Solution:

Let us first find the equations of the two lines. Using slope-intercept form, we know they are the form $y = ax + b.$

For the first line, we know that $a = \dfrac{1}{2},$ so we get $$2 = \dfrac{1}{2} \cdot 2 + b$$$$b = 1.$$

Similarly, for the second line, we get that $a = 2,$ which gives us $$2 = 2 \cdot 2 + b$$$$b = -2.$$

Our two lines are now $y = \dfrac{1}{2}x + 1$ and $y = 2x - 2.$

We can rewrite $x + y = 10$ as $y = 10 - x.$ Substituting this into the first line yields $$10 - x = \dfrac{1}{2}x + 1$$$$x = 6, y = 4.$$

Similarly, for the second line, $$10 - x = 2x - 2$$$$x = 4, y = 6.$$

The three vertices of the triangle are therefore $(2, 2), (6, 4),$ and $(4, 6).$

Note that these vertices form an isosceles triangle (distance formula yields the three sides as $2\sqrt{5}, 2\sqrt{5},$ and $2\sqrt{2}.$

The midpoint of the base is $(5, 5),$ and applying the distance formula again tells us that the height is $3\sqrt{2}.$

The area is therefore $$\dfrac{1}{2} \cdot 2\sqrt{2} \cdot 3 \sqrt{2} = 6.$$ Thus, C is the correct answer.