Question

Please help me see solve this problem. This is so confusing

Answer 1

Given that the triangle is right-angled at vertex Q, it follows that,

`m\angle Q=90^(\circ)`

The other two angles' measures are given as,

`\begin{gathered} m\angle R=(5x-39)^(\circ) \\ m\angle S=(2x+31)^(\circ) \end{gathered}`

The Angles Sum Property of a triangle suggests that the sum of the degree measure of the three angles formed by the sides of the triangle at the vertices is always 180.

Applying this in the given triangle QRS,

`m\angle Q+m\angle R+m\angle S=180^(\circ)`

Substitute the values and simplify,

`\begin{gathered} 90^(\circ)+(5x-39)^(\circ)+(2x+31)^(\circ)=180^(\circ) \\ 90+5x-39+2x+31=180 \\ 7x+82=180 \\ 7x=180-82 \\ 7x=98 \\ x=14 \end{gathered}`

Substitute this value to obtain the degree measure of unknown angles as follows,

`\begin{gathered} m\angle R=(5(14)-39)^(\circ)=(70-39)^(\circ)=31^(\circ) \\ m\angle S=(2(14)+31)^(\circ)=(28+31)^(\circ)=59^(\circ) \end{gathered}`

Thus, the measure of angles of the given triangle are,

`\begin{gathered} m\angle Q=90^(\circ) \\ m\angle R=31^(\circ) \\ m\angle S=59^(\circ) \end{gathered}`