Question

A parallelogram upper A upper B upper D upper C has expressions for each angle marked and labeled.
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

In a parallelogram, the opposite angles are equal, and the sum of the adjacent angles is equal to \(180^\circ\).

Let's denote the measures of the angles as follows:

- \(\angle A = m\)
- \(\angle B = n\)
- \(\angle C = m\) (because \(\angle A\) is equal to \(\angle C\))
- \(\angle D = n\) (because \(\angle B\) is equal to \(\angle D\))

Since adjacent angles in a parallelogram must add up to \(180^\circ\), we can set up the following equations based on the angles:

1. \(\angle A + \angle B = 180^\circ\)
\[
m + n = 180^\circ
\]

2. \(\angle C + \angle D = 180^\circ\) (which will naturally hold true since \(\angle C\) and \(\angle D\) are equal to \(\angle A\) and \(\angle B\) respectively).

Thus, the condition that Tiana is guaranteed that quadrilateral \(ABCD\) is a parallelogram is:

\[
m + n = 180^\circ
\]

So to answer your question:

If \( m = m \) and \( n = 180^\circ - m \), then Tiana is guaranteed that ABCD is a parallelogram.

In summary:
- \( m = m \)
- \( n = 180^\circ - m \)

Please note that \(m\) must be between \(0^\circ\) and \(180^\circ\) for realistic angles in a quadrilateral.