Question

Use the image to answer the question.



Tiana draws quadrilateral ABCD
𝐴
𝐵
𝐶
𝐷
as shown, with the measures of ∠A, ∠B, ∠C, and ∠D
∠
𝐴
,

∠
𝐵
,

∠
𝐶
,

𝑎
𝑛
𝑑

∠
𝐷
represented. For what values of m and n is Tiana guaranteed that ABCD
𝐴
𝐵
𝐶
𝐷
is a parallelogram?

(1 point)
If m=
𝑚
=
and n=
𝑛
=
, then Tiana is guaranteed that ABCD
𝐴
𝐵
𝐶
𝐷
is a parallelogram.

Answer 1

To determine the values of \( m \) and \( n \) that guarantee quadrilateral \( ABCD \) is a parallelogram, we can use the properties of the angles in a parallelogram.

For a quadrilateral to be a parallelogram, one of the following conditions must be met:

1. Both pairs of opposite angles are equal: \( \angle A = \angle C \) and \( \angle B = \angle D \).
2. The sum of the measures of adjacent angles is \( 180^\circ \): \( \angle A + \angle B = 180^\circ \) and \( \angle C + \angle D = 180^\circ \).
3. The measures of one pair of opposite angles are equal or the sum of one pair of adjacent angles is \( 180^\circ \).

Assuming \( \angle A = m \), \( \angle B = n \), \( \angle C = m \), and \( \angle D = n \), or if \( n = 180 - m \) guarantees that adjacent angles sum up to \( 180^\circ \).

So, we can set up the equations based on any of the conditions.

1. Setting \( m = n \) leads to:
- \( \angle A + \angle B = 180^\circ \) implies \( m + n = 180^\circ \).

Thus, if \( m + n = 180^\circ \), Tiana is guaranteed that quadrilateral \( ABCD \) is a parallelogram.

So the values of \( m \) and \( n \) that guarantee \( ABCD \) to be a parallelogram would be:

- If \( m = x \) and \( n = 180-x \)

In summary, if

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

Can you please confirm or provide values for \( m \) and \( n \) so I can answer accordingly?