Chapter 5, Problem 1RVS

To determine

To calculate: The measures of the angles if the degree measures of the angles of a triangle are three consecutive integers.

Answer to Problem 1RVS

Solution:

The measures of the angles of a triangle are 59, 60 and 61.

Explanation of Solution

Given information:

The degree measures of the angles of a triangle are three consecutive integers.

Formula used:

The sum of all the angles of a triangle is equal to ${180}^{\circ }$.

Calculation:

From the statement, the degree measures of the angles of a triangle are three consecutive integers.

So, let the consecutive angles be $x,\left(x+1\right)\text{ and }\left(x+2\right)$.

Since, the sum of all the angles of a triangle is equal to ${180}^{\circ }$.

So,

$x+\left(x+1\right)+\left(x+2\right)=180$

Now, simplify the above equation for $x$ as,

$\begin{array}{c}x+\left(x+1\right)+\left(x+2\right)=180\\ 3x+3=180\\ 3x=180-3\\ 3x=177\end{array}$

Further simplified,

$\begin{array}{c}3x=177\\ x=\frac{177}{3}\\ x=59\end{array}$

Since, three consecutive angles are $x,\left(x+1\right)\text{ and }\left(x+2\right)$

So, substitute the value of $x$ in the $\left(x+1\right)\text{ and }\left(x+2\right)$

$\begin{array}{c}\left(x+1\right)=\left(59+1\right)\\ =60\end{array}$

And

$\begin{array}{c}\left(x+2\right)=\left(59+2\right)\\ =61\end{array}$

Now, to check the answer put the values in equation $x+\left(x+1\right)+\left(x+2\right)=180$ as,

$\begin{array}{r}x+\left(x+1\right)+\left(x+2\right)=180\\ \left(59\right)+\left(\left(59\right)+1\right)+\left(\left(59\right)+2\right)\stackrel{?}{=}180\\ 59+60+61\stackrel{?}{=}180\\ 180=180\end{array}$

Thus, the solutions are correct.

Hence, the measures of the angles of a triangle are 59, 60 and 61.