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.

Answer 1

To determine the values of \( m \) and \( n \) for which quadrilateral \( ABCD \) is guaranteed to be a parallelogram, we can use the properties of opposite angles in a parallelogram.

A quadrilateral is a parallelogram if:
1. Each pair of opposite angles are equal, or
2. The sum of the measures of two adjacent angles is \( 180^\circ \).

If we label the angles as follows:
- \( \angle A = m \)
- \( \angle B = n \)
- \( \angle C = m \)
- \( \angle D = n \)

The conditions that would guarantee that \( ABCD \) is a parallelogram are:
1. \( m + n = 180^\circ \)
2. \( m + m = 180^\circ \) (or equivalently, \( 2m = 180^\circ \))

Given these properties, we can set the following equations:
1. For adjacent angles, we have \( m + n = 180^\circ \).
2. Opposite angles must also hold the same relationship, so if \( \angle A = \angle C \) and \( \angle B = \angle D \), we also should consider opposite angles being equal.

Based on the property mentioned, Tiana is guaranteed that quadrilateral ABCD is a parallelogram when:

- If \( m + n = 180^\circ \) (for adjacent angles), then:
- \( n = 180^\circ - m \).

This means any specific values for \( m \) or \( n \) must satisfy this relationship, ensuring that when one angle is known, the other can be deduced accordingly.

Therefore, Tiana is guaranteed that ABCD is a parallelogram if \( m \) and \( n \) satisfy the equation:

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

In a specific example or case, if we pick \( m = 60^\circ \), then:

\[
n = 180^\circ - 60^\circ = 120^\circ.
\]

So generalized, if \( m = x \) then \( n = 180 - x \) for any value of \( x \) such that \( x + n = 180 \).

Thus, for Tiana's quadrilateral ABCD to be a parallelogram, Tiana can use:

\[
m = m, \quad n = 180 - m.
\]

Summarizing:

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