Question

the figure shown is a parallelogram. solve for the variables. m=68, n=110; m=68, n=70; m=35, n=70; m=35, n=110

Explanation:

Step1: Recall parallelogram properties

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \(180^\circ\)). Also, the diagonal bisects the angles? Wait, looking at the angles: one angle is \(70^\circ\), another is \(n^\circ\), and another is \(2m^\circ\). Wait, maybe the angles: in a parallelogram, alternate interior angles? Wait, maybe the triangle formed? Wait, no, the figure is a parallelogram, so adjacent angles are supplementary, and opposite angles are equal. Wait, maybe the angles given: let's assume that \(n\) and \(70^\circ\) are related, and \(2m\) and another angle. Wait, maybe the angle \(70^\circ\) and \(n\): if there's a triangle, but no, it's a parallelogram. Wait, maybe the angles: in a parallelogram, if one angle is \(70^\circ\), then its opposite angle is \(70^\circ\), and consecutive angles are \(110^\circ\). But the options have \(m = 35\), \(n = 70\) or \(m = 35\), \(n = 110\)? Wait, no, let's re - examine. Wait, maybe the angle \(2m\) and \(n\): if the angle is \(70^\circ\), and maybe \(2m=70^\circ\)? No, \(2m = 70^\circ\) would give \(m = 35^\circ\), and \(n = 70^\circ\)? Wait, no, maybe the consecutive angles. Wait, if one angle is \(70^\circ\), then the consecutive angle is \(180 - 70=110^\circ\). But the options: let's check the options. The options are:

1. \(m = 68\), \(n = 110\)
2. \(m = 68\), \(n = 70\)
3. \(m = 35\), \(n = 70\)
4. \(m = 35\), \(n = 110\)

Wait, maybe the angle \(70^\circ\) and \(n\): if \(n = 70^\circ\) (opposite angle), and \(2m\) and the angle supplementary to \(70^\circ\). Wait, no, if \(n = 70^\circ\), then the consecutive angle is \(110^\circ\), but \(2m\) should be equal to \(70^\circ\)? No, \(2m = 70^\circ\) would be \(m = 35^\circ\). Wait, let's think again. In a parallelogram, the diagonal divides it into two congruent triangles. So if we have a diagonal, then the angles formed: the angle \(70^\circ\), and the angle \(2m\) and \(n\). Wait, maybe the triangle has angles \(70^\circ\), \(2m\), and \(n\), but in a parallelogram, the diagonal makes the angles equal. Wait, no, maybe the angle \(n\) is equal to \(70^\circ\) (opposite angle), and \(2m\) is equal to \(70^\circ\)? No, \(2m = 70^\circ\) gives \(m = 35^\circ\), and \(n = 70^\circ\). Wait, but then the consecutive angle would be \(110^\circ\). But the option \(m = 35\), \(n = 70\) is there (third option) and \(m = 35\), \(n = 110\) (fourth option). Wait, maybe I made a mistake. Wait, let's check the angle sum. If \(2m\) and \(n\) are angles in a triangle? No, the figure is a parallelogram. Wait, maybe the angle \(70^\circ\) and \(n\) are alternate interior angles? No, let's look at the options. Let's check the option \(m = 35\), \(n = 70\): if \(2m=70^\circ\), then \(m = 35^\circ\), and \(n = 70^\circ\) (opposite angle to \(70^\circ\)). That fits the parallelogram property of opposite angles being equal.

Step2: Verify the values

If \(m = 35\), then \(2m=2\times35 = 70^\circ\). And \(n = 70^\circ\), which is equal to the given \(70^\circ\) angle (opposite angles in a parallelogram are equal). So this satisfies the parallelogram angle properties.

Answer:

\(m = 35\), \(n = 70\) (the option with \(m = 35\), \(n = 70\))