A rhombus has four congruent sides. Let's find the measure of each numbered angle one at a time.
Measures of ∠ 2 and ∠ 3
To help with the process of solving, we will label the angle next to ∠ 2 as ∠ 2' and the angle next to ∠ 3 as ∠ 3'.
In a parallelogram, the diagonals bisect opposite angles. Therefore, since a rhombus is a parallelogram, we know that ∠ 3≅ ∠ 3' and therefore their measures are equal.
∠ 3 ≅ ∠ 3' ⇔ m∠ 3=m∠ 3'
Notice that ∠ 3 and ∠ 2' are alternate interior angles. Since a rhombus is a parallelogram, we know that its opposite sides are parallel. Therefore, by the Alternate Interior Angles Theorem ∠ 3 and ∠ 2' are congruent. By the definition of congruent angles, we know that their measures are equal.
∠ 3 ≅ ∠ 2' ⇔ m ∠ 3 = m ∠ 2'
Now, notice that ∠ 2', ∠ 3', and the angle of measure 128 are three angles in a triangle. By the Triangle Angle-Sum Theorem, we can conclude that their measures add to 180.
m ∠ 2' + m ∠ 3' + 128 = 180
We already know that m ∠ 2' = m ∠ 3 and m ∠ 3' = m ∠ 3. So, let's substitute m∠ 3 for m∠ 2' and m∠ 3' into our equation.
The measure of ∠ 3 is 26. Therefore, the measure of ∠ 3' is also 26. Since ∠ 2 and ∠ 3' are alternate interior, their measures are the same. This means that m∠ 2=26.
Measure of ∠ 1
Notice that ∠ 1, ∠ 2, and ∠ 3 are three angles in a triangle. By the Triangle Angle-Sum Theorem, we can conclude that their measures add to 180.
m ∠ 2 + m ∠ 3 + m ∠ 1 = 180
We already know that both m ∠ 2 and m ∠ 3 are equal to 26. Let's substitute m ∠ 2=26 and m ∠ 3=26 into our equation to find m ∠ 1.
