Question

what is the sum of the measures of the angles of a convex quadrilateral? will this property hold is a quadrilateral is not convex (make a non -convex quadrilateral and try)​

Answer 1

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Here is your answer

The convex quadrilateral has all the angles less than 180 or equal to 180

The concave quadrilateral has all the angles greater than 180

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Answer 2

Let ABCD be a convex quadrilateral....

From the figure, we infer that the quadrilateral ABCD is formed by two triangles,

that is: ∆ADC & ∆ABC

Since, we know that sum of the interior angle of triangle is 180°, the sum of the measures of the angles is 180°+180°=360°.

Let us take another quadrilateral ABCD, which is not a convex.

Join BC, such that it divides ABCD into two triangles ∆ABC & ∆BCD.

In ∆ABC,

<1+<2+<3=180° (Angle sum property of triangle)

In ∆BCD,

<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°

✰Thus, this property hold if the quadrilateral is not convex.