Triangles and its Properties Ex-6.3

NCERT solutions for Maths Triangles and its Properties 

NCERT Solutions for Class 7 Maths Triangles and its Properties

Class –VII Mathematics (Ex. 6.3)

Question 1.Find the value of unknown $x$ in the following diagrams:

Answer:

(i) In $\Delta {\text{ABC}},$

$\angle$BAC + $\angle$ACB + $\angle$ABC = $180^\circ$ [By angle sum property of a triangle]

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

$\Rightarrow$ $x + 110^\circ = 180^\circ$ $\Rightarrow$ $x = 180^\circ – 110^\circ = 70^\circ$

(ii) In $\Delta$PQR,

$\angle$RPQ + $\angle$PQR + $\angle$RPQ = $180^\circ$ [By angle sum property of a triangle]

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

$\Rightarrow$ $x + 120^\circ = 180^\circ$ $\Rightarrow$ $x = 180^\circ – 120^\circ = 60^\circ$

(iii) In $\Delta$XYZ,

$\angle$ZXY + $\angle$XYZ + $\angle$YZX = $180^\circ$ [By angle sum property of a triangle]

$\Rightarrow$ ${30^ \circ } + {110^ \circ } + x = 180^\circ$

$\Rightarrow$ $x + 140^\circ = 180^\circ$ $\Rightarrow$ $x = 180^\circ – 140^\circ = 40^\circ$

(iv) In the given isosceles triangle,

$x + x + 50^\circ = 180^\circ$ [By angle sum property of a triangle]

$\Rightarrow$ $2x + 50^\circ = 180^\circ$

$\Rightarrow$ $2x = 180^\circ – 50^\circ$ $\Rightarrow$ $2x = 130^\circ$

$\Rightarrow$ $x = \frac{{130^\circ }}{2} = 65^\circ$

(v) In the given equilateral triangle,

$x + x + x = 180^\circ$ [By angle sum property of a triangle]

$\Rightarrow$ $3x = 180^\circ$

$\Rightarrow$ $x = \frac{{180^\circ }}{3} = 60^\circ$

(vi) In the given right angled triangle,

$x + 2x + 90^\circ = 180^\circ$ [By angle sum property of a triangle]

$\Rightarrow$ $3x + 90^\circ = 180^\circ$

$\Rightarrow$ $3x = 180^\circ – 90^\circ$ $\Rightarrow$ $3x = 90^\circ$

$\Rightarrow$ $x = \frac{{90^\circ }}{3} = 30^\circ$

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NCERT Solutions for Class 7 Maths Exercise 6.3

Question 2.Find the values of the unknowns $x$ and $y$ in the following diagrams:

Answer:

(i) $50^\circ + x = 120^\circ$ [Exterior angle property of a $\Delta$ ]

$\Rightarrow$ $x = 120^\circ – 50^\circ = 70^\circ$

Now, $50^\circ + x + y = 180^\circ$ [Angle sum property of a $\Delta$ ]

$\Rightarrow$ $50^\circ + 70^\circ + y = 180^\circ$

$\Rightarrow$ $120^\circ + y = 180^\circ$ $\Rightarrow$ $y = 180^\circ – 120^\circ = 60^\circ$

(ii) $y = 80^\circ$ ……….(i) [Vertically opposite angle]

Now, $50^\circ + x + y = 180^\circ$ [Angle sum property of a $\Delta$ ]

$\Rightarrow$ $50^\circ + 80^\circ + y = 180^\circ$

[From eq. (i)]

$\Rightarrow$ $130^\circ + y = 180^\circ$ $\Rightarrow$ $y = 180^\circ – 130^\circ = 50^\circ$

(iii) $50^\circ + 60^\circ = x$ [Exterior angle property of a $\Delta$ ]

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

Now $50^\circ + 60^\circ + y = 180^\circ$ [Angle sum property of a $\Delta$ ]

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

$\Rightarrow$ $y = 180^\circ – 110^\circ$ $\Rightarrow$ $y = 70^\circ$

(iv) $x = {60^ \circ }$ ……….(i) [Vertically opposite angle]

Now, $30^\circ + x + y = 180^\circ$ [Angle sum property of a $\Delta$ ]

$\Rightarrow$ $50^\circ + 60^\circ + y = 180^\circ$ [From eq. (i)]

$\Rightarrow$ $90^\circ + y = 180^\circ$ $\Rightarrow$ $y = 180^\circ – 90^\circ = 90^\circ$

(v) $y = {90^ \circ }$ ……….(i) [Vertically opposite angle]

Now, $y + x + x = 180^\circ$ [Angle sum property of a $\Delta$ ]

$\Rightarrow$ $90^\circ + 2x = 180^\circ$ [From eq. (i)]

$\Rightarrow$ $2x = 180^\circ – 90^\circ$ $\Rightarrow$ $2x = 90^\circ$

$\Rightarrow$ $x = \frac{{{{90}^ \circ }}}{2} = {45^ \circ }$

(vi) $x = y$ ……….(i) [Vertically opposite angle]

Now, $x + x + y = 180^\circ$ [Angle sum property of a $\Delta$ ]

$\Rightarrow$ $2x + x = 180^\circ$ [From eq. (i)]

$\Rightarrow$ $3x = 180^\circ$ $\Rightarrow$ $x = \frac{{180^\circ }}{3} = 60^\circ$

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