Question

Select One
A=3n+3
B=5m
C=3m+30
D=105
Tiana draws quadrilateral ABCD as shown, with the measures of ∠A, ∠B, ∠C, and ∠D represented. For what values of m and n is Tiana guaranteed that ABCD is a parallelogram? (1 point) If m= and n= , then Tiana is guaranteed that ABCD is a parallelogram.

Answer 1

To determine the values of \(m\) and \(n\) that guarantee quadrilateral \(ABCD\) is a parallelogram, we need to use the property that opposite angles in a parallelogram are equal; therefore, the sum of the interior angles must equal 360 degrees.

Given the measures:
- \( \angle A = 3n + 3 \)
- \( \angle B = 5m \)
- \( \angle C = 3m + 30 \)
- \( \angle D = 105 \)

To satisfy the properties of a parallelogram, we can set up the following equations based on the angles:

1. \( \angle A + \angle C = 180^\circ \)
2. \( \angle B + \angle D = 180^\circ \)

Starting with the first equation:
\[
(3n + 3) + (3m + 30) = 180
\]
\[
3n + 3m + 33 = 180
\]
\[
3n + 3m = 180 - 33
\]
\[
3n + 3m = 147
\]
\[
n + m = 49 \quad \text{(Equation 1)}
\]

Now, for the second equation:
\[
(5m) + 105 = 180
\]
\[
5m + 105 = 180
\]
\[
5m = 180 - 105
\]
\[
5m = 75
\]
\[
m = 15 \quad \text{(Equation 2)}
\]

Substituting Equation 2 into Equation 1:
\[
n + 15 = 49
\]
\[
n = 49 - 15
\]
\[
n = 34 \quad \text{(Equation 3)}
\]

Thus, the values of \(m\) and \(n\) for which Tiana can guarantee that quadrilateral \(ABCD\) is a parallelogram are:
\[
m = 15, \quad n = 34
\]

So, if \( m = 15 \) and \( n = 34 \), then Tiana is guaranteed that \(ABCD\) is a parallelogram.