Chapter 6

medians. • The perpendicular line segment from a vertex of a triangle to its opposite side is called an altitude of the triangle. A triangle has 3 altitudes. • An exterior angle of a triangle is formed, when a side of a triangle is produced. • The measure of any exterior angle of a triangle is equal to the sum of the measures of its two interior opposite angles. • The sum of the three angles of a triangle is 180°. • A triangle is said to be equilateral, if each of its sides has the same length. • In an equilateral triangle, each angle has measure 60°. • A triangle is said to be isosceles if at least two of its sides are of same length. • The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. • The difference of the lengths of any two sides of a triangle is always smaller than the length of the third side. 15-04-2018 • In a right-angled triangle, the side opposite to the right angle is called the hypotenuse and the other two sides are called its legs or arms. • In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares on its legs. • Two plane figures, say, F 1 and F2 are said to be congruent, if the trace-copy of F1 fits exactly on that of F2. We write this as F1 ≅ F2. • Two line segments, say AB and CD , are congruent, if they have equal lengths. We write this as AB ≅ CD . However, it is common to write it as AB = CD . • Two angles, say ∠ABC and ∠PQR, are congruent, if their measures are equal. We write this as ∠ABC ≅ ∠ PQR or as m ∠ABC = m∠PQR or simply as ∠ ABC = ∠ PQR. • Under a given correspondence, two triangles are congruent, if the three sides of the one are equal to the three sides of the other (SSS). • Under a given correspondence, two triangles are congruent if two sides and the angle included between them in one of the triangles are equal to the two sides and the angle included between them of the other triangle (SAS). • Under a given correspondence, 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 (ASA). • Under a given correspondence, two right-angled triangles are congruent if the hypotenuse and a leg (side) of one of the triangles are equal to the hypotenuse and one of the leg (side) of the other triangle (RHS). In Examples 1 to 5, there are four options, out of which only one is correct. Write the correct one.

In a right-angled triangle, the angles other than the right angle are

This question refers to a figure in the original PDF.

• (a)obtuse
• (b)right
• (c)acute
• (d)straight Fig. 6.7

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 15-04-2018 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)

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)

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

In Fig. 6.8, PB = PD. The value of x is

This question refers to a figure in the original PDF.

• (a)85°
• (b)90°
• (c)25°
• (d)35°

In ∆PQR,

• (a)PQ – QR > PR
• (b)PQ + QR < PR
• (c)PQ – QR< PR
• (d)PQ + PR< QR

In ∆ ABC, Fig. 6.8

This question refers to a figure in the original PDF.

• (a)AB + BC > AC
• (b)AB + BC < AC
• (c)AB + AC < BC
• (d)AC + BC < AB 1. Explain what it means for two polygons to be congruent. 2. Tell how to write a congruence statement for two triangles. 15-04-2018

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

The trianlge ABC formed by AB = 5 cm, BC = 8 cm, 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

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

If in an isosceles triangle, each of the base angles is 40°, then the triangle is

• (a)Right-angled triangle
• (b)Acute angled triangle
• (c)Obtuse angled triangle
• (d)Isosceles right-angled triangle

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

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

In ∆PQR, if PQ = QR and ∠Q = 100°, then ∠R is equal to

• (a)40°
• (b)80°
• (c)120°
• (d)50°

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

In Fig. 6.9, BC = CA and ∠A = 40. Then, ∠ACD is equal to

This question refers to a figure in the original PDF.

• (a)40°
• (b)80°
• (c)120°
• (d)60° Fig. 6.9 15-04-2018

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

This question refers to a figure in the original PDF.

• (a)1 cm and 10 cm
• (b)2 cm and 8 cm
• (c)3 cm and 16 cm Fig. 6.10
• (d)1 cm and 16 cm

From Fig. 6.10, the value of x is

This question refers to a figure in the original PDF.

• (a)75°
• (b)90°
• (c)120°
• (d)60°

In Fig. 6.11, the value of ∠A + ∠B + ∠C + ∠D + ∠E + ∠F is

This question refers to a figure in the original PDF.

• (a)190°
• (b)540°
• (c)360°
• (d)180° Fig. 6.11

In Fig. 6.12, PQ = PR, RS = RQ and ST || QR. If the exterior angle RPU is 140°, then the measure of angle TSR is

This question refers to a figure in the original PDF.

• (a)55°
• (b)40°
• (c)50°
• (d)45°

In Fig. 6.13, ∠BAC = 90°, AD ⊥ BC and ∠BAD = 50°, then ∠ACD is

This question refers to a figure in the original PDF.

• (a)50°
• (b)40°
• (c)70°
• (d)60°

If one angle of a triangle is equal to the sum of the Fig. 6.12 other two angles, the triangle is

This question refers to a figure in the original PDF.

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

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

This question refers to a figure in the original PDF.

• (a)55°
• (b)65°
• (c)50°
• (d)60° Fig. 6.13 15-04-2018

If one of the angles 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°

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

In ∆PQR, 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°

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°

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 cm

How many altitudes does a triangle have?

• (a)1
• (b)3
• (c)6
• (d)9

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

The measures of ∠x and ∠y in Fig. 6.14 are respectively

This question refers to a figure in the original PDF.

• (a)30°, 60°
• (b)40°, 40°
• (c)70°, 70°
• (d)70°, 60°

If length of two sides of a triangle are Fig. 6.14 6 cm and 10 cm, then the length of the third side can be

This question refers to a figure in the original PDF.

• (a)3 cm
• (b)4 cm
• (c)2 cm
• (d)6 cm

In a right-angled triangle ABC, if angle 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 15-04-2018

In a right-angled triangle ABC, if angle B = 90°, then which of the following is true?

• (a)AB2 = BC2 + AC2
• (b)AC2 = AB2 + BC2
• (c)AB = BC + AC
• (d)AC = AB + BC

Which of the following figures will have it’s altitude outside the triangle? Fig. 6.15

This question refers to a figure in the original PDF.

In Fig. 6.16, if AB || CD, then Fig. 6.16 15-04-2018

This question refers to a figure in the original PDF.

• (a)∠ 2 = ∠ 3
• (b)∠ 1 = ∠ 4
• (c)∠ 4 = ∠ 1 + ∠ 2
• (d)∠ 1 + ∠ 2 = ∠ 3 + ∠ 4

In ∆ABC, ∠Α = 100°, AD bisects ∠A and AD⊥BC. Then, ∠B is equal to

• (a)80°
• (b)20°
• (c)40°
• (d)30°

In ∆ABC, ∠Α = 50°, ∠B = 70° and bisector of ∠C meets AB in D (Fig. 6.17). Measure of ∠ADC is. Fig. 6.17

This question refers to a figure in the original PDF.

• (a)50°
• (b)100°
• (c)30°
• (d)70°

If for ∆ABC and ∆DEF, 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

In Fig. 6.18, M is the mid-point of both AC and BD. Then

This question refers to a figure in the original PDF.

• (a)∠1 = ∠2
• (b)∠1 = ∠4
• (c)∠2 = ∠4
• (d)∠1 = ∠3

If D is the mid-point of the side BC in ∆ABC where AB = AC, then ∠ADC is Fig. 6.18

This question refers to a figure in the original PDF.

• (a)60°
• (b)45°
• (c)120s°
• (d)90° 15-04-2018

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

By which congruency criterion, the two triangles in Fig. 6.19 are congruent?

This question refers to a figure in the original PDF.

• (a)RHS
• (b)ASA
• (c)SSS
• (d)SAS

By which of the following criterion two triangles Fig. 6.19 cannot be proved congruent?

This question refers to a figure in the original PDF.

• (a)AAA
• (b)SSS
• (c)SAS
• (d)ASA

If ∆PQR is congruent to ∆STU (Fig. 6.20), then what is the length of TU?

This question refers to a figure in the original PDF.

• (a)5 cm
• (b)6 cm
• (c)7 cm
• (d)cannot be determined Fig. 6.20

If ∆ABC and ∆DBC are on the same base BC, AB = DC and AC = DB (Fig. 6.21), then which of the following gives a congruence relationship?

This question refers to a figure in the original PDF.

• (a)∆ ABC ≅ ∆ DBC
• (b)∆ ABC ≅∆CBD
• (c)∆ ABC ≅∆ DCB
• (d)∆ ABC ≅∆BCD 15-04-2018 Fig. 6.21 In questions 50 to 69, fill in the blanks to make the statements true.

The triangle always has altitude outside itself.

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

The longest side of a right angled triangle is called its .

Median is also called in an equilateral triangle.

Measures of each of the angles of an equilateral triangle is .

In an isosceles triangle, two angles are always .

In an isosceles triangle, angles opposite to equal sides are .

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

Every triangle has at least acute angle (s).

Two line segments are congruent, if they are of lengths.

Two angles are said to be , if they have equal measures.

Two rectangles are congruent, if they have same and .

Two squares are congruent, if they have same .

If ∆ PQR and ∆ XYZ are congruent under the correspondence QPR ↔ XYZ, then

• (i)∠R =
• (ii)QR =
• (iii)∠P =
• (iv)QP = (v) ∠Q = (vi) RP = 15-04-2018

In Fig. 6.22, ∆PQR ≅ ∆ Fig. 6.22

This question refers to a figure in the original PDF.

In Fig. 6.23, ∆PQR ≅ ∆ Fig. 6.23

This question refers to a figure in the original PDF.

In Fig. 6.24, ∆ ≅ ∆ PQR Fig. 6.24

This question refers to a figure in the original PDF.

In Fig. 6.25, ∆ ARO ≅ ∆ Fig. 6.25

This question refers to a figure in the original PDF.

In Fig. 6.26, AB = AD and ∠ BAC = ∠ DAC. Then

This question refers to a figure in the original PDF.

• (i)∆ ≅ ∆ ABC.
• (ii)BC = . 15-04-2018
• (iii)∠ BCA = .
• (iv)Line segment AC bisects and . Fig. 6.26

In Fig. 6.27,

This question refers to a figure in the original PDF.

• (i)∠ TPQ = ∠ _____ + ∠ _____
• (ii)∠ UQR = ∠ _____ + ∠ _____
• (iii)∠ PRS = ∠ _____ + ∠ _____ Fig. 6.27 In questions 70 to 106 state whether the statements are True or False.

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

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

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

In ∆ABC, AB = 3.5 cm, AC = 5 cm, BC = 6 cm and in ∆PQR, PR= 3.5 cm, PQ = 5 cm, RQ = 6 cm. Then ∆ABC ≅ ∆PQR. 15-04-2018

Sum of any two angles of a triangle is always greater than the third angle.

The sum of the measures of three angles of a triangle is greater than 180°.

It is possible to have a right-angled equilateral triangle.

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

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

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

It is possible to have a triangle in which two angles are acute.

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

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

It is possible to have a triangle in which each angle is equal to 60°.

A right-angled triangle may have all sides equal.

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

In Fig. 6.28, two triangles are congruent by RHS.

This question refers to a figure in the original PDF.

The congruent figures super impose each other completely. Fig. 6.28

This question refers to a figure in the original PDF.

A one rupee coin is congruent to a five rupee coin.

The top and bottom faces of a kaleidoscope are congruent.

Two acute angles are congruent.

Two right angles are congruent.

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

If the areas of two squares is same, they are congruent.

If the areas of two rectangles are same, they are congruent.

If the areas of two circles are the same, they are congruent.

Two squares having same perimeter are congruent.

Two circles having same circumference are congruent. 15-04-2018

If three angles of two triangles are equal, triangles are congruent.

If two legs of a right triangle are equal to two legs of another right triangle, then the right triangles are congruent.

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.

If two triangles are congruent, then the corresponding angles are equal.

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.

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

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

AAS congruence criterion is same as ASA congruence criterion.

In Fig. 6.29, AD ⊥ BC and AD is the bisector of angle BAC. Then, ∆ABD ≅ ∆ACD by RHS. Fig. 6.29

This question refers to a figure in the original PDF.

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

In Fig. 6.30, find the value of x. Fig. 6.30 15-04-2018

This question refers to a figure in the original PDF.

In Fig. 6.31(i) and (ii), find the values of a, b and c. (i) (ii) Fig. 6.31

This question refers to a figure in the original PDF.

In triangle XYZ, the measure of angle X is 30° greater than the measure of angle Y and angle Z is a right angle. Find the measure of ∠Y.

In a triangle ABC, the measure of angle A is 40° less than the measure of angle B and 50° less than that of angle C. Find the measure of ∠ A.

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?

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

Jayanti takes shortest route to her home by walking diagonally across a rectangular park. The park measures 60 metres × 80 metres. How much shorter is the route across the park than the route around its edges? Understand the Problem • If you write a problem in your own words, you may understand it better. Before writing a problem in your own words, you may need to read it over several times – perhaps aloud, so you can hear yourself say the words. • Once you have written the problem in your own words, you may want to make sure you included all of the necessary information to solve the problem. 15-04-2018

In ∆PQR of Fig. 6.32, PQ = PR. Find the measures of ∠Q and ∠R. Fig. 6.32

This question refers to a figure in the original PDF.

In Fig. 6.33, find the measures of ∠ x and ∠ y. Fig. 6.33

This question refers to a figure in the original PDF.

In Fig. 6.34, find the measures of ∠ PON and ∠ NPO. Fig. 6.34

This question refers to a figure in the original PDF.

In Fig. 6.35, QP || RT. Find the values of x and y. Fig. 6.35 15-04-2018

This question refers to a figure in the original PDF.

Find the measure of ∠ A in Fig. 6.36. Fig. 6.36

This question refers to a figure in the original PDF.

In a right-angled triangle if an angle measures 35°, then find the measure of the third angle.

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

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

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

In ∆ABC, if ∠A = ∠C, and exterior angle ABX = 140°, then find the angles of the triangle. Fig. 6.37

This question refers to a figure in the original PDF.

Find the values of x and y in Fig. 6.37. Plan a Strategy • Concept maps are visual tools for organising information. A concept map shows how key concepts are related and can help you summarise and analyse information in lessons or chapters. Create a Concept Map • Give your concept map a title. • Identify the main idea of your concept map. • List the key concepts. • Link the concepts to show the relationships between the concepts and the main idea. 15-04-2018

This question refers to a figure in the original PDF.

Find the value of x in Fig. 6.38. Fig. 6.38

This question refers to a figure in the original PDF.

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.

In ∆ ABC, DE || BC (Fig. 6.39). Find the values of x, y and z. Fig. 6.39

This question refers to a figure in the original PDF.

In Fig. 6.40, find the values of x, y and z. Fig. 6.40

This question refers to a figure in the original PDF.

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

In ∆PQR, if 3∠P = 4∠Q = 6∠R, calculate the angles of the triangle.

In ∆DEF, ∠D = 60°, ∠E = 70° and the bisectors of ∠E and ∠F meet at O. Find

• (i)∠F
• (ii)∠EOF.

In Fig. 6.41, ∆PQR is right-angled at P. U and T are the points on line QRF. If QP || ST and US || RP, find ∠S. Fig. 6.41 15-04-2018

This question refers to a figure in the original PDF.

In each of the given pairs of triangles of Fig. 6.42, applying only ASA congruence criterion, determine which triangles are congruent. Also, write the congruent triangles in symbolic form.

This question refers to a figure in the original PDF.

• (a)
• (b)
• (c)
• (d)15-04-2018 (e) (f) Fig. 6.42

In each of the given pairs of triangles of Fig. 6.43, using only RHS congruence criterion, determine which pairs of triangles are congruent. In case of congruence, write the result in symbolic form:

This question refers to a figure in the original PDF.

• (a)
• (b)
• (c)
• (d)15-04-2018 (e) (f) Fig. 6.43

In Fig. 6.44, if RP = RQ, find the value of x. Fig. 6.44

This question refers to a figure in the original PDF.

In Fig. 6.45, if ST = SU, then find the values of x and y. Fig. 6.45

This question refers to a figure in the original PDF.

Check whether the following measures (in cm) can be the sides of a right-angled triangle or not. 1.5, 3.6, 3.9

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. 15-04-2018

In Fig. 6.46, if y is five times x, find the value of z. Fig. 6.46

This question refers to a figure in the original PDF.

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

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

• (a)∆STU ∆DEF
• (b)∆ABC ∆LMN
• (c)∆YZX ∆PQR
• (d)∆XYZ ∆MLN

In the following pairs of triangles of Fig. 6.47, 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.

This question refers to a figure in the original PDF.

• (a)
• (b)
• (c)
• (d)15-04-2018 (e) (f) (g) (h) Fig. 6.47

ABC is an isosceles triangle with AB = AC and D is the mid-point of base BC (Fig. 6.48).

This question refers to a figure in the original PDF.

• (a)State three pairs of equal parts in the triangles ABD and ACD.
• (b)Is ∆ABD ≅ ∆ACD. If so why? Fig. 6.48 15-04-2018

In Fig. 6.49, it is given that LM = ON and NL = MO

This question refers to a figure in the original PDF.

• (a)State the three pairs of equal parts in the triangles NOM and MLN.
• (b)Is ∆NOM ≅ ∆MLN. Give reason? Fig. 6.49

Triangles DEF and LMN 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?

If ∆PQR and ∆SQR are both isosceles triangle on a common base QR such that P and S lie on the same side of QR. Are triangles PSQ and PSR congruent? Which condition do you use?

In Fig. 6.50, which pairs of triangles are congruent by SAS congruence criterion (condition)? If congruent, write the congruence of the two triangles in symbolic form.

This question refers to a figure in the original PDF.

• (i)
• (ii)15-04-2018
• (iii)
• (iv)(v) (vi) 15-04-2018 (vii) (viii) Fig. 6.50

State which of the following pairs of triangles are congruent. If yes, write them in symbolic form (you may draw a rough figure).

• (a)∆ PQR : PQ = 3.5 cm, QR = 4.0 cm, ∠ Q = 60° ∆ STU : ST = 3.5 cm, TU = 4 cm, ∠ T = 60°
• (b)∆ABC : AB = 4.8 cm, ∠ A = 90°, AC = 6.8 cm ∆XYZ : YZ = 6.8 cm, ∠ X = 90° , ZX = 4.8 cm

In Fig. 6.51, PQ = PS and ∠ 1 = ∠ 2.

This question refers to a figure in the original PDF.

• (i)Is ∆PQR ≅ ∆PSR? Give reasons.
• (ii)Is QR = SR? Give reasons.

In Fig. 6.52, DE = IH, EG = FI and ∠ E = ∠ I. Is ∆DEF ≅ ∆HIG? If yes, by Fig. 6.51 which congruence criterion? Fig. 6.52 15-04-2018

This question refers to a figure in the original PDF.

In Fig. 6.53, ∠1 = ∠ 2 and ∠ 3 = ∠ 4.

This question refers to a figure in the original PDF.

• (i)Is ∆ADC ≅ ∆ ABC? Why ?
• (ii)Show that AD = AB and CD = CB.

Observe Fig. 6.54 and state the three pairs of equal parts in triangles ABC and DBC.

This question refers to a figure in the original PDF.

• (i)Is ∆ABC ≅ ∆DCB? Why?
• (ii)Is AB = DC? Why?
• (iii)Is AC = DB? Why? Fig. 6.53 Fig. 6.54

In Fig. 6.55, QS ⊥ PR, RT ⊥ PQ and QS = RT.

This question refers to a figure in the original PDF.

• (i)Is ∆ QSR ≅ ∆ RTQ? Give reasons.
• (ii)Is ∠ PQR = ∠ PRQ? Give reasons. Fig. 6.55

Points A and B are on the opposite edges of a pond as shown in Fig. 6.56. To find the distance between the two points, the surveyor makes a right-angled triangle as shown. Find the distance AB. 15-04-2018 Fig. 6.56

This question refers to a figure in the original PDF.

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

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?

In Fig. 6.57, state the three pairs of equal parts in ∆ABC and ∆EOD. Is ∆ABC ≅ ∆ EOD? Why? Fig. 6.57 1. Draw an equilateral triangle of side 6 cm, an isosceles triangle of base 3 cm and equal sides 6 cm each and a scalene triangle of sides 3 cm, 6 cm and 7 cm. Now draw a median and an altitude in each triangle from the top vertex, measure and tabulate the lengths of all the medians and altitude’s of respective triangles. What can you conclude from this activity (This activity can also be done by paper folding)? 2. Draw two triangles which have a pair of corresponding sides equal but are not congruent. 3. Draw two triangles which have two pairs of corresponding sides equal but are not congruent. 15-04-2018 4. Draw two triangles, which have one pair of corresponding angles equal and one pair of corresponding sides equal but are not congruent. 5. Draw two triangles which have three pairs of corresponding angles equal but are not congruent. Solve the given cross number/word and then fill up the given boxes in activities 6 and 7. Clues are given below for across as well as downward fillings. For across and downward clue numbers are written at the corner of boxes. Answers of clues have to fill up in their respective boxes. Cross Number Puzzle 6 Across

This question refers to a figure in the original PDF.

• (a)If 6, 8, m are the sides of a right triangle, then the value of m is ______.
• (b)In ∆ABC, AC is the longest side, then what can be the measure of angle B (in degree), if the three angles of triangle are 120°, 40°, 20°?
• (c)In a right-angled triangle, one acute angle measures twice the other angle, then the smaller angle shall measure _________.
• (d)If three angles in ∆ABC are in the ratio 2 : 3 : 5, then measure of ∠B is _____. (e) Length of third side of a triangle whose two sides are 5 cm and 6 cm, must be less than _______. (f) The perimeter of ∆ABC in Fig. 6.58 is _________. Fig. 6.58 15-04-2018 Down (a) In an isosceles triangle if one of the equal angles measures 35°, then the third angle is __________. (b) In Fig. 6.59, the value of x is _________. Fig. 6.59 (c) The sum of the angles in a quadrilateral is ___________. (d) In ∆ ABC, ∠B = 80°, ∠ A = 30°, the bisectors of ∠B and ∠C meet at O. The measure of ∠BOC is ___________. 15-04-2018 Cross Word Puzzle 7 Across 1. A triangle with all its sides unequal. 2. The longest side of a right-angled triangle. 3. Two squares having same side lengths. 4. Line segment drawn from a vertex of a triangle perpendicular to its opposite side. Down 5. A type of triangle in which altitude falls outside the triangle. 6. A line segment joining vertex with the mid-point of the opposite side. 7. A regular triangle. 8. In a parallelogram, the line segment that divides it into two congruent triangles. 15-04-2018