Gujarat BoardĀ GSEB Textbook Solutions Class 7 Maths Chapter 6 The Triangles and Its Properties InText Questions and Answers.

## Gujarat Board Textbook Solutions Class 7 Maths Chapter 6 The Triangles and Its Properties InText Questions

Try These (Page 113)

Question 1.

Write the six elements (i.e. the 3 sides and the 3 angles) of ĪABC.

Solution:

Six elements of āABC are: ā A, ā B, ā C,

\(\overline{A B}\), \(\overline{B C}\) and \(\overline{C A}\).

Question 2.

Write the:

(i) Side opposite to the vertex Q of āPQR

(ii) Angle opposite to the side LM of āLMN

(iii) Vertex opposite to the side RT of āRST

Solution:

(i) The Side opposite to the vertex Q of \(\overline{P R }\)

(ii) The Angle opposite to the side LM of ā N

(iii) Vertex opposite to the side RT is ‘S’ .

Question 3.

Look at figures and classify each of the triangles according to its –

(a) Sides

(b) Angles

Solution:

(i) (a) āµ \(\overline{AC}\) = \(\overline{B C }\) = 8 cm

ā“ āABC is an isosceles triangle.

(b) Since all angles of āABC are less than 90Ā°

ā“ It is an acute triangle.

(ii) (a) Since PQ ā QR ā RP

ā“ āPQR is a scalene triangle.

(b) Since ā R = 90Ā°

ā“ āPQR is a right triangle.

(iii) (a) In āLMN, LN = MN = 7 cm

ā“ āLMN is an isosceles triangle.

(b) In āLMN, ā N > 90Ā°

ā“ āLMN is an obtuse triangle.

(iv) (a) In āRST, RS = ST = TR = 5.2 cm

ā“ It is an equilateral triangle.

(b) All the angles of āRST are acute.

ā“ It is an acute triangle.

(v) (a) In āABC, \(\overline{A B}\) = \(\overline{B C}\) = 3 cm

ā“ It is an isosceles triangle.

(b) In āABC, ā B > 90Ā°

ā“ It is an obtuse triangle

(vi) (a) In āPQR, \(\overline{P Q}\) = \(\overline{Q R}\) = 6 cm

ā“ It is an isosceles triangle.

(b) In āPQR, ā Q = 90Ā°

ā“ It is a right triangle.

Think, Discuss and Write (Page 114)

Question 1.

How many medians can a triangle have?

Solution:

Since a triangle has three sides and by joining the mid-point of each side to the opposite vertex, we can draw three line segments. Thus, a triangle can have three medians.

Question 2.

Does a median lie wholly in the interior of the triangle? (If you think that this is not true, draw a figure to show such a case.)

Solution:

Yes, a median lies wholly in the interior of the triangle.

Think, Discuss and Write (Page 115)

Question 1.

How many altitudes can a triangle have?

Solution:

An altitude can be drawn from each vertex of a triangle.

ā“ A triangle can have three altitudes.

Question 2.

Draw rough sketches of altitudes from A to \(\overline{B C}\) for the following triangles.

Solution:

We have \(\overline{A L}\) as an altitude in each case as shown below:

Question 3.

Will an altitude always lie in the interior of a triangle? If you think that this need not be true, draw a rough sketch to show such a case.

Solution:

No, the altitude does not always lie in the interior of a triangle.

Example: In the adjoining figure, the altitude AD is not in the interior of ĪABC.

Question 4.

Can you think of a triangle in which two altitudes of the triangle are two of its sides?

Solution:

Yes, a right triangle has two of its sides as its altitudes. In a right triangle ABC, \(\overline{A C}\) and \(\overline{B C}\) are its altitudes.

Question 5.

Can the altitude and median be same for a triangle?

Solution:

Yes, in a triangle, its A median and altitude can be same line segment. In the adjoining figure, AL is an altitude as well as a median of āABC.

Think, Discuss and Write (Page 117)

Question 1.

Exterior angles can be formed for a triangle in many ways. Three of them are shown here.

There are three more ways of getting exterior angles. Try to produce those rough sketches.

Solution:

Three other exterior angles can be as under:

Question 2.

Are the exterior angles formed at each vertex of a triangle equal?

Solution:

No.

Question 3.

What can you say about the sum of an exterior angle of a triangle and its adjacent interior angle?

Solution:

The sum of an exterior angle and its adjacent interior angle form a linear pair.

ā“ [Exterior angle] + [Interior adjacent angle] = 180Ā°.

Think, Discuss and Write (Page 118)

Question 1.

What can you say about each of the interior opposite angles, when the exterior angle is –

(i) a right angle?

(ii) an obtuse angle?

(iii) an acute angle?

Solution:

(i) Each of the interior angle is an acute angle.

(ii) At least one of the interior opposite angle must be an acute angle.

(iii) Each of the interior angles is an acute angle.

Question 2.

Can the exterior angle of a triangle be a straight angle?

Solution:

No.

Try These (Page 118)

Question 1.

An exterior angle of a triangle is of measure 70Ā° and one of its interior opposite angles is of measure 25Ā°. Find the measure of the other interior opposite angle.

Solution:

Exterior angle = 70Ā°

Interior opposite angles are 25Ā° and x.

ā“ x + 25Ā° = 70Ā°

[Using the exterior angle property of a triangle]

or x = 70Ā° – 25Ā° = 45Ā°

ā“ The required interior opposite angle = 45Ā°.

Question 2.

The two interior opposite angles of an exterior angle of triangle are 60Ā° and 80Ā°. Find the measure of the exterior angle.

Solution:

Interior angles are 60Ā° and 80Ā°.

āµ [Exterior angle] = [Sum of the interior opposite angles]

ā“ [Exterior angle] = 60Ā° + 80Ā° = 140Ā°

Question 3.

Is something wrong in this diagram? Comment.

Solution:

We know that an exterior angle of a triangle is equal to the sum of interior opposite angles.

Here interior angles are 50Ā° angle is 50Ā°.

ā“ This triangle cannot be formed. [āµ 50Ā° ā 50Ā° + 50Ā°]

Try These (Page 122)

Question 1.

Two angles of a triangle are 30Ā° and 80Ā°. Find the third angle.

Solution:

Let the third angle be x.

ā“ Using the angle sum property of a triangle, we have

30Ā° + 80Ā° + x = 180Ā°

or x + 110Ā° = 180Ā°

or x = 180Ā° – 110Ā° = 70Ā°

ā“ The measure of the required third angle is 70Ā°.

Question 2.

One of the angles of a triangle is 80Ā° and the other two angles are equal. Find the measure of each of the equal angles.

Solution:

Let each of the equal angles be x.

Using the angle sum property of a triangle, we have

x + x + 80Ā° = 180Ā°

or 2x + 80Ā° = 180Ā°

or 2x = 180Ā° – 80Ā° = 100Ā°

or \(\frac { 2x }{ 2 }\) = \(\frac { 100Ā° }{ 2 }\) or x = 50Ā°

ā“ The required measure of each of the equal angles is 50Ā°.

Question 3.

The three angles of a triangle are in the ratio 1:2:1. Find all the angles of the triangle. Classify the triangle in two different ways.

Solution:

Let the angles of the triangle be x, 2x, x.

ā“ Using the angle sum property, we have

x + 2x + x = 180Ā°

or 4x = 180Ā°

or \(\frac { 4x }{ 4 }\) = \(\frac { 180Ā° }{ 4 }\)

or x = 45Ā°

ā“ 2x = 2 x 45Ā° = 90Ā°

Thus, the three angles of the triangle are 45Ā°, 90Ā°, 45Ā°.

āµ Its two angles are equal.

āµ It is an isosceles triangle.

āµ Its one angle is 90Ā°.

ā“ It is a right-angled triangle.

Think, Discuss and Write (Page 122)

Question 1.

Can you have a triangle with two right angles?

Solution:

No, because the sum of two right angles is 180Ā°. On adding the measure of the third angle, the total of three angles will be more than 180Ā°, which is not true for a triangle.

Question 2.

Can you have a triangle with two obtuse angles?

Solution:

No, because an obtuse angle has its measure more than 90Ā°. Therefore, the sum of two obtuse angles is greater than 180Ā°, which is not possible in a triangle.

Question 3.

Can you have a triangle with two acute angles?

Solution:

Yes, because the sum of two acute angles can be less than 180Ā°. Therefore, a triangle can have two acute angles.

Question 4.

Can you have a triangle with all the three angles greater than 60Ā°?

Solution:

No, because the sum of three angles (each of them being greater than 60Ā°) is greater than 180Ā°, which is not possible in a triangle.

Question 5.

Can you have a triangle with all the three angles equal to 60Ā°2

Solution:

Yes, because sum of three angles, each being equal to 60Ā°, is 180Ā°, which is true for a triangle.

Question 6.

Can you have a triangle with all the three angles less than 60Ā°?

Solution:

No, because the sum of the three angles, each being less than 60Ā°, is less than 180Ā°, which is not possible in a triangle.

Try These (Page 123)

Question 1.

Find angle x in each figure:

Solution:

(i) In the figure, two sides of the triangle are equal.

ā“ The base angles opposite to equal sides are equal.

or x = 40Ā°

Thus, the required value of x is 40Ā°.

(ii) In the figure, two sides of the triangle are equal.

ā“ The base angles opposite to equal sides are equal.

ā“ The other base angle 45Ā°

Now, the sum of three angles = x + 45Ā° + 45Ā°

But the sum of three angles of a triangle is 180Ā°

ā“ x + 90Ā° = 180Ā°

or x = 180Ā° – 90Ā° = 90Ā°.

(iii) In the figure, two sides of the triangle are equal.

ā“ The base angles opposite to equal sides are equal,

i.e. x = 50Ā°

Thus, the required value of x is 50Ā°.

(iv) In the figure, two sides of the triangle are equal.

ā“ The base angles opposite to the equal sides are equal.

ā“ The other base angle is x.

Now, sum of three angles of a triangle = 180Ā°

or x + x + 100Ā° = 180Ā°

or 2x + 100Ā° = 180Ā°

or 2x = 180Ā° – 100Ā° = 80Ā°

or \(\frac { 2x }{ 2 }\) = \(\frac { 80Ā° }{ 2 }\)

Thus, the required value of x is 40Ā°.

(v) In the figure, two sides of the triangle are equal and it is a right triangle.

ā“ Its base angles opposite to the equal sides are equal.

ā“ The other base angle = x

āµ Sum of the three angles of a triangle is 180Ā°

ā“ x + x + 90Ā° = 180Ā°

[āµ The triangle is a right triangle]

or 2x + 90Ā° = 180Ā° or 2x = 180Ā° – 90Ā° = 90Ā°

or \(\frac { 2x }{ 2 }\) = \(\frac { 90Ā° }{ 2 }\)

or x = 45Ā°

Thus, the required value of x is 45Ā°.

(vi) In the figure, the two sides of the triangle are equal.

ā“ The base angles opposite to the equal sides are equal.

Since one of the base angle is x.

ā“ The other base angle is also x.

āµ Sum of the three angles of a triangle is 180Ā°

ā“ x + x + 40Ā° = 180Ā°

or 2x + 40Ā° = 180Ā°

or 2x = 180Ā° – 40Ā° = 140Ā°

or \(\frac { 2x }{ 2 }\) = \(\frac { 140Ā° }{ 2 }\)

or x = 70Ā°

Thus, the required value of x = 70Ā°.

(vii) In the figure, two sides of the triangle are equal.

ā“ The base angles opposite to equal sides are equal.

One of the base angle = x

ā“ Other base angle = x

Now, x and 120Ā° form a linear pair = 180Ā°

ā“ x + 120Ā° = 180Ā°

or x = 180Ā° – 120Ā° = 60Ā°

Thus the value of x = 60Ā°.

(viii) In the figure, two sides of the triangle are equal.

ā“ The base angles opposite to the equal sides are equal.

Since, one of the base angles = x

ā“ The other base angle = x

Since, exterior angle is equal to sum of the interior opposite angles.

ā“ x + x – 110Ā°

or 2x = 110Ā°

or x = \(\frac { 110Ā° }{ 2 }\)

(ix) Two sides of the triangle are equal.

ā“ The base angles opposite to the equal sides are equal.

Since, one of the base angle = x.

ā“ The other base angle = x

Also, the vertically opposite angles 30Ā° and x are equal.

ā“ x = 30Ā°

Question 2.

Solution:

(i) In the figure, two sides of the triangle are equal.

ā“ The base angles opposite to the equal sides are equal.

Since, one of the base angles is y,

ā“ Other base angle = y

Now, y and 120Ā° form a linear pair,

ā“ y + 120Ā° = 180Ā°

or y = 180Ā° – 120Ā° = 60Ā°

Now, sum of the three angles = 180Ā°

ā“ x + y + y =180Ā°

or x + 60Ā° + 60Ā° = 180Ā°

or x + 120Ā° = 180Ā°

or x = 180Ā° – 120Ā° = 60Ā°

Thus, x = 60Ā° and y = 60Ā°.

(ii) In the given figure, two sides of the triangle are equal.

ā“ The base angles opposite to equal sides are equal.

Since, one of the base angles is x,

ā“ The other base angle = x.

Also, the triangle is a right-angled triangle.

ā“ Third angle of the triangle = 90Ā°

Now, sum of the three angles of the triangle = 180Ā°

ā“ x + x + 90Ā° = 180Ā°

or 2x + 90Ā° = 180Ā°

or 2x = 180Ā° – 90Ā° = 90Ā°

or \(\frac { 2x }{ 2 }\) = \(\frac { 90Ā° }{ 2 }\) or x = 45Ā°

Now, x and y form a linear pair,

ā“ x + y =180Ā°

or 45Ā° + y = 180Ā°

or y =180Ā° – 45Ā° = 135Ā°

Thus, x = 45Ā° and y = 135Ā°.

(iii) In the figure, two sides of the triangle are equal.

ā“ Base angles are x and x.

The third angle of the triangle = The vertically opposite angle of 92Ā° = 92Ā°

Now, sum of the three angles of the triangle = 180Ā°

ā“ x + x + 92Ā° = 180Ā°

or 2x + 92Ā° = 180Ā°

or 2x =180Ā° – 92Ā° = 88Ā°

or \(\frac { 2x }{ 2 }\) = \(\frac { 88Ā° }{ 2 }\) or x = 44Ā°

Now, x and y form a linear pair,

ā“ x + y = 180Ā°

or 44Ā° + y = 180Ā°

or y = 180Ā° – 44Ā° = 136Ā°

Thus, x = 44Ā° and y = 136Ā°.

Think, Discuss and Write (Page 127)

Question 1.

Is the sum of any two angles of a triangle always greater than the third angle?

Solution:

No, the sum of any two angles of a triangle is not always greater than the third angle.

Try These (Page 129)

Question 1.

Find the unknown length x in the following figures.

Solution:

(i) In the given right triangle, the longest side (hypotenuse) = x.

ā“ 3Ā² + 4Ā² = xĀ².

[Using Pythagoras property]

or 9 + 16 = xĀ²

or 25 = xĀ²

xĀ² = 5Ā² ā x = 5

(ii) The given figure is a right-angled triangle.

ā“ Using the Pythagoras property, we have xĀ² = 6Ā² + 8Ā²

or xĀ² = 36 + 64 = 100

xĀ² = 10Ā² ā x = 10

(iii) The given figure is a right-angled triangle.

ā“ Using the Pythagoras property, we have

xĀ² = 8Ā² + 15Ā²

or xĀ² = 64 + 225 = 289

or xĀ² = 17Ā² ā x = 17 cm

(iv) Using the Pythagoras property in the given right triangle, we have

xĀ² = 7Ā² + 24Ā²

or xĀ² =49 + 576 = 625

xĀ² = 25Ā² ā x = 25

(v) The given figure can be labelled as:

In right ā-I, using the Pythagoras property, we have

yĀ² + 12Ā² = 37Ā²

or yĀ² + 144 = 1369

or yĀ² = 1369 – 144 = 1225

yĀ² = 35Ā² ā y = 35

In right ā-II, using the Pythagoras property, we have

(x – y)Ā² + 12Ā² = 37Ā²

or (x – 35)Ā² + 144 = 1369

or (x – 35)Ā² = 1369 – 144 = 1225

or (x – 35)Ā² = (35)Ā²

or x – 35 = 35

x = 35 + 35 = 70

(vi) Using the Pythagoras property in rt āABC, we have

122Ā² + 5Ā² = xĀ²

or 144 + 25 = xĀ²

or 169 = xĀ²

(13)Ā² = xĀ²

or 13 = x

ā x = 13

Think, Discuss and Write (Page 131)

Question 1.

Which is the longest side in the triangle PQR, right-angled at P?

Solution:

The vertex containing 90Ā° R is P.

ā“ Arms of the right angle are PQ and PR

ā“ Hypotenuse is QR.

Thus, the longest side is QR.

Question 2.

Which is the longest side in the triangle ABC, right-angled at B?

Solution:

āµ The right angle is at B.

ā“ The legs of the right triangle are AB and BC.

ā“ Hypotenuse is AC.

So, the longest side is AC.

Question 3.

Which is the longest side of a right triangle?

Solution:

In a right triangle, the longest side is its hypotenuse.

Question 4.

āThe diagonal of a rectangle produce by itself the same area as produced by its length and breadth ā This is Baudhayan Theorem. Compare it with the Pythagoras property.

Solution:

In the figure, ABCD is a rectangle and BD is its one of the diagonals.

According to the Baudhayan Theorem, we have

(Diagonal)Ā² = (Length)Ā² + (Breadth)Ā²

i.e. (DB)Ā² = (CD)Ā² + (BC)Ā² … (1)

Now, BCD is a right triangle, and DB is its hypotenuse,

ā“ Using the Pythagoras property, we have

(DB)Ā² = Sum of the squares of the legs

or (DB)Ā² = (DC)Ā² + (CB)Ā²

or (DB)Ā² = (CD)Ā² + (BC)Ā² … (2)

From (1) and (2), we can say that the Baudhayan Theorem and Pythagoras property are same.