Question

ABCD is shown.
m
∠
A
B
C
=
110
°
,
m
∠
B
C
D
=
70
°
,
m
∠
D
A
B
=
(
5
x
−
8
)
°
,
m∠ABC=110°, m∠BCD=70°, m∠DAB=(5x−8)°, and
m
∠
C
D
A
=
(
8
y
+
12
)
°
.
m∠CDA=(8y+12)°.







Determine the values that would make
A
B
C
D
ABCD a parallelogram.



The value of
x
x is
and the value of
y
y is
.

Answer 1

To determine the values of \( x \) and \( y \) that make quadrilateral \( ABCD \) a parallelogram, we can use the properties of a parallelogram, specifically that opposite angles are equal.

Given the angles in the quadrilateral:
- \( m\angle ABC = 110^\circ \)
- \( m\angle BCD = 70^\circ \)
- \( m\angle DAB = (5x - 8)^\circ \)
- \( m\angle CDA = (8y + 12)^\circ \)

First, since \( ABCD \) is a parallelogram, we know that:
\[
m\angle ABC + m\angle CDA = 180^\circ
\]

Substituting the given angles:
\[
110^\circ + (8y + 12)^\circ = 180^\circ
\]

Now, simplify and solve for \( y \):
\[
110 + 8y + 12 = 180
\]
\[
8y + 122 = 180
\]
\[
8y = 180 - 122
\]
\[
8y = 58
\]
\[
y = \frac{58}{8} = 7.25
\]

Next, we also need to check the opposite pair of angles:
\[
m\angle DAB + m\angle BCD = 180^\circ
\]
Substituting the values:
\[
(5x - 8) + 70 = 180
\]
\[
5x - 8 + 70 = 180
\]
\[
5x + 62 = 180
\]
\[
5x = 180 - 62
\]
\[
5x = 118
\]
\[
x = \frac{118}{5} = 23.6
\]

Thus, the values that make \( ABCD \) a parallelogram are:
- The value of \( x \) is \( 23.6 \).
- The value of \( y \) is \( 7.25 \).