# Ex.6.3 Q1 The-Triangle-and-its-Properties Solution - NCERT Maths Class 7

## Question

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

## Text Solution

**What is known?**

Measure of two interior angles of triangles.

**What is unknown?**

Value of one of the angles of the given triangles.

**Reasoning:**

We make use of the angle sum property of a triangle according to which the sum of the interior angles of a triangle is always equal to \(180^\circ.\) If the given unknown interior angle of a triangle is \(x\), then it can be obtained by subtracting sum of the other two angles from \(180^\circ\)

**Steps:**

(i) Sum of interior angles of a triangle \( = 180^\circ\)

\[\begin{align}∠A + ∠B + ∠C&=180^\circ\\ x+50^\circ+60^\circ&=180^\circ\\x=180^\circ-110^\circ&=70^\circ\\\end{align}\]

(ii) Sum of interior angles of a triangle \( = 180^\circ\)

\(\begin{align} \angle P+\angle Q+\angle R=180^\circ \\ 90^\circ +30^\circ +x=180^\circ \\ (\angle P=90^\circ \text{from figure}) \\ x=180^\circ -120^\circ =60^\circ \\ \end{align}\)

(iii) Sum of interior angles of a triangle \( = 180^\circ\)

\[\begin{align}∠X+ ∠Y+ ∠Z&=180^\circ\\ 30^\circ+110^\circ+x&=180^\circ\\ x=180^\circ-140^\circ&=40^\circ\\\end{align}\]

(iv) Sum of interior angles of a triangle \( = 180^\circ\) (From step\(1\))

\[\begin{align} x+x+50^\circ&=180^\circ\\2 x+50^\circ&=180^\circ\\2x=180^\circ&-50^\circ\\x\frac{130}{2}&=65^\circ\end{align}\]

(v) Sum of interior angles of a triangle \( = 180^\circ\) (From step\(1\))

\[\begin{align}x+x+x&=180^\circ\\ 3x&=180^\circ\\x\frac{180}{3}&=60^\circ\end{align}\]

(vi) Sum of interior angles of a triangle \( = 180^\circ\) (From step \(1\))

\[\begin{align} x+2x+90^\circ&=180^\circ\\3x+90^\circ&=180^\circ\\ 3x&=180^\circ-90^\circ\\x=\frac{90}{3}&=30{}^\circ \end{align}\]