Question

In given figure find the value of x and y​

Answer 1

Given :-

• ∠SPX = 148°
• ∠PSQ = 103°
• SQ = RQ

To Find :-

• value of x and y ?

Solution :-

∠SPX is an exterior angle of ∆SPQ

and we know that,

• exterior angle of a triangle is equal to the sum of opposite angles of a triangle

$\bold{ : \implies∠SPX = ∠PSQ + ∠PQS} \\ \\ \bold{: \implies 148 \degree = 103 \degree + x \degree} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{: \implies x \degree = 148 \degree - 103 \degree } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{: \implies x \degree = 45 \degree} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

so, the value of x is 45

___________________________

now, we know that

• sum of angles of a straight line or linear pair is equal to 180°

$\bold{: \implies∠PSQ + ∠RSQ = 180° }$

$\bold{: \implies103 \degree + ∠RSQ = 180 \degree } \\ \\ \bold{: \implies ∠RSQ = 180 \degree - 103 \degree } \\ \\ \bold{: \implies ∠RSQ = 77 \degree} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

SQ = RQ (given)

and we know that,

• angles opposite to the equal sides are equal to each other.

$\bold{: \implies ∠RSQ =∠SRQ} \\ \\ \bold{: \implies ∠SRQ = 77 \degree} \: \: \: \: \: \:$

now, in ∆QRS we know that,

• sum of all interior angles of the triangle is equal to 180°

$\bold{ : \implies∠RSQ + ∠SRQ + ∠RQS = 180 \degree } \\ \\ \bold{: \implies 77 \degree + 77 \degree + y \degree = 180 \degree} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{: \implies 154 \degree + y \degree = 180 \degree} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{: \implies y \degree = 180 \degree - 154 \degree } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bold{: \implies y \degree = 26 \degree }\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

so, the value of y is 26

hence,

• value of x is 45 and value of y is 26