Question

Of the three angles of a triangle, the second is one third the first and the third is 26 degrees more than the first finds the measure of all the angles.

Answer 1

Hint: In this question the relation between the three angles of the triangle is given so we will add all these angles and equate it with \[{180^ \circ }\] and then we will find the measure of each angle of the triangle.

Complete step-by-step answer:
Let the three angle of the triangle be \[\angle A\], \[\angle B\] and \[\angle C\]
We know that the sum of the internal angles of a triangle is equal to \[{180^ \circ }\], hence we can write
\[\angle A + \angle B + \angle C = {180^ \circ } - - (i)\]
Now it is said that the second is one third the first, so we get
\[\angle B = \dfrac{1}{3}\angle A - - (ii)\]
And also the third angle is 26 degree more than the first angle and this can be written as
\[\angle C = {26^ \circ } + \angle A - - (iii)\]
Now we substitute the values of equation (ii) and (iii) in the equation (i) to find the measure of first angle, therefore we get
\[
\Rightarrow \angle A + \dfrac{1}{3}\angle A + \left( {{{26}^ \circ } + \angle A} \right) = {180^ \circ } \\
\Rightarrow \angle A + \dfrac{1}{3}\angle A + \angle A = 180 - 26 \\
\Rightarrow 3\angle A + \angle A + 3\angle A = 3 \times 154 \\
\Rightarrow 7\angle A = 462 \\
\Rightarrow \angle A = {66^ \circ } \;
 \]
Hence the measure of first angle \[\angle A = {66^ \circ }\]
Now substitute the value of \[\angle A\] in equation (ii) to find \[\angle B\], so we get
\[
\Rightarrow \angle B = \dfrac{1}{3}\angle A \\
   = \dfrac{1}{3} \times 66 \\
   = {22^ \circ } \;
 \]
Now we substitute the value of \[\angle A\] in equation (iii) to find \[\angle C\],we get
\[
\Rightarrow \angle C = {26^ \circ } + {66^ \circ } \
   = {92^ \circ } \;
 \]
Hence the measure of all the angles is
\[\angle A = {66^ \circ }\]
\[\angle B = {22^ \circ }\]
\[\angle C = {92^ \circ }\]

Note: Sum of the internal angles of a triangle is equal to \[{180^ \circ }\]. Students must note that if it is said that all the angles of the triangle are equal then the triangle is said to be an equilateral triangle with each angle being measured as \[{60^ \circ }\].