Question

Tiana draws quadrilateral ABCD as shown, with the measures of angle A, angle B, angle C, and Angle D represented. For what values of m and n is Tiana guaranteed that ABCD is a parallelogram?
If m= ___ and n= ___, then Tiana is guaranteed that ABCD is a parallelogram.
A = 3n + 3
B= 5m
C = 3m + 30
D = 105

Answer 1

To determine the values of \(m\) and \(n\) that guarantee that quadrilateral \(ABCD\) is a parallelogram, we need to use the properties of a parallelogram: opposite angles are equal and the sum of consecutive angles is equal to \(180^\circ\).

Given the measures of the angles:

- \(A = 3n + 3\)
- \(B = 5m\)
- \(C = 3m + 30\)
- \(D = 105\)

1. **Checking that opposite angles are equal:**
Since in a parallelogram angles \(A\) and \(C\) are opposite to each other and \(B\) and \(D\) are opposite to each other, we set up the equations:

- \(A + C = 180^\circ\)
- \(B + D = 180^\circ\)

First, consider angles \(A\) and \(C\):
\[
3n + 3 + (3m + 30) = 180
\]
Simplifying the equation gives:
\[
3n + 3m + 33 = 180
\]
\[
3n + 3m = 147
\]
\[
n + m = 49 \quad \text{(Equation 1)}
\]

Now, for angles \(B\) and \(D\):
\[
5m + 105 = 180
\]
Simplifying gives:
\[
5m = 75
\]
\[
m = 15 \quad \text{(Equation 2)}
\]

2. **Substituting \(m\) back into Equation 1:**
Now substitute \(m = 15\) into Equation 1:
\[
n + 15 = 49
\]
Solving for \(n\):
\[
n = 49 - 15 = 34
\]

Thus, the values of \(m\) and \(n\) that guarantee quadrilateral \(ABCD\) is a parallelogram are:
\[
m = 15 \quad \text{and} \quad n = 34
\]

Therefore, the final answer is:
\[
\text{If } m = 15 \text{ and } n = 34, \text{ then Tiana is guaranteed that } ABCD \text{ is a parallelogram.}
\]