a In a polygon, an interior angle and its corresponding exterior angle form a linear pair. This means the angles are supplementary. Examining the diagram, we see that the exterior angle to the triangle's unknown interior angle, θ, is 120^(∘). With this information we can calculate θ.
120^(∘)+m∠ θ =180^(∘) ⇔ m∠ θ =60^(∘)
If the exterior angle is 120^(∘), the interior angle must be 60^(∘). However, according to the Triangle Angle Sum Theorem the three angles of any triangle must sum to 180^(∘). Now that we know the triangle's third angle, we can try this for the three angles.
60^(∘) + 64^(∘) + 56^(∘) ? = 180^(∘) ⇔ 180^(∘) = 180^(∘)
This means the figure is possible.
62^(∘) +117^(∘)? = 180^(∘) ⇔ 179^(∘) ≠ 180^(∘)
Since the angles are not supplementary, the figure is not possible.
c Examining the diagram, we see two triangles. We also notice that they have one pair of congruent angles. However, we also see that the unknown angles of the triangles form vertical angles. According to the Vertical Angles Theorem, they are congruent.
Since two pairs of angles are congruent, these triangles must be similar according to the AA (Angle-Angle) Similarity Theorem. However, similar triangles have three pairs of congruent angles. This is however not the case, since one pair of angles is not congruent. Therefore, this is not a possible figure.
