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 ABCD be a convex quadrilateral.

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

i.e. ΔADC and ΔABC.

Since, we know that sum of interior angles of triangle is 180^\circ,

the sum of the measures of the angles is 180^\circ+180^\circ=360^\circ

Let us take another quadrilateral ABCD which is not convex .

Join BC, Such that it divides ABCD into two triangles ΔABC and ΔBCD.

In ΔABC,

∠1 + ∠2 + ∠3 = 180^\circ (angle sum property of triangle)

In ΔBCD,

∠4 + ∠5 + ∠6 = 180^\circ (angle sum property of triangle)

\beginarrayl\therefore \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=180^\circ+180^\circ \\\Rightarrow \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=360^\circ \\\Rightarrow \angle A+\angle B+\angle C+\angle D=360^\circ\endarray

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

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Science

Chapters in NCERT Solutions - Mathematics , Class 8

Rational Numbers

Linear Equations in One Variable

Understanding Quadrilaterals

Practical Geometry

Data Handling

Squares and Square Roots

Cubes and Cube Roots

Comparing Quantities

Algebraic Expressions and Identities

Visualising Solid Shapes

Mensuration

Exponents and Powers

Direct and Inverse Proportions

Factorisation

Introduction to Graphs

Playing with Numbers

Exercises in Understanding Quadrilaterals

Exercise 3.1

Exercise 3.2

Exercise 3.3

Exercise 3.4

Question 10

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!)

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Let ABCD be a convex quadrilateral.

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

i.e. ΔADC and ΔABC.

Since, we know that sum of interior angles of triangle is 180^\circ,

the sum of the measures of the angles is 180^\circ+180^\circ=360^\circ

Let us take another quadrilateral ABCD which is not convex .

Join BC, Such that it divides ABCD into two triangles ΔABC and ΔBCD.

In ΔABC,

∠1 + ∠2 + ∠3 = 180^\circ (angle sum property of triangle)

In ΔBCD,

∠4 + ∠5 + ∠6 = 180^\circ (angle sum property of triangle)

\beginarrayl\therefore \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=180^\circ+180^\circ \\\Rightarrow \angle 1+\angle 2+\angle 3+\angle 4+\angle 5+\angle 6=360^\circ \\\Rightarrow \angle A+\angle B+\angle C+\angle D=360^\circ\endarray

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

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