Excercise 6.2 The-Triangle-and-its-Properties- NCERT Solutions Class 7

Go back to  'The Triangle and its Properties'

Chapter 6 Ex.6.2 Question 1

Find the value of the unknown exterior angle \( x\) in the following diagrams:

Solution

Video Solution

What is known:

Measurement of interior opposite angles.

What is unknown:

Value of the unknown exterior angle \(x\).

Reasoning:

We know that an exterior angle of a triangle is the sum of interior opposite angles. By using this fact, we can find out the unknown exterior angle \(x.\)

Steps:

(i) Interior angles are \(50^\circ\) and \(70^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\  \text{opposite angles} \\ \end{array} \right] \\  x=&50^\circ + 70^\circ \\ x=&120^\circ  \end{align}\]

(ii) Interior angles are \(65^\circ\) and \(45^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\  \text{opposite angles} \\ \end{array} \right] \\  x=&65^\circ + 45^\circ \\ x=&110^\circ  \end{align}\]

(iii) Interior angles are \(30^\circ\) and \( 70^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\  x=&30^\circ + 70^\circ \\ x=&100^\circ  \end{align}\] 

(iv) Interior angles are \(60^\circ\) and \(60^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\  \text{opposite angles} \\ \end{array} \right] \\  x=&60^\circ + 60^\circ \\ x=&120^\circ  \end{align}\]

(v) Interior angles are \(50^\circ\) and \(50^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\  x=&50^\circ + 50^\circ \\ x=&100^\circ  \end{align}\]

(vi) Interior angles are \(30^\circ\) and \(60^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\  x=&30^\circ + 60^\circ \\ x=&90^\circ  \end{align}\]

Chapter 6 Ex.6.2 Question 2

Find the value of the unknown interior angle \(x\) in the following figures:

Solution

Video Solution

What is known?

One of the interior opposite angle and exterior angle.

What is unknown?

interior opposite angle \(x\).

Reasoning:

We know that the sum of interior opposite angle is equal to the exterior angle. By using this fact, we can find the unknown interior angle.

Steps: 

(i) Exterior angle is \(115^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\ 115^\circ =&x+ 50^\circ \\x=&115^\circ- 50^\circ \\ x=&65^\circ \end{align}\]

(ii) Exterior angle is \(100^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\ 100^\circ =&x+ 70^\circ \\x=&100^\circ- 70^\circ \\ x=&30^\circ \end{align}\]

(iii) Exterior angle is \(125^\circ\) 

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\ 125^\circ =&x+ 90^\circ (90^\circ \rm{given})\\x=&125^\circ- 90^\circ \\ x=&35^\circ \end{align}\]

(iv) Exterior angle is \(120^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\ 120^\circ =&x+ 60^\circ \\x=&120^\circ- 60^\circ\\ x=&60^\circ \end{align}\]

(v) Exterior angle is \(80^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\ 80^\circ =&x+ 30^\circ \\x=&80^\circ- 30^\circ \\ x=&50^\circ \end{align}\]

(vi) Exterior angle is \(75^\circ\)

\[\begin{align} \text{Exterior angle}=&\left[ \begin{array} \, \text{Sum of interior} \\ \text{opposite angles} \\ \end{array} \right] \\ 75^\circ =&x+ 35^\circ \\x=&75^\circ- 35^\circ \\ x=&40^\circ \end{align}\]

  
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