Question

Jonah drew quadrilateral ABCD.

If side AB = 20y, side BC = 1 + 3x, side CD = 4y + 32, and side AD = 6x - 20, for what values of x and y can Jonah be sure his quadrilateral is a parallelogram?

Answer 1

The figure shown is a rhombus. If the measure of angle DAB = 115 degrees, what is the measure of angle ADE?

Answer 2

To determine the values of \( x \) and \( y \) for which quadrilateral ABCD is a parallelogram, we need to use the property that in a parallelogram, opposite sides are equal in length.

We have the following side lengths for quadrilateral ABCD:
- \( AB = 20y \)
- \( BC = 1 + 3x \)
- \( CD = 4y + 32 \)
- \( AD = 6x - 20 \)

For quadrilateral ABCD to be a parallelogram, the following equalities must hold:
1. \( AB = CD \)
2. \( BC = AD \)

Now, let's set up the equations based on these equalities:

**1. From \( AB = CD \):**
\[
20y = 4y + 32
\]

To solve for \( y \), we can isolate \( y \):
\[
20y - 4y = 32
\]
\[
16y = 32
\]
\[
y = \frac{32}{16} = 2
\]

**2. From \( BC = AD \):**
\[
1 + 3x = 6x - 20
\]

Now, let's solve for \( x \):
\[
1 + 20 = 6x - 3x
\]
\[
21 = 3x
\]
\[
x = \frac{21}{3} = 7
\]

**Final Values:**
The values of \( x \) and \( y \) for which quadrilateral ABCD is a parallelogram are:
\[
x = 7, \quad y = 2
\]

Thus, Jonah can be sure that his quadrilateral \( ABCD \) is a parallelogram when \( x = 7 \) and \( y = 2 \).