Question

The angle A of a triangle ABC is equal to the sum of the other two angles . Also , the ratio of the angle B to the angle C is 4 : 5 . Determine the three angles of the triangle ABC
A.$A = {90^ \circ },B = {40^ \circ },C = {50^ \circ }$
B.$A = {50^ \circ },B = {20^ \circ },C = {60^ \circ }$
C.$A = {40^ \circ },B = {50^ \circ },C = {20^ \circ }$
D.$A = {25^ \circ },B = {45^ \circ },C = {35^ \circ }$

Answer 1

Hint: As we are given that the ratio of angle B to angle C is 4 : 5 then the angles can be written as $\angle B = 4x,\angle C = 5x$.As we are given that angle A is the sum of the other two angles we get $\angle A = 9x$ and now by using the property that sum of all angles in a triangle is ${180^ \circ }$ we can get the value of x and using that we can find the angles.

Complete step-by-step answer:
We are given a triangle ABC

We are given that the angle A is equal to the sum of the angles B and C
$ \Rightarrow \angle A = \angle B + \angle C$ ………(1)
Next , we are given that the ratio of angle B to angle C is 4 : 5
As we are given in ratio we can write the angles as
$ \Rightarrow \angle B = 4x,\angle C = 5x$
Using this in (1) we get
$
   \Rightarrow \angle A = 4x + 5x \\
   \Rightarrow \angle A = 9x \\
$
By using the angle sum property of the triangle we know that the sum of the angles of a triangle is ${180^ \circ }$
$ \Rightarrow \angle A + \angle B + \angle C = {180^ \circ }$
Using the values found above we get
$
   \Rightarrow 9x + 4x + 5x = {180^ \circ } \\
   \Rightarrow 18x = {180^ \circ } \\
   \Rightarrow x = \dfrac{{180}}{{18}} = {10^ \circ } \\
$
Using the value of x we can find the value of the angles
$
   \Rightarrow \angle A = 9x = 9\left( {10} \right) = {90^ \circ } \\
   \Rightarrow \angle B = 4x = 4\left( {10} \right) = {40^ \circ } \\
   \Rightarrow \angle C = 5x = 5\left( {10} \right) = {50^ \circ } \\
$
Therefore the correct option is A
Note: Points to remember while solving this problem:-
The sum of the length of two sides of a triangle is always greater than the length of the third side.
A triangle with vertices P, Q, and R is denoted as $\vartriangle PQR$ .
The area of a triangle is equal to half of the product of its base and height.