Question

In the given figure, in ∆PQR if PQ = PR and the exterior angle ∠PRS = 135° then find the value of ∠P​

Answer 1

Given Question :-

In the given figure, in ∆PQR if PQ = PR and the exterior angle ∠PRS = 135° then find the value of ∠P

Answer

Given :-

• In ∆ PQR

• PQ = PR

• exterior angle ∠PRS = 135°

To Find :-

• The value of ∠P

Concept Used :-

• Angle opposite to equal sides are always equal.

• Exterior angle of a triangle is always equal to the sum of the interior opposite angles.

$\large\underline{\bold{Solution :- }}$

Given that,

• QRS is a straight line.

So,

• ∠PRS + ∠PRQ = 180°

• 135° + ∠PRQ = 180°

• ∠PRQ = 180° - 135°

• ∠PRQ = 45°

Now,

$\rm :\longmapsto\:In \: \triangle \: PQR$

$\rm :\longmapsto\:PQ = PR$

$\rm :\implies\:\angle \:PRQ = \angle \:PQR$

$\rm :\implies\:\angle \:PQR = 45\degree$

Now,

• Again,

$\rm :\longmapsto\:In \: \triangle \: PQR$

We know that,

• Exterior angle of a triangle is always equal to the sum of the interior opposite angles.

So,

$\rm :\longmapsto\:\angle \:PRS = \angle \:PQR + \angle \:RPQ$

$\rm :\longmapsto\:135\degree \: = 45\degree \: + \angle \:RPQ$

$\rm :\longmapsto\:\angle \:RPQ = 135\degree - 45\degree$

$\rm :\implies\: \boxed{ \bf \: \angle \:RPQ \: = \: 90\degree \: }$

Additional Information :-

Properties of a triangle

• A triangle has three sides, three angles, and three vertices.

• The sum of all internal angles of a triangle is always equal to 180°. This is called the angle sum property of a triangle.

• The sum of the length of any two sides of a triangle is greater than the length of the third side.

• The side opposite to the largest angle of a triangle is the largest side.

• Any exterior angle of the triangle is equal to the sum of its interior opposite angles. This is called the exterior angle property of a triangle.

Based on the angle measurement, there are three types of triangles:

• Acute Angled Triangle : A triangle that has all three angles less than 90° is an acute angle triangle.

• Right-Angled Triangle : A triangle that has one angle that measures exactly 90° is a right-angle triangle.

• Obtuse Angled Triangle : triangle that has one angle that measures more than 90° is an obtuse angle triangle.