Explanation:
The diagram for this problem is shown below. We know some facts:
[tex]\angle CED=35^{\circ} \\ \\ \angle ADC=50^{\circ}[/tex]
From the figure, [tex]\angle ADE=180^{\circ}[/tex], so we can write this relationship:
[tex]\angle ADC+\angle CDE=180^{\circ} \\ \\ \\ Isolating \ \angle CDE: \\ \\ \angle CDE=180^{\circ}-\angle ADC \\ \\ \\ Substituting \ angle ADC=50^{\circ} \\ \\ \angle CDE=180^{\circ}-50^{\circ} \\ \\ \angle CDE=130^{\circ}[/tex]
Since C, E and D form a triangle, then the internal angles of any triangle add up to 180 degrees, so:
[tex]\angle CED + \angle EDC + \angle DCE=180^{\circ} \\ \\ \\ Substituting \ know \ values: \\ \\ 35^{\circ}+ 130^{\circ}+ \angle DCE=180^{\circ} \\ \\ \\ Isolating \ \angle DCE: \\ \\ \angle DCE=180^{\circ} -35^{\circ}-130^{\circ} \\ \\ \boxed{\angle DCE=15^{\circ}}[/tex]
