Question

problem 16: (first taught in lesson 30) find the measures of the angles of a triangle whose angles have measures of x, $\frac{1}{2}x$, and $\frac{1}{6}x$. also, what kind of triangle is it?

Explanation:

Step1: Use angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. So, $x+\frac{1}{2}x+\frac{1}{6}x = 180$.

Step2: Combine like terms

First, find a common denominator for the left - hand side. The common denominator of 1, 2, and 6 is 6. So, $\frac{6x + 3x+x}{6}=180$, which simplifies to $\frac{10x}{6}=180$.

Step3: Solve for x

Multiply both sides of the equation $\frac{10x}{6}=180$ by $\frac{6}{10}$. We get $x=\frac{180\times6}{10}=108$.

Step4: Find the other angle measures

If $x = 108$, then $\frac{1}{2}x=\frac{1}{2}\times108 = 54$ and $\frac{1}{6}x=\frac{1}{6}\times108 = 18$.

Step5: Classify the triangle

Since one of the angles ($x = 108^{\circ}$) is greater than $90^{\circ}$, the triangle is an obtuse - angled triangle.

Answer:

The angle measures are $x = 108^{\circ}$, $\frac{1}{2}x=54^{\circ}$, $\frac{1}{6}x = 18^{\circ}$. It is an obtuse - angled triangle.