Examining the diagram, we can identify a quadrilateral and a triangle. According to the Triangle Angle Sum Theorem, the sum of a triangle's angles equals 180^(∘). With this information we can solve for the triangle's unknown angle, which we will label v.
m∠ v +23^(∘)+48^(∘) = 180^(∘) ⇔ m∠ v = 109^(∘)
Examining the diagram, we notice two pairs of vertical angles. We also see a pair of consecutive interior angles. According to the Vertical Angles Theorem, the vertical angles are congruent.
Since the two lines cut by the transversal are parallel, we can claim that they are supplementary by the Consecutive Interior Angles Theorem.
m∠ z+81^(∘)= 180^(∘) ⇔ m∠ z =99^(∘)
When we know three angles of the quadrilateral, we can find the fourth angle by equating the sum of the quadrilaterals angles with 360^(∘) and solving for m∠ w.
m∠ w+81^(∘)+109^(∘)+99^(∘)= 360^(∘)
⇕
m∠ w =71^(∘)
Since ∠ y≅ ∠ w, we know that m∠ y =71^(∘). Now we know all of the unknown angles and can complete the diagram.
