Question

What is the ratio of the measure of angle $B$ to the measure of angle $A$?

Answer 1

Hint: We will use the property of triangle that states that the sum of all interior angles of triangle is 180. Since, the given triangle is right angled-triangle so one angle will be 90. We will put the value of each angle and solve the equation to find the angles.

Complete step by step answer:
We have given that in a right-angled triangle ABC angle A is y+10 and angle B is y. We know the angle sum property of a triangle that is the sum of all interior angles of a triangle is 180. So, we can write
$ \Rightarrow \angle A + \angle B + \angle C = {180^ \circ }$
We will put the value of each angle
$ \Rightarrow y + {10^ \circ } + y + {90^ \circ } = {180^ \circ }$

We added y term and constant
$ \Rightarrow 2y + {100^ \circ } = {180^ \circ }$
We have subtracted 100 from both side
$ \Rightarrow 2y = {80^ \circ }$
We have divided whole equation by 2
$ \Rightarrow y = {40^ \circ }$ (1)
So, the measure of angle A
$ \Rightarrow \angle A = y + {10^ \circ }$
We put the value of y , we find in equation 1
$ \Rightarrow \angle A = {40^ \circ } + {10^ \circ }$
$ \Rightarrow \angle A = {50^ \circ }$

And, the measure of angle B is,
$ \Rightarrow \angle B = y$
$ \Rightarrow \angle B = {40^ \circ }$
So, the ratio of measure of angle B to angel A
$ \Rightarrow \dfrac{{\angle B}}{{\angle A}} = \dfrac{{{{40}^ \circ }}}{{{{50}^ \circ }}}$
We have divided numerator and denominator of RHS by 10
$ \Rightarrow \dfrac{{\angle B}}{{\angle A}} = \dfrac{4}{5}$
We can write above equation in ratio form as
$ \therefore \angle B:\angle A = 4:5$

Hence, the ratio of measure of angle B to measure of angle A is $4:5$.

Note: The theorem that the sum of a triangle's three angles is always equal to 180 holds true for all triangles. We should also know that a fraction $\dfrac{a}{b}$ can be written in ratio as \[a:b\] . We will use a general method of solving the equation to get the desired result.