Question

Use the image to answer the question. Solve the question very carefully to make the answer correct

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.
Remaining Attempts : 3
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Answer 1

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

In a parallelogram:
1. Opposite angles are equal.
2. The sum of the angles in any quadrilateral is \( 360^\circ \).
3. Consecutive angles are supplementary (their sum is \( 180^\circ \)).

If we label the angles of the quadrilateral as follows:
- Let \( \angle A = m \)
- Let \( \angle B = n \)
- Then by the property of opposite angles, \( \angle C = m \) (since \( \angle C \) is opposite \( \angle A \))
- And \( \angle D = n \) (since \( \angle D \) is opposite \( \angle B \))

For ABCD to be a parallelogram:
- We need \( m + n = 180^\circ \).

This is because consecutive angles in a parallelogram are supplementary. Therefore, if angles \( A \) and \( B \) are consecutive angles, their measures must add up to \( 180^\circ \).

Thus, the guarantee that \( ABCD \) is a parallelogram can be written as:
- \( m + n = 180^\circ \)

In summary, Tiana is guaranteed that ABCD is a parallelogram if \( m + n = 180 \).

If \( m = 180 - n \) and \( n = 180 - m \) are used, the condition remains satisfied.

So the answer is:
- If \( m = \text{any angle value} \)
- \( n = 180 - m \)

That ensures \( ABCD \) is a parallelogram.

Please choose specific values for \( m \) or \( n \) in your problem context or clarify the relationship needed to fit any particular conditions you might have.