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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Mathematics
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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