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Pearson Geometry Common Core, 2011

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Pearson Geometry Common Core, 2011 View details

Chapter Test

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Exercise 7 Page 425

What makes this quadrilateral a parallelogram? What are the properties of the angles in a parallelogram?

x^(∘)=57^(∘)
y^(∘)=57^(∘)
z^(∘)=66^(∘)

Practice makes perfect

Let's take a look at the given figure. Notice that the quadrilateral has four congruent sides. In particular, the two pairs of opposite sides that are congruent.

The related theorem states that if both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.

First, notice that the diagonal of the parallelogram lies on the transversal between the parallel lines. By using the Alternate Interior Angles Theorem, we know that the angles of measure x^(∘) and 57^(∘) are congruent.

Also, notice that the diagonal divides the figure into two isosceles triangles. The base angles of an isosceles triangle have the same measure.

Since we already know the values of x^(∘) and y^(∘), we now need to find z^(∘). For that, we will use the Triangle Angle Sum Theorem.

57^(∘)+57^(∘)+z^(∘)=180^(∘)

Add terms

114^(∘)+z^(∘)=180^(∘)

LHS-114^(∘)=RHS-114^(∘)

z^(∘)=66^(∘)

We found that x^(∘)=57^(∘), y^(∘)=57^(∘), and z^(∘)=66^(∘).