We want to find the angle measures of the numbered angles in the diagram. To do this we will first find the angle measures of the two unknown angles in the triangle formed by the two congruent segments and one of the sides of the parallelogram. Let's call the angles A and B.
Measure of ∠ A and ∠ B
Since ∠ A and ∠ B are base angles in an isosceles triangle, the Isosceles Triangle Theorem states that the angles are congruent.
∠ A ≅ ∠ B
The definition of congruent angles tells us that their measures are equal, so m ∠ A = m ∠ B. The third angle in the triangle is right. Therefore, its measure is 90 ^(∘). By the Triangle Angle-Sum Theorem, the measures of the three angles add up to 180.
m∠ A + m∠ B + 90 =180
The unknown angles are congruent, so we can find their measure.
Since ∠ A and ∠ B are congruent, we know that m∠ B=45.
Measure of ∠ 2
We see that ∠ 2 and ∠ B are alternate interior angles. Opposite sides of a parallelogram are parallel, so by the Alternate Interior Angles Theorem we can conclude that they are congruent.
∠ 2 ≅ ∠ B
The definition of congruent angles tells us that their measures are equal, so m ∠ 2 = m ∠ B. We already know that m ∠ B = 45, so we can conclude that m ∠ 2 = 45.
Measure of ∠ 1
Next we will need to find the measures of the two angles that forms a linear pair with two other angles. Let's can call them L_1 and L_2.
To find the measures of L_1 we can use that angles that form a linear pair add up to a measure of 180.
m∠ L_1 + 90 =180 ⇔ m∠ L_1 = 90
Thus, ∠ L_1 measures to 90^(∘), which means that it is a right angle. Since ∠ L_2 also forms a linear pair with the right angle we now know that m∠ L_2=90^(∘). Let's recall that the diagonals of a parallelogram bisect each other. Therefore, the two parts of both diagonals must be congruent.
We see that ∠ 1 is one of the base angles in an isosceles triangle. The third angle in that triangle is a right angle. By using the same argument we used to find the measures of ∠ A and ∠ B, we find that m∠ 1=45^(∘).
Measure of ∠ 3
∠ 3 is a base angle in an isosceles triangle in which the third angle is a right angle. Therefore, we know that m∠ 3=45^(∘). Let's gather the results we found!
