3. What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex? (Make a non-convex quadrilateral and try!)
Let us draw a convex quadrilateral ABCD.
From the figure, we understand that the quadrilateral ABCD is formed by two triangles,
i.e. ΔADC and ΔABC.
So, according to angle sum property of triangle we know that sum of interior angles of triangle is 180°,
The sum of the measures of the angles is 180° + 180° = 360°
Hence, the sum of all angles of a convex quadrilateral = 360°
Let us take another non-convex quadrilateral ABCD.
We can Join BC, Such that it divides ABCD into two triangles ΔABC and ΔBCD.
∠1 + ∠2 + ∠3 = 180° (angle sum property of triangle)
∠4 + ∠5 + ∠6 = 180° (angle sum property of triangle)
∴, ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 180° + 180°
⇒ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 + ∠6 = 360°
⇒ ∠A + ∠B + ∠C + ∠D = 360°
Hence, this property also holds true for non-convex quadrilateral.