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!)
Solution:
Let ABCD be a convex quadrilateral. We draw a diagonal AC which divides the quadrilateral into two triangles.
ABCD is a convex quadrilateral made of two triangles ∆ABC and ∆ADC.
We know that the sum of the angles of a triangle is 180 degrees. So:
∠6 + ∠5 + ∠4 = 180° [sum of the angles of ΔABC =180°]
∠1 + ∠2 + ∠3 = 180° [sum of the angles of ΔADC =180°]
Adding we get,
∠6 + ∠5 + ∠4 + ∠1 + ∠2 + ∠3 = 180° +180° = 360°
Hence, the sum of measures of the angles of a convex quadrilateral is 360°. Yes, even if a quadrilateral is not convex then, this property applies.
Let ABCD be a non-convex or a concave quadrilateral. Join BD, which also divides the quadrilateral into two triangles.
Using the angle sum property of a triangle, on triangles ΔABD and ΔBCD
= 180° +180° [sum of angles of triangles ΔABD and ΔBCD]
= 360°
Therefore, the sum of all the interior angles of this quadrilateral will also be 360°.
☛ Check: NCERT Solutions for Class 8 Maths Chapter 3
Video Solution:
What is the sum of the measures of the angles of a convex quadrilateral? Will this property hold if the quadrilateral is not convex?
NCERT Solutions Class 8 Maths Chapter 3 Exercise 3.1 Question 3
Summary:
The sum of the measures of the angles of a convex quadrilateral is 360 degrees. This property will hold even if the quadrilateral is not convex.
☛ Related Questions:
- Examine the table. (Each figure is divided into triangles and the sum of the angles deduced from that.) What can you say about the angle sum of a convex polygon with number of sides? (a) 7 (b) 8 (c) 10 (d) n
- What is a regular polygon? State the name of a regular polygon of (i) 3 sides (ii) 4 sides (iii) 6 sides
- Find the angle measure x in the following figures:
- a) Find x +y +z b) Find x + y + z + w