Let's find the measures of the numbered angles in the given parallelogram one at a time.
Measure of ∠3
Recall the theorem referenced in the book that states that if a quadrilateral is a parallelogram, then its opposite angles are congruent. Therefore ∠3 is congruent with the angle that measures 99^(∘).
∠3 ≅ 99^(∘)
The definition of congruent angles tells us that their measures are equal, so m ∠3 = 99^(∘).
Measure of ∠2
We recall that consecutive angles in a parallelogram are supplementary. Using this information we can state an equation for ∠2 and ∠3.
m ∠3 + 38 + m ∠2 = 180
Let's substitute 99 for m ∠3 and solve the resulting equation.
Next, notice that ∠1, ∠2, and the angle that measures 99^(∘) are three interior angles of a triangle. Therefore, by the Triangle Angle-Sum Theorem their measures add up to 180.
m ∠1 + m ∠2 + 99 =180
We know that m ∠2=43, so we can find m ∠1 by substituting 43 for m ∠2 and solving the equation.
