Let's start by analyzing the given diagram. For the purposes of the solution, we will name the vertices of the triangles.
As we can see, ∠ 10 is an exterior angle to △ BCD. Because ∠ 2 and ∠ 6 do not share a vertex or corner of the triangle with ∠ 10, these are the corresponding remote interior angles. We can find a few more corresponding remote angles. Moreover, ∠ 10 is also an exterior angle to △ ABC.
Therefore, ∠ 4 and the sum of ∠ 2 and ∠ 3 are the corresponding remote interior angles of ∠ 10. Let's now recall the Exterior Angle Inequality Theorem.
Exterior Angle Inequality Theorem
The measure of an exterior angle of a triangle is greater than the measure of either of its corresponding remote interior angles.
By this theorem, we can immediately conclude that the measure of ∠ 10 is greater than the measures of ∠ 2 and ∠ 6. Besides that we have two more inequalities.
lcm∠ 10 > m∠ 2+ m∠ 3 & (I) m∠ 10 > m∠ 4 & (II)
By the first inequality, we can also say that m∠ 10 is greater than m ∠ 3.
m∠ 10 > m∠ 2+ m∠ 3
⇓
m∠ 10 > m∠ 2 and m∠ 10 >m∠ 3
As a result, the measure of ∠ 10 is greater than the measures of ∠ 2, ∠ 3, ∠ 4, and ∠ 6.
