x =30°
Step-by-step explanation:
Step 1: To find [tex]\angle \mathrm{KNL}[/tex].
Given N ∈ KO, where KO is a line.
Sum of the adjacent angles in a straight line is 180°.
[tex]\Rightarrow \angle \mathrm{KNL}+\angle \mathrm{LNM}+\angle \mathrm{MNO}=180^{\circ}[/tex]
[tex]\Rightarrow \angle \mathrm{KNL}+45^{\circ}+105^{\circ}=180^{\circ}[/tex]
[tex]\Rightarrow \angle \mathrm{KNL}+150^{\circ}=180^{\circ}[/tex]
[tex]\Rightarrow \angle \mathrm{KNL}=180^{\circ}-150^{\circ}[/tex]
[tex]\Rightarrow \angle \mathrm{KNL}=30^{\circ}[/tex]
Step 2: To find [tex]\angle \mathrm{NLK}[/tex].
Sum of the adjacent angles in a straight line is 180°.
Given[tex]\angle \mathrm{NLM}=90^{\circ}[/tex]
[tex]\Rightarrow \angle \mathrm{NLM}+\angle \mathrm{NLK}=180^{\circ}[/tex]
[tex]\Rightarrow \angle \mathrm{NLK}=180^{\circ}-90^{\circ}[/tex]
[tex]\Rightarrow \angle \mathrm{NLK}=90^{\circ}[/tex]
Step 3: To find x.
Sum of the interior angles in a triangle is 180°.
Let us take the triangle NLK.
[tex]\Rightarrow \angle \mathrm{KNL}+\angle \mathrm{NLK}+\angle \mathrm{LKN}=180^{\circ}[/tex]
⇒ 30° + 90° + 2x = 180°
⇒ 2x = 180° – 30° – 90°
⇒ 2x = 60°
⇒ x = 30°
Hence, x = 30°.
