Question

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.


A= 3n+3
B= 5m
C= 3m+30
D= 105

Answer 1

For a quadrilateral to be a parallelogram, the sum of the opposite angles must be equal. That means ∠A + ∠C must equal 180 degrees and ∠B + ∠D must also equal 180 degrees.

Given the measures:

\[
\angle A = 3n + 3
\]
\[
\angle B = 5m
\]
\[
\angle C = 3m + 30
\]
\[
\angle D = 105
\]

We can start with the equation for angles A and C:

\[
\angle A + \angle C = 180
\]
\[
(3n + 3) + (3m + 30) = 180
\]
\[
3n + 3m + 33 = 180
\]
\[
3n + 3m = 147
\]
\[
n + m = 49 \quad \text{(1)}
\]

Next, let's use the equation for angles B and D:

\[
\angle B + \angle D = 180
\]
\[
5m + 105 = 180
\]
\[
5m = 75
\]
\[
m = 15 \quad \text{(2)}
\]

Now substitute \( m = 15 \) into equation (1):

\[
n + 15 = 49
\]
\[
n = 34
\]

Thus, Tiana is guaranteed that ABCD is a parallelogram for:

\[
m = 15 \quad \text{and} \quad n = 34.
\]