RBSE Solutions for Class 7 Maths Chapter 6 The Triangles and Its Properties Intext Questions

Rajasthan Board RBSE Solutions for Class 7 Maths Chapter 6 The Triangles and Its Properties Intext Questions Textbook Exercise Questions and Answers.

RBSE Class 7 Maths Solutions Chapter 6 The Triangles and Its Properties Intext Questions

(Try These - Page 113-114)

Class 7 Maths Chapter 6 Try These Solutions Question 1.
Write the six elements (i.e., the 3 sides and the 3 angles) of ΔABC.
Answer:
∠A, ∠B, ∠C, \(\overline{A B}, \overline{B C}\) and \(\overline{C A}\)

Class 7 Maths Chapter 6 Think Discuss And Write Question 2.
Write the :
(i) Side opposite to the vertex Q of ΔPQR.
Answer:
PR

(ii) Angle opposite to the side LM of ΔLMN.
Answer:
∠MNL

(iii) Vertex opposite to the side RT of ΔRST.
Answer:
S

Try These Solutions Class 7 Chapter 6 Question 3.
Look at figure and elassify each of the triangle according to its :
(a) Sides
(b) Angles

Answer:
(a) Sides: Isosceles
(b) Angles: Acute-angled

Answer:
(a) Sides: Scalene
(b) Angles: Right-angled

Answer:
(a) Sides: Isosceles
(b) Angles: Obtuse-angled

Answer:
(a) Sides: Equilateral
(b) Angles: Acute-angled

Answer:
(a) Sides: Isosceles
(b) Angles: Obtuse-angled

Answer:
(a) Sides: Isosceles
(b) Angles: Right-angled

(Think, Discuss and Write - Page 144)

Try These Solutions Class 7 Triangle And Its Properties Question 1.
How many medians can a triangle have?
Answer:
Three medians.

Think Discuss And Write Class 7 Chapter 6 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 cas&.)
Answer:
Yes, a median lies wholly in the interior of the triangle.

(Think, Discuss and Write - Page 115)

Class 7 Maths Triangle And Its Properties Try These Solutions Question 1.
How many altitudes can a triangle have?
Answer:
A triangle can have three altitudes.

Try These Solutions Class 7 Maths Chapter 6 Question 2.
Draw rough sketches of altitudes from A to \(\overline{B C}\) for the following triangles:

Answer:

Answer:

Answer:

Class 7 Maths Chapter 6 Try These 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.
Answer:
No, the altitude does not always lie in the interior of the triangle.

In the above triangle, the altitude AL is not in the interior of ΔABC.

Chapter 6 Maths Class 7 Try These Solutions Question 4.
Can you think of a triangle in which two altitudes of the triangle are two of its sides?
Answer:
Right-angled triangle has two of its sides as its altitude. In MBC, \(\overline{A B}\) and \(\overline{B C}\) are its altitudes.

Triangle And Its Properties Class 7 Try These Question 5.
Can the altitude and median be same of a triangle?
Answer:
Yes, in i4BC, \(\overline{A L}\) is median and altitude both. In equilateral triangles median and altitude are same.

(Think, Discuss and Write - Page 117)

The Triangle And Its Properties Try These 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.
Answer:

Class 7 Chapter 6 Try These Question 2.
Are the exterior angles formed at each vertex of a triangle equal?
Answer:
Yes, the exterior angles at a vertex of a triangle are equal.

Think Discuss And Write Class 7 Question 3.
What can you say about the sum of an exterior angle of a triangle and its adjacent interior angle?
Answer:
Sum of exterior angle and its adjacent interior angle of a triangle is 180°.

(Think, Discuss and Write - Page 118)

Class 7 Triangle And Its Properties Try These Question 1.
What can you say about each of the interior opposite angles, when the exterior angle is:
(i) a right angle
Answer:
Each of the interior opposite angles is acute.

(ii) an obtuse angle
Answer:
At least one of the interior opposite angles is acute.

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

(Try These - Page 118)

Question 1.
An exterior angle of a triangle is of measure 70° and one of its interior opposite angles is of measure 25s. Find the measure of the other interior opposite angle.
Answer:
The exterior angle of a triangle is equal to sum of its interior opposite angles.
Let the other interior opposite angles = x
∴ 70° = 25°+x
or x = 70°- 25°
⇒ x = 45°
∴ Required opposite angle = 45°

Question 2.
The two interior opposite angles of an exterior angle of a triangle are 60° and 80°. Find the measure of the exterior angle.
Answer:
Exterior angle = Sum of the its interior opposite angles
= 60° + 80°= 140°

Question 3.
Is something wrong in this diagram? Comment.
Answer:

Such a triangle cannot be formed as exterior angle of a triangle is equal to sum of its opposite angles. [∵ 50° ≠ 50°+ 50°]

(Try These - Page 122)

Question 1.
Two angles of a triangle are 30° and 80°. Find the third angle.
Answer:
Let the third angle be x.
∴ x + 30° + 80° = 180°
[By angles sum property of a triangle]
or x = 180° - 110° = 70°
∴ The third angle is 70°.

Question 2.
One of the angles of a triangle is 80° and the other two angles are equal. Find the measure of each of the equal angles.
Answer:
Let the other two angles be of measure x
∴ x + x + 80° = 180°
⇒ 2x = 180° - 80° (Angles sum property)
⇒ 2x = 100° ⇒ x = \(\frac{100}{2}\)
⇒ x = 50°
∴ The other two angles are of measure 50° each.

Question 3.
The three angles of a triangle are in the ratio 1 : 2: 1 Find aU the angles of the triangle. Classify the triangle in two different ways.
Answer:
Let the three angles of the triangle be x, 2x and x.
∴ x + 2x + x = 180° (Angles sum property of a triangle)
4x = 180° ⇒ x = \(\frac{180^{\circ}}{4}\)
⇒ x = 45°
The angles of the triangle are 45°, 90° and 45°.

(Think, Discuss and Write - Page 122)

Question 1.
Can you have a triangle with two right angles?
Answer:
No

Question 2.
Can you have a triangle with two obtuse angles?
Answer:
No

Question 3.
Can you have a triangle with two acute angles?
Answer:
Yes

Question 4.
Can you have a triangle with all the three angles greater than 60°?
Answer:
No

Question 5.
Can you have a triangle with all the three angles equal to 60°?
Answer:
Yes

Question 6.
Can you have a triangle with all the three angles less than 60°?
Answer:
No

(Try These - Page 123)

Try These Page 123 Class 7 Question 1.
Find angle x in each figure :


Answer:
(i) The base angles opposite to equal sides are also equal.
∴ x = 40°

(ii) The base angles opposite to equal sides are equal.
Other base angle = 45°
Using angles sum property of a triangle, 45° + 45° + x = 180°
⇒ x = 180° - 90° ⇒ x = 90°

(iii) The base angle opposite to equal sides are equal.
∴ x = 50°

(iv) The base angle opposite to equal sides are equal.
Other base angle = x
Using angles sum property of a triangle,
x + x + 100° = 180° ⇒ 2x = 180° - 100°
⇒ 2x = 80° ⇒ x = 40°

(v) The base angle opposite to equal sides are equal.
∴ Other base angle = x
Also, the given triangle is a right-angled triangle.
Now, using angles sum property of a triangle,
x + x + 90° = 180p
⇒ 2x = 180° - 90° = 90°
⇒ x = \(\frac{90^{\circ}}{2}\)
⇒ x = 45°

(vi) The base angles opposite to equal sides are equal.
∴ Other base angle = x
Using angles sum property of a triangle,
x + x + 40° = 180°
⇒ 2x = 180° - 40° = 140°
⇒ x = \(\frac{140^{\circ}}{2}\) or x = 70

(vii) The base angle opposite to equal sides are equal.
∴ Other base angle = x
Also, 120° and x form a linear pair, x + 120° = 180°
⇒ x = 180° - 120° ⇒ x = 60°

(viii) The base angle opposite to equal sides are equal.
∴ Other base angle = x
As exterior angle is equal to sum of the
interior opposite angles.
∴ x + x = 110° ⇒ 2x = 110° ⇒ x = 55°

(ix) The base angle opposite to equal sides are equal.
∴ Other base angle = x
Also, the vertically opposite angles are equal.
∴ x = 30°

Class 7 Maths Ch 6 Try These Solutions Question 2.
Find angles x and y ¡n each figure:

Answer:
The base angle opposite to equal sides are equal.
∴ Other base angle = y
Now, y and 120° form a linear pair.
∴ y + 120° = 180°
⇒ y = 180°-120°
⇒ y = 60°

Using angles sum property of a triangle,
x + y + y = 180°
⇒ x + 2y = 180°
⇒ x + 2 × 60° = 180°
⇒ x = 180° - 120°
⇒ x = 60°

Answer:
The base angle opposite to equal sides are equal.
∴ Other base angle = x
Also, the given triangle is a right-angled triangle.
∴ x + x + 90° = 180°
(By angles sum property of a triangle)
⇒ 2x = 180° - 90°
⇒ 2x = 90°
⇒ x = \(\frac{90^{\circ}}{2}\)
∴ x = 45°
Also, the angles x andy form a linear pair,
∴ x +y = 180°
⇒ y = 180°-x
⇒ y = 180° - 45°
∴ y = 135°

Answer:
The base angle opposite to equal sides are equal.
∴ Other base angle = x
Also, the vertically opposite angles are equal.
∴ Third angle of triangle = 92°
Using angles sum property of a triangle,
x + x + 92° = 180°
⇒ 2x = 180° - 92°
⇒ 2x = 88°
⇒ x = \(\frac{88^{\circ}}{2}\) ⇒ x = 44°.
Now, x and y form a linear pair.
∴ x + y = 180°
⇒ 44° + y = 180°
⇒ y = 180° - 44°
⇒ y = 136°

(Think, Discuss and Write - Page 127)

Try These Class 7 Chapter 6 Question 1.
Is the sum of any two angles of a triangle always greater than the third angle?
Answer:
No, the sum of any two angles of a triangle is not always greater than the third angle.

(Try These - Page 129-130)

Question 1.
Find the unknown length x in the following figures:

Answer:
x² = 3² + 4² (By Pythagoras property)
⇒ x² = 9 + 16
⇒ x² = 25
⇒ x = 5
∴ x = 5

Answer:
x² = 6² + 8² (By Pythagoras property)
⇒ x² = 36 + 64
⇒ x² = 100
∴ x = 10

Answer:
x² = 15² + 8² (By Pythagoras property)
⇒ x² = 225 + 64
⇒ x² = 289
∴ x = 17

Answer:
x² = 7² + 24² (By Pythagoras property)
⇒ x² = 49 + 576
⇒ x² = 625
∴ x = 25

Answer:

a² + 12² = 37² (By Pythagoras property)
⇒ a² + 144 =136
⇒ a² = 1369 - 144
⇒ a² = 1225
∴ a = 35
⇒ x - a = x - 35
or (x - 35)² + 122 = 37² (By Pythagoras property)
⇒ (x - 35)² + 144 = 1369
⇒ (x-35)² = 1369-144
⇒ (x- 35)² = 1225
⇒ (x - 35)² = 352
⇒ x - 35 = 35
∴ x = 35 + 35 = 70

Answer:

In ΔABC,
3² + BC² = 5² (By Pythagoras property)
⇒ 9 + BC² = 25
BC² = 25 - 9
⇒ BC² = 16
∴ BC = 4
Again, in ΔABD,
5² + 12² = BD² (By Pythagoras property)
⇒ BD² = 144 + 25
⇒ BD² = 169
∴ BD = 13
x = BD - BC = 13 - 4 = 9

(Think, Discuss and Write - Page 131)

Class 7th Maths Chapter 6 Try These Solutions Question 1.
Which is the longest side in the triangle PQR right-angled at P?
Answer:
QR

Question 2.
Which is the longest side in the triangle ABC right-angled at B?
Answer:
AC

Question 3.
Which is the longest side of a right triangle?
Answer:
Hypotenuse

Question 4.
“The diagonal of a rectangle produce by itself the same area as produced by its length and breadth.” This is Baudhayana Theorem. Compare it with the Pythagoras property.
Answer:
Given that
Area produced by diagonal = Area produced by length + Area produced by breadth
⇒ BD² = AB² + AD²
which is Pythagoras property.