPQR $\cong$ ΔEFD, then ∠E =

(a) ∠P

(b) ∠Q

(c) ∠R

(d) None of these

If

Then we have to find

From the given congruence, as equal angles or equal sides are decided by the location of the letters in naming the triangles.

Hence (a)

#### Question 8:

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

(a) 20°

(b) 40°

(c) 60°

(d) 80°

In the triangle ABC it is given that

We have to find

Now (linear pair)

Since

So, (by isosceles triangle)

This implies that

Now,

(Property of triangle)

Hence (a).

#### 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°

Let be isosceles triangle

Then

Now it is given that vertex angle is 2 times the sum of base angles

(As)

Now

(Property of triangle)

(Since, and )

Hence (b).

#### Question 10:

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

(a) SAS

(b) SSA

(c) ASA

(d) SSS

(b) ,as it does not follow the congruence criteria.

#### Question 11:

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

(a) 50°

(b) 60°

(c) 70°

(d) 80°

We have to find

Since triangles are congruent

So

Now in

(By property of triangle)

Hence (b) .

#### Question 12:

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

(a) DF = 5cm, ∠F = 60°

(b) DE = 5cm, ∠E = 60°

(c) DF = 5cm, ∠E = 60°

(d) DE = 5cm, ∠D = 40°

It is given thatand

So and

Now, in triangle ABC,

Therefore,

Hence the correct option is (c).

#### Question 13:

In the given figure, ABBE and FEBE. If BC = DE and AB = EF, then ΔABD is congruent to

(a) ΔEFC

(b) ΔECF

(c) ΔCEF

(d) ΔFEC

It is given that

And

(Given)

(Given)

So (from above)

Hence

From (d).

#### Question 14:

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

(a) 70°

(b) 110°

(c) 120°

(d) 130°

We have to find the value of in the following figure.

It is given that

(Vertically apposite angle)

Now (linear pair) …… (1)

Similarly (linear pair) …… (2)

From equation (1) we have

Now (same exterior angle)

(Interior angle)

Now

So

Since

Hence (b).

#### Question 15:

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

(a) 52°

(b) 76°

(c) 156°

(d) 104°

We are given that;

, is isosceles

And

We are asked to find angle x

From the figure we have

Therefore,

Since, so

Now

Hence (d) .

#### Question 16:

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

(a) 2 cm

(b) 3 cm

(c) 4 cm

(d) 5 cm

It is given that

, is bisector of

We are to find the side CD

Analyze the figure and conclude that

(As in the two triangles are congruent)

In

So

Hence (c) .

#### Question 17:

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

(a) ABC

(b) AEF

(c) BFD, CDE

(d) AFE, BFD, CDE

It is given that, and are the mid points of the sides, andrespectively of

(By mid point theorem)

(As it is mid point)

Now in and

(Common)

(Mid point)

(Mid point)

Hence (d)

#### Question 18:

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

(a) 55°

(b) 70°

(c) 35°

(d) 110°

It is given that, AB=AC and Ad is the median of BC

We know that in isosceles triangle the median from he vertex to the unequal side divides it into two equal part at right angle.

Therefore,

(Property of triangle)

Hence (a) .

#### Question 19:

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 =

(a) 5 cm

(b) 6 cm

(c) 7 cm

(d) 8 cm

In the following figure we are given

Where ABCD is a square and AXYZ is also a square

We are asked to find BY

From the above figure we have XY=YZ=AZ=AX

Now in the given figure

So,

Now inn triangle ΔAXB

So

Hence (c) .

#### Question 20:

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

(a) 72°

(b) 73°

(c) 74°

(d) 95°

It is given that

AB = CD

We have to find

Now AB = CD

AB = BD

Now the triangle is isosceles

Let

So

Now

Since

Hence (a) .

#### Question 1:

If AB = QR, BC = PR and CA = PQ, then triangle _________ ≅ triangle _________.

It is given that, AB = QR, BC = PR and CA = PQ.

So, A ↔ Q, B ↔ R and C ↔ P

∴ ∆ABC ≅ ∆QRP

If AB = QR, BC = PR and CA = PQ, then triangle __ABC__ ≅ triangle __QRP__.

#### Question 2:

In triangles ABC and DEF, AB = FD and ∠A = ∠D. The two triangles will be congruent by SAS axiom, if __________.

SAS congruence axiom states that two triangles are congruent if two sides and the included angle of one are equal to the corresponding sides and the included angle of the other triangle.

In ∆ABC, ∠A is included between the sides AB and AC.

In ∆DEF, ∠D is included between the sides DF and DE.

∴ ∆ABC ≅ ∆DEF by SAS axiom if AC = DE.

In triangles ABC and DEF, AB = FD and ∠A = ∠D. The two triangles will be congruent by SAS axiom, if __AC = DE__.

#### Question 3:

In ∆ABC and PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are __________ but not __________.

If two sides of a triangle are equal, then it is an isosceles triangle.

In ∆ABC, AB = AC

∴ ∆ABC is an isosceles triangle.

In ∆ABC,

AB = AC    (Given)

∴ ∠C = ∠B       (In a triangle, equal sides have equal angles opposite to them)

It is given that, ∠C = ∠P and ∠B = ∠Q.

∴ ∠P = ∠Q

In ∆PQR,

∠P = ∠Q        (Proved)

∴ QR = PR     (In a triangle, equal sides have equal angles opposite to them)

So, ∆PQR is an isosceles triangle.

However, it cannot be proved that the corresponding sides of ∆ABC are congruent to the corresponding sides of ∆PQR. Hence, the triangles are not congruent.

In ∆ABC and ∆PQR, AB = AC, ∠C = ∠P and ∠B = ∠Q. The two triangles are __isosceles__ but not __congruent__.

#### Question 4:

In ∆PQR, ∠P = ∠R, QR = 4 cm and PR = 5 cm. Then PQ = ________.

In ∆PQR,

∠P = ∠R       (Given)

∴ QR = PQ    (Sides opposite to the equal angles of a triangle are equal)

⇒ PQ = 4 cm       (QR = 4 cm)

In ∆PQR, ∠P = ∠R, QR = 4 cm and PR = 5 cm. Then PQ = ___4 cm___.

#### Question 5:

In ∆ABC, AB = AC and ∠B = 50°. Then, ∠C = __________.

In ∆ABC,

AB = AC           (Given)

∴ ∠C = ∠B        (Angles opposite to the equal sides of a triangle are equal)

⇒ ∠C = 50°       (∠B = 50°)

In ∆ABC, AB = AC and ∠B = 50°. Then, ∠C = __50°__.

#### Question 6:

In ∆ABC, AB = BC and ∠B = 80°. Then ∠A = __________.

In ∆ABC,

AB = BC           (Given)

∴ ∠C = ∠A        ..... (1)         (Angles opposite to the equal sides of a triangle are equal)

Now,

∠A + ∠B + ∠C = 180°       (Angle sum property of triangle)

∴ ∠A + 80° + ∠A = 180°

⇒ 2∠A = 180° − 80° = 100°

⇒ ∠A = $\frac{100°}{2}$ = 50°

In ∆ABC, AB = BC and ∠B = 80°. Then ∠A = __50°__.

#### Question 7:

If in ∆PQR, ∠P = 70° and ∠R = 30°, then the longest side of ∆PQR is ___________.

In ∆PQR,

$\angle \mathrm{P}+\angle \mathrm{Q}+\angle \mathrm{R}=180°$          (Angle sum property of a triangle)

$\therefore 70°+\angle \mathrm{Q}+30°=180°$

$⇒\angle \mathrm{Q}+100°=180°$

$⇒\angle \mathrm{Q}=180°-100°=80°$

So, ∠Q is the greatest angle in the ∆PQR.

We know that, in a triangle the greater angle has the longer side opposite to it.

∴ PR is the longest side of ∆PQR.

If in ∆PQR, ∠P = 70° and ∠R = 30°, then the longest side of ∆PQR is ___PR___.

#### Question 8:

In ∆PQR, if ∠R > ∠Q, then __________.

In ∆PQR,

∠R > ∠Q        (Given)

∴ PQ > PR     (In a triangle, the greater angle has the longer side opposite to it)

In ∆PQR, if ∠R > ∠Q, then ___PQ > PR___.

#### Question 9:

D is a point on side BC of a ∆ABC such that AD bisects ∠BAC. Then ________.

In ∆ABC, AD is the bisector of ∠A.

We know that exterior angle of a triangle is greater than each of interior opposite angle.

In ∆ABD,

∴ AC > CD           (In a triangle, the greater angle has the longer side opposite to it)

Similarly, AB > BD

D is a point on side BC of a ∆ABC such that AD bisects ∠BAC. Then ___AC > CD and AB > BD___.

#### Question 10:

Two sides of a triangle are of lengths 5 cm and 1.5 cm. The length of the third side of the triangle lies between _________ and _________.

In a triangle, the sum of two sides is greater than the third side.

∴ 5 cm + 1.5 cm > Third side

Or Third side < 6.5 cm

In a triangle, the difference of two sides is less than the third side.

∴ 5 cm − 1.5 cm < Third side

Or Third side > 3.5 cm

So,

3.5 cm < Third side < 6.5 cm

Thus, the length of the third side of the triangle lies between 3.5 cm and 6.5 cm.

Two sides of a triangle are of lengths 5 cm and 1.5 cm. The length of the third side of the triangle lies between __3.5 cm__ and _6.5 cm _.

#### Question 11:

If AD is a median of ∆ABC, then the perimeter of ∆ABC cannot be less than or equal to ___________.

AD is the median of the ∆ABC.

In ∆ABD,

AB + BD > AD      .....(1)          (In a triangle, the sum of any two sides is greater than the third side)

In ∆ACD,

CD + CA > AD      .....(2)          (In a triangle, the sum of any two sides is greater than the third side)

Adding (1) and (2), we get

⇒ AB + BC + CA > 2AD                     (BC = BD + CD)

⇒ Perimeter of ∆ABC > 2AD

Or the perimeter of ∆ABC cannot be less than or equal to 2AD

If AD is a median of ∆ABC, then the perimeter of ∆ABC cannot be less than or equal to ___2AD___.

#### Question 12:

If ∆PQR ≅ ∆EDF, then PR = ___________.

If ∆PQR ≅ ∆EDF, then

P ↔ E, Q ↔ D and R ↔ F

So, PR = EF,  PQ = ED and QR = DF       (If two triangles are congruent, then their corresponding sides are congruent)

If ∆PQR ≅ ∆EDF, then PR = ____EF____.

#### Question 13:

It is given that ∆ABC ≅ ∆RPQ, then BC = __________.

If ∆ABC ≅ ∆RPQ, then

A ↔ R, B ↔ P and C ↔ Q

So, AB = RP, BC = PQ and CA = QR       (If two triangles are congruent, then their corresponding sides are congruent)

It is given that ∆ABC ≅ ∆RPQ, then BC = ___PQ___.

#### Question 14:

In ∆ABC and ∆PQR, if ∠A = ∠Q, ∠B = ∠R and PR = AC, then two triangles are congruent by __________ criterion.

In ∆ABC and ∆PQR,

∠A = ∠Q

∠B = ∠R

AC = PR

∴ ∆ABC ≅ ∆QRP      (AAS congruence criterion)

AAS congruence criterion states that if any two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are congruent.

In ∆ABC and ∆PQR, if ∠A = ∠Q, ∠B = ∠R and PR = AC, then two triangles are congruent by __AAS congruence__ criterion.

#### Question 15:

In ∆ABC and ∆PQR, ∠A = ∠Q, ∠B = ∠R and AB = QR, then these triangles are congruent by _______.

In ∆ABC and ∆PQR,

∠A = ∠Q

∠B = ∠R

AB = QR

∴ ∆ABC ≅ ∆QRP      (ASA congruence criterion)

ASA congruence criterion states that two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle.

In ∆ABC and ∆PQR, ∠A = ∠Q, ∠B = ∠R and AB = QR, then these triangles are congruent by __ASA congruence criterion__.

#### Question 1:

In two congruent triangles ABC and DEF, if AB = DE and BC = EF. Name the pairs of equal angles.

It is given that

Since, the triangles ABC and DEF are congruent, therefore,

#### Question 2:

In two triangles ABC and DEF, it is given that ∠A = ∠D, ∠B = ∠E and ∠C =∠F. Are the two triangles necessarily congruent?

It is given that

For necessarily triangle to be congruent, sides should also be equal.

#### Question 3:

If ABC and DEF are two triangles such that AC = 2.5 cm, BC = 5 cm, ∠C = 75°, DE = 2.5 cm, DF = 5cm and ∠D = 75°. Are two triangles congruent?

It is given that

Since, two sides and angle between them are equal, therefore triangle ABC and DEF are congruent.

#### Question 4:

In two triangles ABC and ADC, if AB = AD and BC = CD. Are they congruent?

The given information and corresponding figure is given below

From the figure, we have

And,

Hence, triangles ABC and ADC are congruent to each other.

#### Question 5:

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

For the triangles ABC and ECD, we have the following information and corresponding figure:

In triangles ABC and ECD, we have

The SSA criteria for two triangles to be congruent are being followed. So both the triangles are congruent.

#### Question 6:

ABC is an isosceles triangle in which AB = AC. BE and CF are its two medians. Show that BE = CF.

In the triangle ABC it is given that

, and are medians.

We have to show that

To show we will show that

In triangle ΔBFC and ΔBEC

As, so

.........(1)

BC=BC (common sides)   ........(2)

Since,

As F and E are mid points of sides AB and AC respectively, so

BF = CE         ..........(3)

From equation (1), (2), and (3)

HenceProved.

#### Question 7:

Find the measure of each angle of an equilateral triangle.

In equilateral triangle we know that each angle is equal

So

Now (by triangle property)

Hence.

#### Question 8:

CDE is an equilateral triangle formed on a side CD of a square ABCD. Show that ΔADE ΔBCE.

We have to prove that

Given is a square

So

Now in is equilateral triangle.

So

In and

(Side of triangle)

(Side of equilateral triangle)

And,

So

Hence from congruence Proved.

#### Question 9:

Prove that the sum of three altitudes of a triangle is less than the sum of its sides.

We have to prove that the sum of three altitude of the triangle is less than the sum of its sides.

In we have

, and

We have to prove

As we know perpendicular line segment is shortest in length

Since

So     ........(1)

And

........(2)

Adding (1) and (2) we get

........(3)

Now, so

.......(4)

And again, this implies that

........(5)

Adding (3) & (4) and (5) we have

HenceProved.

#### Question 10:

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

It is given that

, and

We have to prove that

We basically will prove to show

In and

(Given)

(Given)

And is common in both the triangles

So all the properties of congruence are satisfied

So

Hence Proved.

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