NCERT Solutions for Class 7 Maths Chapter 6: The Triangle and its Properties - Exercise 6.3

Exercise 6.3

1. Find the value of \[x\] in the following diagrams.

i. Find the value of \[x\] in the following diagram.

Ans: The sum of the internal angles of a triangle is $180^\circ $. In $\vartriangle ABC$, $\angle BAC + \angle ACB + \angle ABC = 180^\circ $.

$ \Rightarrow x + 60^\circ  + 50^\circ  = 180^\circ  $

$ \Rightarrow x + 110^\circ  = 180^\circ  \\  $

Subtract $110^\circ $ from both sides and simplify.

$\Rightarrow x + 110^\circ  - 110^\circ  = 180^\circ  - 110^\circ  $ 

$\Rightarrow x = 70^\circ $

The value of \[x\] is $70^\circ $.

ii. Find the value of \[x\] in the following diagram.

Ans: The sum of the internal angles of a triangle is $180^\circ $. In $\vartriangle PQR$, $\angle RPQ + \angle PQR + \angle QRP = 180^\circ $.

$ \Rightarrow 90^\circ  + 30^\circ  + x = 180^\circ  $

$ \Rightarrow x + 120^\circ  = 180^\circ $

Subtract $120^\circ $ from both sides and simplify.

\[ \Rightarrow x + 120^\circ  - 120^\circ  = 180^\circ  - 120^\circ \]

\[ \Rightarrow x = 60^\circ  \]

The value of \[x\] is \[60^\circ \].

iii. Find the value of \[x\] in the following diagram.

Ans: The sum of the internal angles of a triangle is $180^\circ $. In $\vartriangle XYZ$, $\angle ZXY + \angle XYZ + \angle YZX = 180^\circ $.

\[ \Rightarrow 30^\circ  + 110^\circ  + x = 180^\circ \]

  \[ \Rightarrow x + 140^\circ  = 180^\circ  \]

Subtract \[140^\circ \] from both sides and simplify.

\[\Rightarrow x + 140^\circ  - 140^\circ  = 180^\circ  - 140^\circ \] 

\[   \Rightarrow x = 40^\circ \]

The value of \[x\] is \[40^\circ \].

iv. Find the value of \[x\] in the following diagram.

Ans: The sum of the internal angles of a triangle is $180^\circ $. In the given triangle, \[50^\circ  + x + x = 180^\circ \].

Hence,

\[50^\circ  + 2x = 180^\circ \]

Subtract \[50^\circ \] from both sides and simplify.

\[\Rightarrow 50^\circ  - 50^\circ  + 2x = 180^\circ  - 50^\circ \] 

\[\Rightarrow 2x = 130^\circ \]

Divide both sides by 2 and simplify.

\[ \Rightarrow \dfrac{{2x}}{2} = \dfrac{{130^\circ }}{2} \]

 \[\Rightarrow x = 65^\circ \]

The value of \[x\] is \[65^\circ \].

v. Find the value of \[x\] in the following diagram.

Ans: The sum of the internal angles of a triangle is $180^\circ $. In the given triangle, \[x + x + x = 180^\circ \].

Hence,

\[3x = 180^\circ \]

Divide both sides by 3 and simplify.

\[ \Rightarrow \dfrac{{3x}}{3} = \dfrac{{180^\circ }}{3} \]

\[ \Rightarrow x = 60^\circ   \]

The value of \[x\] is \[60^\circ \].

vi. Find the value of \[x\] in the following diagram.

Ans: The sum of the internal angles of a triangle is $180^\circ $. In the given triangle, \[90^\circ  + x + 2x = 180^\circ \].

\[ \Rightarrow 90^\circ  + 3x = 180^\circ \]

Subtract \[90^\circ \] from both sides and simplify.

\[\Rightarrow 90^\circ  - 90^\circ  + 3x = 180^\circ  - 90^\circ  \]

\[\Rightarrow 3x = 90^\circ \]

Divide both sides by 3 and simplify.

\[\Rightarrow \dfrac{{3x}}{3} = \dfrac{{90^\circ }}{3} \]

 \[\Rightarrow x = 30^\circ  \]

The value of \[x\] is \[30^\circ \].

2. Find the value of \[x\] and $y$ in the following diagrams.

i. Find the value of \[x\] and $y$ in the following diagram.

Ans: The exterior angle property of a triangle states that the exterior angle is equal to the sum of the opposite non-adjacent interior angles.

$x + 50^\circ  = 120^\circ $

Subtract $50^\circ $ from both sides of the equation.

$ \Rightarrow x = 120^\circ  - 50^\circ  $ 

$ \Rightarrow x = 70^\circ  $

The value of \[x\] is $70^\circ $.

The sum of the internal angles of a triangle is $180^\circ $. In the given triangle, \[50^\circ  + 70^\circ  + y = 180^\circ \].

\[ \Rightarrow 120^\circ  + y = 180^\circ \]

Subtract $120^\circ$ from both sides and simplify.

\[ \Rightarrow 120^\circ  + y - 120^\circ  = 180^\circ  - 120^\circ  \]

\[\Rightarrow y = 60^\circ   \]

The value of \[y\] is \[60^\circ \].

ii. Find the value of \[x\] and $y$ in the following diagram.

Ans: Since the vertical opposite angles are equal, $y = 80^\circ $.

The value of \[y\] is $80^\circ $.

The sum of the internal angles of a triangle is $180^\circ $. In the given triangle, \[50^\circ  + 80^\circ  + x = 180^\circ \].

\[ \Rightarrow 130^\circ  + x = 180^\circ \]

Subtract \[130^\circ \] from both sides and simplify.

\[ \Rightarrow 130^\circ  - 130^\circ  + x = 180^\circ  - 130^\circ \]

 \[  \Rightarrow x = 50^\circ   \]

The value of \[y\] is \[50^\circ \].

iii. Find the value of \[x\] and $y$ in the following diagram.

Ans: The exterior angle property of a triangle states that the exterior angle is equal to the sum of the opposite non-adjacent interior angles.

$50^\circ  + 60^\circ  = x$.

Hence,

$x = 110^\circ $

The value of \[x\] is $110^\circ $.

The sum of the internal angles of a triangle is $180^\circ $. In the given triangle, \[50^\circ  + 60^\circ  + y = 180^\circ \].

Hence,

\[110^\circ  + y = 180^\circ \]

Subtract $110^\circ $ from both sides and simplify.

\[\Rightarrow 110^\circ  + y - 110^\circ  = 180^\circ  - 110^\circ  \]

\[\Rightarrow y = 70^\circ  \]

The value of \[y\] is \[70^\circ \].

iv. Find the value of \[x\] and $y$ in the following diagram.

Ans: Since the vertical opposite angles are equal, $x = 60^\circ $.

The value of \[x\] is $60^\circ $.

The sum of the internal angles of a triangle is $180^\circ $. In the given triangle,

 \[\Rightarrow 60^\circ  + 30^\circ  + y = 180^\circ  \]

 \[ \Rightarrow 90^\circ  + y = 180^\circ  \].

Subtract \[{90^\circ }\] from both sides and simplify.

\[\Rightarrow 90^\circ  - 90^\circ  + y = 180^\circ  - 90^\circ \]

\[\Rightarrow y = 90^\circ   \]

The value of \[y\] is \[90^\circ \].

v. Find the value of \[x\] and $y$ in the following diagram.

Ans: Since the vertical opposite angles are equal, $y = 90^\circ $.

The value of \[y\] is \[90^\circ \].

The sum of the internal angles of a triangle is $180^\circ $. In the given triangle, \[x + x + 90^\circ  = 180^\circ \].

Hence,

\[90^\circ  + 2x = 180^\circ \]

Subtract \[90^\circ \] from both sides and simplify.

\[ \Rightarrow 90^\circ  + 2x - 90^\circ  = 180^\circ  - 90^\circ  \]

\[ \Rightarrow 2x = 90^\circ   \]

Divide both sides by 2 and simplify.

\[ \Rightarrow \dfrac{{2x}}{2} = \dfrac{{90^\circ }}{2} \]

\[   \Rightarrow x = 45^\circ   \]

The value of \[x\] is \[45^\circ \].

vi. Find the value of \[x\] and $y$ in the following diagram.

Ans: Since the vertical opposite angles are equal, $y = x$.

The sum of the internal angles of a triangle is $180^\circ $.

In the given triangle, \[x + x + x = 180^\circ \].

Hence,

\[3x = 180^\circ \]

Divide both sides by 3 and simplify.

\[ \Rightarrow \dfrac{{3x}}{3} = \dfrac{{180^\circ }}{3} \]

 \[ \Rightarrow x = 60^\circ   \]

The value of \[x\] is \[60^\circ \] and the value of \[y\] is \[60^\circ \].

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