## NCERT Exemplar Class 7 Maths Book PDF Download Chapter 6 Triangles Solutions

**Multiple Choice Questions (MCQs)**

**Question 1:
**The sides of a triangle have lengths (in cm) 10, 6.5 and a, where a is a whole number. The minimum value that a can take is

(a) 6 (b) 5 (c) 3 (d) 4

**Solution :**

(d) As we know, sum of any two sides in a triangle is always greater than the third side.

So, only 4 is the minimum value that satisfies as a side in triangle.

\(\left\{ \begin{matrix} 10<6.5+4 \\ 6.5<10+4 \\ 4<10+6.5 \end{matrix} \right\}\)

**Question 2:
**ΔDEF of following figure is a right angled triangle with ∠E = 90°.

What type of angles are ∠D and ∠F

(a) They are equal angles (b) They form a pair of adjacent angles

(c) They are complementary angles (d) They are supplementary angles

**Solution :**

(c) Since, ∠D and ∠F are complementary angles.

**Question 3:
**In the given figure, PQ = PS. The value of x is

**Solution :**

**Question 4:
**In a right angled triangle, the angles other than the right angle are

(a) obtuse (b) right (c) acute (d) straight

**Solution :**

**Question 5:
**In an isosceles triangle, one angle is 70°. The other two angles are of

(i) 55° and 55°

(ii) 70° and 40°

(iii) any measure

In the given option(s) which of the above statement(s) are true?

(a) (i) only (b) (ii) only

(c) (iii) only (d) (i) and (ii)

**Solution :**

(d) As we know, the sum of the interior angles of a triangle is 180°.

**Question 6:
**In a triangle, one angle is of 90°. Then,

(i) the other two angles are of 45° each.

(ii) in remaining two angles, one angle is 90° and other is 45°.

(iii) remaining two angles are complementary.

In the given option(s) which is true?

(a) (i) only (b) (ii) only (c) (iii) only (d) (i) and (ii)

**Solution :**

(c) In a right angled ΔABC,

**Question 7:
**Lengths of sides of a triangle are 3 cm, 4 cm and 5 cm. The triangle is

(a) obtuse angled triangle (b) acute angled triangle

(c) right angled triangle (d) an isosceles right triangle

**Solution :**

(c) Since, these sides satisfy the Pythagoras theorem, therefore it is right angled triangle. Lengths of the sides of a triangle are 3 cm, 4 cm and 5 cm.

According to Pythagoras theorem,

**Note:**The area of the square built upon the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares upon the remaining sides is known as Pythagoras theorem.

**Question 8:
**In the given figure, PB = PD. The value of x is

**Solution :**

**Question 9:
**In A PQR,

(a) PQ – QR > PR (b) PQ +QR<PR

(c) PQ-QR< PR (d) PQ +<PR<QR

**Solution :**

(c) As we know, sum of the lengths of any two sides of a triangle is always greater than the length of the third side.

**Question 10:
**In Δ ABC,

(a) AB+BC> AC (b) AB + BC< AC

(c) AB+AC < BC (d) AC + BC < AB

**Solution :**

(a) As we know, sum of any two sides in a triangle is always greater than the third side.

**Question 11:
**The top of a broken tree touches the ground at a distance of 12 m from its base. If the tree is broken at a height of 5 m from the ground, then the actual height of the tree is

(a) 25 m (b) 13 m (c) 18 m (d) 17 m

**Solution :**

(c) Let AB be the given that tree of height h m, which is broken at D which is 12 m away from its base and the height of remaining part, i.e. CS is 5 m.

**Question 12:
**The ΔABC formed by AB = 5 cm, BC = 8 cm and AC = 4 cm is

(a) an isosceles triangle only (b) a scalene triangle only

(c) an isosceles right triangle (d) scalene as well as a right triangle

**Solution :**

(b) (i) It’s not isosceles triangle as all the sides are of different measure.

(ii) It’s not right triangle, since it does not follow Pythagoras theorem.

**Question 13:
**Two trees 7 m and 4 m high stand upright on a ground. If their bases (roots) are 4 m apart, then the distance between their tops is

(a) 3 m (b) 5 m (c) 4 m (d) 11 m

**Solution :**

(b) Let BE be the smaller tree and AD be the bigger tree. Now, we have to find AB (i.e. the distance between their tops).

**Question 14:
**If in an isosceles triangle, each of the base angle is 40°, then the triangle is

(a) right angled triangle (b) acute angled triangle

(c) obtuse angled triangle (d) isosceles right angled triangle

**Solution :**

(c) As we know, the sum of the interior angles of a triangle is 180°.

**Question 15:
** If two angles of a triangle are 60° each, then the triangle is

(a) isosceles but not equilateral (b) scalene

(c) equilateral (d) right angled

**Solution :**

**Question 16:
**The perimeter of the rectangle whose length is 60 cm and a diagonal is 61 cm is

(a) 120 cm (b) 122 cm (c) 71 cm (d) 142 cm

**Solution :**

(d) Given, length of rectangle = 60 cm and its diagonal = 61 cm,

**Question 17:
**In ΔPQR, if PQ = QR and ∠Q = 100°, then ∠R is equal to

(a) 40° (b) 80° (c) 120° (d) 50°

**Solution :**

**Question 18:
**Which of the following statements is not correct?

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

(b) A triangle can have all its angles acute

(c) A right angled triangle cannot be equilateral

(d) Difference of any two sides of a triangle is greater than the third side

**Solution :**

(d) The difference of the length of any two sides of a triangle is always smaller than the length of the third side.

**Question 19:
**In the given figure, BC = CA and ∠A = 40°. Then, ∠ACD is equal to

**Solution :**

**Question 20:
**The length of two sides of a triangle are 7 cm and 9 cm. The length of the third side may lie between

(a) 1 cm and 10 cm (b) 2 cm and 8 cm

(c) 2 cm and 16 cm (d) 1 cm and 16 cm

**Solution :**

(c) The third side must be greater than the difference between two sides and less than the sum of two sides.

Sum of two sides = 7 + 9 = 16 cm

Difference of two sides = 9 – 7 = 2 cm

So, length of the third side must lie between 2 cm and 16 cm.

**Question 21:
**From the following figure, the value of x is

**Solution :**

**Question 22:
**In the given figure, the value of

**Solution :**

As we know, sum of all the interior angles of a triangle is 180°.

**Question 23:
**In the given figure, PQ = PR, RS = RQ and ST || QR. If the exterior ∠RPU is 140°, then the measure of ∠TSR is

**Solution :**

**Question 24:
**In the given figure ∠BAC = 90°, AD ⊥ BC and ∠BAD = 50°, then ∠ACD is

**Solution :**

**Question 25:
**If one angle of a triangle is equal to the sum of the other two angles, the triangle is

(a) obtuse (b) acute (c) right (d) equilateral

**Solution :**

Let A, B and C be the angles of the triangle. Then, one angle of a triangle is equal to the sum of the other two angles,

i.e. ∠A = ∠B + ∠C …(i)

**Question 26:
**If the exterior angle of a triangle is 130° and its interior opposite angles are equal, then measure of each interior opposite angle is

(a) 55° (b) 65° (c) 50° (d) 60°

**Solution :**

(b) As we know, the measure of any exterior angle is equal to the sum of two opposite interior angles.

Let the interior angle be x.

Given that, interior opposite angles are equal.

**Question 27:
**If one of the angle of a triangle is 110°, then the angle between the bisectors of the other two angles is

(a) 70° (b) 110° (c) 35° (d) 145°

**Solution :**

**Question 28:
**In ∆ABC, AD is the bisector of ∠A meeting BC at D, CF⊥ AB and E is the mid-point of AC. Then, median of the triangle is

(a) AD (b) BE (c) FC (d) DE

**Solution :**

(b) As we know, median of a triangle bisects the opposite sides.

**Question 29:
**In ΔPQR, if ∠P = 60° and ∠Q = 40°, then the exterior angle formed by producing QR is equal to

(a) 60° (b) 120°

(c) 100° (d) 80°

**Solution :**

(c) As we know, the measure of exterior angle is equal to the sum of opposite two interior angles.

**Question 30:
**Which of the following triplets cannot be the angles of a triangle?

(a) 67°, 51°, 62°

(b) 70°, 83°, 27°

(c) 90°, 70°, 20°

(d) 40°, 132°, 18°

**Solution :**

(d) We know that, the sum of the interior angles of a triangle is 180°.

Now, we will verify the given triplets :

(a) 67°+ 51°+ 62° = 180°

(b) 70° + 83° + 27° = 180°

(c) 90° + 70° + 20° = 180°

(d) 40° + 132°+ 18° = 190°

Clearly, triplets in option (d) cannot be the angles of a triangle.

**Question 31:
**Which of the following can be the length of the third side of a triangle whose two sides measure 18 cm and 14 cm?

(a) 4 cm (b) 3 cm (c) 5 cm (d) 32 m

**Solution :**

(c) As we know, sum of any two sides of a triangle is always greater than the third side. Hence, option (c) satisfies the given condition.

Verification

18+14 >5

18 + 5 > 14

5+ 14> 18

**Question 32:
**How many altitudes does a triangle have?

(a) 1 (b) 3 (c) 6 (d) 9

**Solution :**

(b) A triangle has 3 altitudes

**Question 33:
**If we join a vertex to a point on opposite side which divides that side in the ratio 1:1, then what is the special name of that line segment?

(a) Median (b) Angle bisector (c) Altitude (d) Hypotenuse

**Solution :**

(a) Consider ΔABC in which AD divides BC in the ratio 1:1.

**Note:**The line segment joining a vertex of a triangle to the mid-point of its opposite side is called a median.

**Question 34:
**The measures of ∠x and ∠y in the given figure are respectively

**Solution :**

(d) As we know,

**Question 35:
**If two sides of a triangle are 6 cm and 10 cm, then the length of the third side can be

(a) 3 cm (b) 4 cm

(c) 2 cm (d) 6 cm

**Solution :**

(d) As we know, sum of any two sides of a triangle is always greater than the third side. So, option (d) satisfy this rule.

Verification

6+6>10

6+ 10> 6

10+ 6> 6

**Question 36:
**In a right angled ΔABC, if ∠B = 90°, BC = 3 cm and AC = 5 cm, then the length of side AB is

(a) 3 cm (b) 4 cm

(c) 5 cm (d) 6 cm

**Solution :**

(b) Since, Δ ABC is a right angled triangle.

**Question 37:
**In a right angled Δ ABC, if ∠B = 90°, then which of the following is true?

(a) AS

^{2}= BC

^{2}+ AC

^{2 }(b) A C

^{2}= AB

^{2}+ BC

^{2 }(c) AB = BC + AC

(d) AC = AB + BC

**Solution :**

(b) According to Pythagoras theorem,

**Question 38:
**Which of the following figures will have it’s altitude outside triangle?

**Solution :**

(d) As we know, the perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle.

**Question 39:
**In the given figure, if AB || CD, then

**Solution :**

(d) Given, AB || CD and AC is the transversal.

;

**Question 40:
**In ΔABC, ∠A = 100°, AD bisects ∠A and AD ⊥ BC. Then, ∠B is equal to

(a) 80° (b) 20° (c) 40° (d) 30°

**Solution :**

**Question 41:
**In ΔABC, ∠A = 50°, ∠B = 70° and bisector of ∠C meets AB in D as shown in the given figure. Measure of ∠ADC is

**Solution :**

**Question 42:
**If for ΔABC and ΔDEF, the correspondence CAB ↔ EDF gives a congruence, then which of the following is not true?

(a) AC = DE (b) AB = EF (c) ∠A = ∠D (d) ∠C = ∠E

**Solution :**

(b) Two figures are said to be congruent, if the trace copy of figure 1 fits exactly on that of

**Question 43:
** In the given figure, M is the mid-point of both AC and BD. Then,

**Solution :**

**Question 44:
**If D is the mid-point of side BC in ΔABC, where AB = AC, then ∠ADC is

(a) 60° (b) 45° (c) 120° (d) 90°

**Solution :**

**Question 45:
**Two triangles are congruent, if two angles and the side included between them in one of the triangles are equal to the two angles and the side included between them of the other triangle. This is known as the

(a) RHS congruence criterion (b) ASA congruence criterion

(c) SAS congruence criterion (d) AAA congruence criterion

**Solution :**

(b) Under ASA congruence criterion, two triangles are congruent, if two angles and the side included between them in one of the triangles are equal to the two angles and the side included between them of the other triangle.

**Question 46:
**By which congruency criterion, the two triangles in the given figure are congruent?

**Solution :**

**Question 47:
**By which of the following criterion two triangles cannot be proved congruent?

(a) AAA (b) SSS

(c) SAS (d) ASA

**Solution :**

(a) AAA is not a congruency criterion, because if all the three angles of two triangles are equal; this does not imply that both the triangles fit exactly on each other.

**Question 48:
**If ΔPQR is congruent to ΔSTU as shown in the given figure, then what is the length of TU?

**Solution :**

**Question 49:
**If ΔABC and ΔDBC are on the same base BC, AB = DC and AC = DB as shown in the given figure, then which of the following gives a congruence relationship?

**Solution :**

**Fill in the Blanks
**

In questions 50 to 69, fill in the blanks to make the statements true.

**Question 50:
**The_______triangle always has altitude outside itself.

**Solution :**

The

**obtuse**angled triangle always has altitude outside itself,

**Question 51:
**The sum of an exterior angle of a triangle and its adjacent angle is always______.

**Solution :**

The sum of an exterior angle of a triangle and its adjacent angle is always,

**180°**, because they form a linear pair.

**Question 52:
**The longest side of a right angled triangle is called its______.

**Solution :**

**Hypotenuse**is the longest side of a right angled triangle.

**Question 53:
**Median is also called______in an equilateral triangle.

**Solution :**

Median is also called

**an altitude**in an equilateral triangle.

**Question 54:
**Measures of each of the angles of an equilateral triangle is______.

**Solution :**

Measures of each of the angles of an equilateral triangle is

**60°**as all the angles in an equilateral triangle are equal.

Let x be the angle of equilateral.

**Question 55:
**In an isosceles triangle, two angles are always______.

**Solution :**

In an isosceles triangle, two angles are always

**equal.**Since, if two sides are equal, then the angles opposite them are equal.

**Question 56:
**In an isosceles triangle, angles opposite to equal sides are______.

**Solution :**

In an isosceles triangle, angles opposite to equal sides are

**equal.**Since, if two angles are equal then the sides opposite to them are also equal.

**Question 57:
**If one angle of a triangle is equal to the sum of other two, then the measure of that angle is______.

**Solution :**

Let the angles of a triangle be a, band c. It is given that,

**Question 58:
**Every triangle has atleast______acute angle (s).

**Solution :**

Every triangle has atleast

**two**acute angles.

**Question 59:
**Two line segments are congruent, if they are of______lengths.

**Solution :**

Two line segments are congruent, if they are of

**equal**lengths.

**Question 60:
**Two angles are said to be______, if they have equal measures.

**Solution :**

Two angles are said to be

**congruent**, if they have equal measures.

**Question 61:
**Two rectangles are congruent, if they have same______and______.

**Solution :**

Two rectangles are congruent, if they have same length and

**breadth**.

**Question 62:
**Two squares are congruent, if they have same………………. .

**Solution :**

Two squares are congruent, if they have same

**side**.

**Question 63:
**If ΔPQR and ΔXYZ are congruent under the correspondence QPR ↔ XYZ, then

(i) ∠R =______ (ii) QR =______

(iii) ∠P =______ (iv) QP = ______

(v) ∠Q =______ (vi) RP =______

**Solution :**

**Question 64:
**In the given figure, ΔPQR ≅ Δ ……….. .

**Solution :**

**Question 65:
**In the given figure, ΔPQR ≅ Δ ……….. .

**Solution :**

**Question 66:
**In the given figure, Δ ……….. ≅ΔPQR.

**Solution :**

**Question 67:
**In the given figure, ΔARO ≅ Δ ……….. .

**Solution :**

**Question 68:
**In the given figure, AB = AD and ∠BAC = ∠DAC. Then,

(i) A____ ≅ABC

(ii) BC =___ .

(iii) ∠BCA=______ .

(iv) Line segment AC bisects___ and_____ .

**Solution :**

**Question 69:
**In the given figure,

(i) ∠TPQ = ∠______+ ∠______

(ii) ∠UQR = ∠______+ ∠______

(iii) ∠PRS = ∠______+ ∠______

**Solution :**

Exterior angle property

The measure of an exterior angle is equal to the sum of the two opposite interior angles.

(i) ∠TPQ= ∠PQR + ∠PRQ

(ii) ∠UQR= ∠QRP + ∠QPR

(iii) ∠PRS

**RPQ + ∠RQP**

*= ∠***True/False **

In questions 70 to 106, state whether the statements are True or False.

**Question 70:
**In a triangle, sum of squares of two sides is equal to the square of the third side.

**Solution :**

**False**

Only in a right angled triangle, the sum of two shorter sides is equal to the square of the third side.

**Question 71:
**Sum of two sides of a triangle is greater than or equal to the third side.

**Solution :**

**False**

Sum of two sides of a triangle is greater than the third side.

**Question 72:
**The difference between the lengths of any two sides of a triangle is smaller than the length of third side.

**Solution :**

**True**

The difference between the lengths of any two sides of a triangle is smaller than the length of third side.

**Question 73:
**In ΔABC, AB = 3.5 cm, AC = 5 cm, BC = 6 cm and in ΔPQR, PR = 3.5 cm, PQ = 5 cm, RQ = 6 cm. Then, ΔABC ≅ ΔPQR.

**Solution :**

**False**

**Question 74:
**Sum of any two angles of a triangle is always greater than the third angle.

**Solution :**

**False**

It is not necessary that sum of any two angles of a triangle is always greater than the third angle, e.g. Let the angles of a triangle be 20°, 50° and 110°, respectively.

Hence, 20° + 50° = 70°, which is less than 110°.

**Question 75:
**The sum of the measures of three angles of a triangle is greater than 180°.

**Solution :**

**False**

The sum of the measures of three angles of a triangle is always equal to 180°.

**Question 76:
**It is possible to have a right angled equilateral triangle.

**Solution :**

**False**

In a right angled triangle, one angle is equal to 90° and in equilateral triangle, all angles are equal to 60°.

**Question 77:
**If M is the mid-point of a line segment AB, then we can say that AM and MB are congruent.

**Solution :**

**True**

**Question 78:
**It is possible to have a triangle in which two of the angles are right angles.

**Solution :**

**False**

If in a triangle two angles are right angles, then third angle = 180° – (90° + 90°) = 0°, which is not possible.

**Question 79:
**It is possible to have a triangle in which two of the angles are obtuse.

**Solution :**

**False**

Obtuse angles are those angles which are greater than 90°. So, sum of two obtuse angles will be greater than 180°, which is not possible as the sum of all the angles of a triangle is 180°.

**Question 80:
**It is possible to have a triangle in which two angles are acute.

**Solution :**

**True**

In a triangle, atleast two angles must be acute angle.

**Question 81:
**It is possible to have a triangle in which each angle is less than 60°.

**Solution :**

**False**

The sum of all angles in a triangle is equal to 180°. So, all three angles can never be less than 60°.

**Question 82:
**It is possible to have a triangle in which each angle is greater than 60°.

**Solution :**

**False**

If all the angles are greater than 60° in a triangle, then the sum of all the three angles with exceed 180°, which cannot be possible in case of triangle

**Question 83:
**It is possible to have a triangle in which each angle is equal to 60°.

**Solution :**

**True**

The triangle in which each angle is equal to 60° is called an equilateral triangle.

**Question 84:
**A right angled triangle may have all sides equal.

**Solution :**

**False**

Hypotenuse is always the greater than the other two sides of the right angled triangle.

**Question 85:
**If two angles of a triangle are equal, the third angle is also equal to each of the other two angles.

**Solution :**

**False**

In an isosceles triangle, always two angles are equal and not the third one.

**Question 86:
**In the given figures, two triangles are congruent by RHS.

**Solution :**

**Question 87:
**The congruent figures superimpose to each other completely.

**Solution :**

**True**

Because congruent figures have same shape and same size.

**Question 88:
**A one rupee coin is congruent to a five rupees coin.

**Solution :**

**False**

Because they don’t have same shape and same size.

**Question 89:
**The top and bottom faces of a kaleidoscope are congruent.

**Solution :**

**True**

Because they superimpose to each other.

**Question 90:
**Two acute angles are congruent.

**Solution :**

**False**

Because the measure of two acute angles may be different.

**Question 91:
**Two right angles are congruent.

**Solution :**

**True**

Since, the measure of right angles is always same.

**Question 92:
**Two figures are congruent, if they have the same shape.

**Solution :**

**False**

Two figures are congruent, if they have the same shape and same size.

**Question 93:
**If the areas of two squares is same, they are congruent.

**Solution :**

**True**

Because two squares will have same areas only if their sides are equal and squares with same sides will superimpose to each other.

**Question 94:
**If the areas of two rectangles are same, they are congruent.

**Solution :**

**False**

Because rectangles with the different length and breadth may have equal areas. But, they will not superimpose to each other.

**Question 95:
**If the areas of two circles are the same, they are congruent.

**Solution :**

**True**

Because areas of two circles will be equal only if their radii are equal and circle with same radii will superimpose to each other.

**Question 96:
**Two squares having same perimeter are congruent.

**Solution :**

**True**

If two squares have same perimeter, then their sides will be equal. Hence, the squares will superimpose to each other.

**Question 97:
**Two circles having same circumference are congruent.

**Solution :**

**True**

If two circles have same circumference, then their radii will be equal. Hence, the circles will superimpose to each other.

**Question 98:
**If three angles of two triangles are equal, triangles are congruent.

**Solution :**

**False**

Consider two equilateral triangles with different sides.

Both ΔABC and ΔDEF have same angles but their size is different. So, they are not congruent

**Question 99:
**If two legs of a right angled triangle are equal to two legs of another right angled triangle, then the right triangles are congruent.

**Solution :**

**True**

If two legs of a right angled triangle are equal to two legs of another right angled triangle, then their third leg will also be equal. Hence, they will have same shape and same size.

**Question 100:
**If two sides and one angle of a triangle are equal to the two sides and angle of another triangle, then the two triangles are congruent.

**Solution :**

**False**

Because if two sides and the angle included between them of the other triangle, then the two triangles will be congruent.

**Question 101:
**If two triangles are congruent, then the corresponding angles are equal.

**Solution :**

**True**

Because if two triangles are congruent, then their sides and angles are equal.

**Question 102:
**If two angles and a side of a triangle are equal to two angles and a side of another triangle, then the triangles are congruent.

**Solution :**

**False**

if two angles and the side included between them of a triangle are equal to two angles and included a side between them of the other triangle, then triangles are congruent.

**Question 103:
**If the hypotenuse of one right triangle is equal to the hypotenuse of another right triangle, then the triangles are congruent.

**Solution :**

**False**

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

**Question 104:
**If hypotenuse and an acute angle of one right angled triangle are equal to the hypotenuse and an acute angle of another right angled triangle, then the triangles are congruent.

**Solution :**

**Question 105:
**AAS congruence criterion is same as ASA congruence criterion.

**Solution :**

**False**

In ASA congruence criterion, the side ‘S’ included between the two angles of the triangle. In AAS congruence criterion, side ‘S’ is not included between two angles.

**Question 106:
**In the given figure, AD⊥BC and AD is the bisector of angle BAC. Then, ΔABD ≅ ΔACD by RHS.

**Solution :**

**False**

**Question 107:
**The measure of three angles of a triangle are in the ratio 5:3:1. Find the measures of these angles.

**Solution :**

**Question 108:
**In the given figure, find the value of x.

**Solution :**

We know that, the sum of all three angles in a triangle is equal to 180°.

**Question 109:
**In the given figures (i) and (ii), find the values of a, b and c.

**Solution :**

**Question 110:
**In ΔXYZ, the measure of ∠X is 30° greater than the measure of ∠Y and ∠Z is a right angle. Find measure of ∠Y.

**Solution :**

**Question 111:
**In a ΔABC, the measure of an ∠A is 40° less than the measure of other ∠B is 50° less than that of ∠C. Find the measure of ∠A.

**Solution :**

**Question 112:
**I have three sides. One of my angle measures 15°. Another has a measure of 60°. What kind of a polygon am I? If I am a triangle, then what kind of triangle am I?

**Solution :**

The polygon with three sides is called triangle.

**Question 113:
**Jiya walks 6 km due east and then 8 km due north. How far is she from her starting place?

**Solution :**

As per the given information, we can draw the following figure, which is a right angled triangle at B.

**Question 114:
**Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 m x 80 m. How much shorter is the route across park than the route around its edges?

**Solution :**

As the park is rectangular, ail the angles are of 90°.

**Question 115:
**In ΔPQR of the given figure, PQ = PR. Find measures of ∠Q and ∠R.

**Solution :**

**Question 116:
**In the given figure, find the measures of ∠x and ∠y.

**Solution :**

**Question 117:
**In the given figure, find the measures of ∠PON and ∠NPO.

**Solution :**

**Question 118:
**In the given figure, QP || RT. Find the values of x and y.

**Solution :**

In the given figure, QP|| RT, where PR is a transversal line.

So, ∠x and ∠TRPare alternate interior angles,

**Question 119:
**Find the measure of ∠A in the given figure.

**Solution :**

As we know, the measure of exterior angle is equal to the sum of opposite interior angles.

**Question 120:
**In a right angled triangle, if an angle measures 35°, then find the measure of the third angle.

**Solution :**

In a right angled ΔABC,

**Question 121:
**Each of the two equal angles of an isosceles triangle is four times the third angle. Find the angles of the triangle.

**Solution :**

Let the third angle be x. Then, the other two angles are 4x and 4x, respectively.

**Question 122:
**The angles of a triangle are in the ratio 2:3:5. Find the angles.

**Solution :**

Let measures of the given angles of a triangle be 2x, 3x and 5x.

**Question 123:
**If the sides of a triangle are produced in an order, show that the sum of the exterior angles so formed is 360°.

**Solution :**

**Question 124:
**In ΔABC, if ∠A

**= ∠**C and exterior ∠ABX

**=**140°, then find the angles of the triangle.

**Solution :**

**Question 125:
**Find the values of x and y in the given figure.

**Solution :**

**Question 126:
**Find the value of x in the given figure.

**Solution :**

**Question 127:
**The angles of a triangle are arranged in descending order of their magnitudes. If the difference between two consecutive angles is 10°, find the three angles.

**Solution :**

Let one of the angles of a triangle be x. If angles are arranged in descending order. Then, angles will be x, (x -10°) and (x – 20°).

**Question 128:
**In ΔABC, DE || BC (see the figure). Find the values of x, y and z.

**Solution :**

**Question 129:
**In the given figure, find the values of x, y and z.

**Solution :**

**Question 130:
**If one angle of a triangle is 60° and the other two angles are in the ratio 1: 2, find the angles.

**Solution :**

**Question 131:
**In ΔPQR, if 3∠P = 4∠Q = 6∠R, calculate the angles of the triangle.

**Solution :**

**Question 132:
**In ΔDEF, ∠D = 60°, ∠E = 70° and the bisectors of ∠E and ∠E meet at 0. Find (i) ∠F (ii) ∠EOF.

**Solution :**

**Question 133:
**In the given figure, ΔPQR is right angled at P. U and T are the points on line QRF. If QP || ST and US || RP, find ∠S.

**Solution :**

If QP || ST and QT is a transversal, then ∠PQR = ∠STU [alternate interior angles]

and if DS || RP and QT is a transversal, then ∠PRQ = ∠SUT [alternate interior angles]

Hence, ∠S must be equal to ∠Pi.e. 90°.

**Question 134:
**In each of the given pairs of triangles in given figures, applying only ASA congruence criterion, determine which triangles are congruent. Also, write the congruent triangles in symbolic form.

**Solution :**

(a) Not possible, because the side is not included between two angles.

(b) ∆ABD ≅ ∆CDB

(c) ∆XYZ ≅ ∆LMN

(d) Not possible, because there is not any included side equal.

(e) ∆MNO ≅ ∆PON

(f) ∆AOD ≅ ∆BOC

**Question 135:
**In each of the given pairs of triangles in given figures, using only RHS congruence criterion, determine which pairs of triangles are congruent. In case of congruence, write the result in symbolic form,

**Solution :**

**Question 136:
**In the given figure, if RP = RQ, find the value of x.

**Solution :**

**Question 137:
**In the given figure, if ST = SU, then find the values of x and y.

**Solution :**

**Question 138:
**Check whether the following measures (in cm) can be the sides of a right angled triangle or not.

1.5, 3.6, 3.9

**Solution :**

For a right angled triangle, the sum of square of two shorter sides must be equal to the square of the third side.

**Question 139:
**Height of a pole is 8 m. Find the length of rope tied with its top from a point on the ground at a distance of 6 m from its bottom.

**Solution :**

Given, height of a pole is 8 m.

Distance between the bottom of the pole and a point on the ground is 6 m.

On the basis of given information, we can draw the following figure:

**Question 140:
**In the given figure, if y is five times x, find the value of z.

**Solution :**

**Question 141:
**The lengths of two sides of an isosceles triangle are 9 cm and 20 cm. What is the perimeter of the triangle? Give reason.

**Solution :**

Third side must be 20 cm, because sum of two sides should be greater than the third side.

∴ Perimeter of the triangle

= Sum of all sides

= (9 + 20 + 20) cm

= 49 cm

**Question 142:
**Without drawing the triangles write all six pairs of equal measures in each of the following pairs of congruent triangles.

(a) ΔSTU ≅ ΔDEF (b) ΔABC ≅ ΔLMN

(c) ΔYZX ≅ APQR (d) ΔXYZ ≅ ΔMLN

**Solution :**

We know that, corresponding parts of congruent triangles are equal.

(a) ΔSTU ≅ ΔDEF

∠S = ∠D, ∠T = ∠E and ∠U = ∠F ST = DE, TU = EF and SU = DF

(b) ΔABC ≅ ΔLMN

∠A = ∠L, ∠B = ∠M and ∠C = ∠N AB = LM, BC = MN and AC = LN

(c) ΔYZX ≅ APQR

∠T = ∠P, ∠Z = ∠Q and ∠X = ∠R YZ = PQ, ZX = QR and YX = PR

(d) ΔXYZ ≅ ΔMLN

∠X = ∠M, ∠Y = ∠L and ∠Z = ∠N XY = ML,YZ = LN and XZ = MN

**Question 143:
**In the following pairs of triangles in below figures, the lengths of the sides are indicated along the sides. By applying SSS congruence criterion, determine which triangles are congruent. If congruent, write the results in symbolic form.

**Solution :**

(a) ΔABC ≅ ΔNLM (b) ΔLMN ≅ ΔGHI

(c) ΔLMN ≅ ΔLON (d) ΔZYX ≅ ΔWXY

(e) ΔOAB ≅ ΔDOE (f) ΔSTU ≅ ΔSVU

(g) ΔPSR ≅ ARQP (h) ΔSTU ≅ ΔPQR

**Question 144:
**ABC is an isosceles triangle with AB = AC and D is the mid-point of base BC (see the figure).

(a) State three pairs of equal parts in the ΔABD and ΔACD.

(b) Is ΔABD ≅ ΔACD? If so why?

**Solution :**

**Question 145:
**In the given figure, it is given that LM = ON and NL = MO.

(a) State the three pairs of equal parts in the ∆NOM and ∆MLN.

(b) Is ∆NOM ≅ ∆MLN? Give reason.

**Solution :**

**Question 146:
**ΔDEF and ΔLMN are both isosceles with DE = DF and LM = LN, respectively. If DE = LM and EF = MN, then are the two triangles congruent? Which condition do you use?

If ∠E = 40°, what is the measure of ∠N?

**Solution :**

**Question 147:
**If ΔPQR and ΔSQR are both isosceles triangle on a common base QR such that P and S lie on the same side of QR. Are ΔPSQ and ΔPSR congruent? Which condition do you use?

**Solution :**

**Question 148:
**In the given figures, which pairs of triangles are congruent by SAS congruence criterion (condition)? If congruent, write the congruence of the two triangles in symbolic form.

**Solution :**

**Question 149:
**State which of the following pairs of triangles are congruent. If yes,

write them in symbolic form (you may draw a rough figure).

(a) ΔPQR : PQ = 3.5 cm, QR 4.0 cm, ∠Q 60°

ΔSTU : ST = 3.5 cm, TU = 4 cm, ∠T = 60°

(b) ΔABC : AB = 4.8 cm, ∠A = 90°, AC = 6.8 cm

ΔXYZ : YZ = 6.8 cm, ∠X = 90°, ZX = 4.8 cm

**Solution :**

**Question 150:
**In the given figure, PQ = PS and ∠1 = ∠2.

(i) Is ΔPQR ≅ ΔPSR? Give reason.

(ii) Is QR =SR? Give reason.

**Solution :**

**Question 151:
**In the given figure, DE = IH, EG = FI and ∠E = ∠I. Is ΔDEF ≅ ΔHIG? If yes, by which congruence criterion?

**Solution :**

**Question 152:
**In the given figure, ∠1 = ∠2 and ∠3 = ∠4.

(i) Is ΔADC ≅ ΔABC Why?

Show that AD = AB and CD = CB.

**Solution :**

**Question 153:
**Observe the following figure and state the three pairs of equal parts in ΔABC and ΔDBC.

(i) Is ΔABC ≅ ΔDCB? Why?

(ii) Is AB = DC? Why?

(iii) Is AC = DB? Why?

**Solution :**

**Question 154:
**In the given figure, QS⊥PR, RT⊥PQ and QS = RT.

(i) Is ΔQSR ≅ ΔRTS? Give reason.

(ii) Is ∠PQR = ∠PRQ? Give reason.

**Solution :**

**Question 155:
**Points A and B are on the opposite edges of a pond as shown in the given figure. To find the distance between the two points, the surveyor makes a rightangled triangle as shown. Find the distance AB.

**Solution :**

**Question 156:
**Two poles of 10 m and 15 m stand upright on a plane ground. If the distance between the tops is 13 m, find distance between their feet.

**Solution :**

**Question 157:
**The foot of a ladder is 6 m away from its wall and its top reaches a window 8 m above the ground, (a) Find the length of the ladder, (b) If the ladder is shifted in such a way that its foot is 8 m away from the wall, to what height does its top reach?

**Solution :**

**Question 158:
**In the given figure, state the three pairs of equal parts in ΔABC and ΔEOD. Is ΔABC ≅ ΔEOD? Why?

**Solution :**

;