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.
