Question

Can a quadrilateral $\mathrm{ABCD}$ be a parallelogram if $\angle \mathrm{D}+\angle \mathrm{B}=180°?$

Answer 1

Given,

Quadrilateral $\mathrm{ABCD}$

Using the angle sum property of a quadrilateral,

$\angle \mathrm{A}+\angle \mathrm{B}+\angle C+\angle D=360°$

$\angle \mathrm{A}+\angle \mathrm{C}+180°=360°$ (Since it's given that $\angle \mathrm{D}+\angle \mathrm{B}=180°$)

$\angle \mathrm{A}+\angle \mathrm{C}=360°-180°$

$\angle \mathrm{A}+\angle \mathrm{C}=180°$ (Opposite angles should also be of the same measures.)

For $\angle \mathrm{D}+\angle \mathrm{B}=180°$, is a parallelogram.

If the following conditions are fulfilled, then ABCD is a parallelogram.

The sum of the measures of the adjacent angles should be $180°$ and opposite angles should also be of the same measure.

Hence, using the given condition$\angle \mathrm{D}+\angle \mathrm{B}={180}^{\mathrm{o}}$ we can say that yes, it may or may not be a parallelogram.