Parallelograms Examples

Quadrilateral FGHJ is a parallelogram. If ∠FJG has a measure of 27°, what is the measure of ∠HGJ?

The parallelogram is divided into two congruent triangles by the diagonal GJ. We know that ∠FJG in ΔFJG corresponds to ∠HGJ in ΔHGJ, and since corresponding parts of congruent triangles are congruent (CPCTC), ∠FJG ≅ ∠HGJ. That means m∠HGJ = 27°.

Is quadrilateral ABCD a parallelogram? Why?

Yes. The diagonals of the parallelogram bisect each other. If the two diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

Quadrilateral FGHJ is a parallelogram and ∠4 measures 90°. What are the measures of the other three angles?

Since FGHJ is a parallelogram, we can use the fact that consecutive angles in a parallelogram are supplementary to find the measure of ∠3. Supplementary angles add up to 180°, which means that:

m∠3 + m∠4 = 180°
m∠3 + 90 = 180°
m∠3 = 90°

Now we can use the fact that opposite angles in a parallelogram are congruent. This means ∠4 ≅ ∠2 and ∠3 ≅ ∠1. So basically, all four angles here have a measure of 90°. In case you were wondering: yes, this is a rectangle.

What are the measures of ∠CAD and ∠ADC?

First, the measures of the angles in the ΔCAD have to add up to 180°. Since ∠ACD is 90°, we can write

m∠CAD + m∠ACD + m∠CDA = 180
2x + 90 + 3x + 5 = 180
5x + 95 = 180
5x = 85
x = 17

Plugging this back into the original equations, m∠CAD = 34° and m∠ADC = 56°.