Question

In a ∆PQR LP + LQ =115°and LP-LQ=25° find the measure of each angle of the traingle?​

Answer 1

$\begin{gathered}\begin{gathered}\bf \:Given - \begin{cases} &\sf{ \triangle \: PQR \: such \: that} \\ &\sf{ \angle \: P + \angle \: Q = 115 \degree} \\ &\sf{\angle \: P - \angle \: Q = 25\degree} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To\:find - \begin{cases} &\sf{\angle \: P}  \\ &\sf{\angle \:Q} \\ &\sf{\angle \:R} \end{cases}\end{gathered}\end{gathered}$

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

• In triangle PQR,

Given that

$\rm :\longmapsto\:\angle \:P + \angle \:Q = 115\degree \: - - (1)$

and

$\rm :\longmapsto\:\angle \:P - \angle \:P = 25\degree \: - - (2)$

Now,

• we use elimination method to get the values of ∠P and ∠Q.

So,

• On adding equation (1) and (2), we get

$\rm :\longmapsto\:\angle \:P + \cancel{ \angle \:Q }\: + \angle \:P \: - \cancel{\angle \:Q} = 115\degree + 25\degree$

$\rm :\longmapsto\:2\angle \:P = 140\degree$

$\rm :\implies\: \boxed{ \bf \: \angle \:P = 70\degree} - - (3)$

Now,

• Substituting the value of ∠P in equation (1), we get

$\rm :\longmapsto\:70\degree + \angle \:Q = 115\degree$

$\rm :\longmapsto\:\angle \:P = 115\degree - 70\degree$

$\rm :\implies\: \boxed{ \bf \: \angle \:Q = 45\degree} - - (4)$

Now,

We know that,

• Sum of all the angles of angles of a triangle is 180°.

So,

• In triangle PQR,

$\rm :\longmapsto\:\angle \:P + \angle \:Q + \angle \:R = 180\degree$

$\rm :\longmapsto\:70\degree + 45\degree + \angle \:R = 180\degree$

$\rm :\longmapsto\:115\degree + \angle \:R = 180\degree$

$\rm :\longmapsto\:\angle \:R = 180\degree - 115\degree$

$\rm :\implies\: \boxed{ \bf \: \angle \:R = 65\degree \: }$

$\begin{gathered}\begin{gathered}\bf \:Hence - \begin{cases} &\sf{\angle \: P = 70\degree}  \\ &\sf{\angle \:Q = 45\degree} \\ &\sf{\angle \:R = 65\degree} \end{cases}\end{gathered}\end{gathered}$

Additional Information

Additional InformationProperties 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.