In a right triangle, the acute angles are complementary angles, which means that their measures add up to 180º. Then:
![m\angle L+m\angle K=180](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/a6qhu2pmc1c47c58e27l.png)
Replace the expressions for the measure of L and K in terms of x. Then, solve for x and substitute its value back into the expressions for L and K to find their values. Since angle J is a right angle, its measure is 90º.
![\begin{gathered} m\angle L=2x+17 \\ m\angle K=3x+28 \\ \Rightarrow(2x+17)+(3x+28)=90 \end{gathered}](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/2pq3xkudv04ua6h881et.png)
Solve for x:
![\begin{gathered} \Rightarrow5x+45=90 \\ \Rightarrow5x=90-45 \\ \Rightarrow5x=45 \\ \Rightarrow x=(45)/(5) \\ \Rightarrow x=9 \end{gathered}](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/owelz1h6f4wf6uhtx481.png)
Replace x=9into the expressions for L and K:
![\begin{gathered} m\angle K=3x+28 \\ =3(9)+28 \\ =27+28 \\ =55 \end{gathered}](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/rnwl6mf3y2fkp0fm4nar.png)
![\begin{gathered} m\angle L=2x+17 \\ =2(9)+17 \\ =18+17 \\ =35 \end{gathered}](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/hc7nuzxbu5jegdjm9cws.png)
Therefore, the measures of the angles are:
![\begin{gathered} m\angle J=90 \\ m\angle K=55 \\ m\angle L=35 \end{gathered}](https://img.qammunity.org/qa-images/2023/formulas/mathematics/college/1g59svxduvumtzdas0bn.png)