Question

What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex?

Answer 1

Step-1: Determine the sum of the measures of the angles of a convex quadrilateral:

Let us assume $ABCD$as a convex quadrilateral.

From the figure, we indicate that the quadrilateral $ABCD$is formed by two triangles, that is $\Delta ADC$ and $\Delta ABC$.

We know that sum of interior angles of a triangle is $180°$, the sum of the measures of the angles of a convex quadrilateral is$180°+180°=360°$.

Step-2: Determine the sum of the measures of the angles of a concave quadrilateral:

Let us consider another quadrilateral $ABCD$ that is not convex.

Join $BC$.

$BC$ divides $ABCD$in such a way that it into two triangles $\Delta ABCand\Delta BCD$ .

In $\Delta ABC,\angle 1+\angle 2+\angle 3=180°$ (angle sum property of triangle)

In $\Delta BCD,\angle 4+\angle 5+\angle 6=180°$ (angle sum property of triangle)

$$\begin{gathered} \therefore \\ \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=180°+180° \\ \Rightarrow \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=360° \\ \Rightarrow \angle A+\angle B+\angle C+\angle D=360° \end{gathered}$$

Hence, the sum of the measures of the angles of a convex or concave quadrilateral is $360°$.