Convex and Concave Quadrilaterals
Convex quadrilateral: A quadrilateral is called a convex quadrilateral, if the line segment joining any two vertices of the quadrilateral is in the same region.
In figure, ABCD is a convex quadrilateral because AB, BC, CD, DA, AC, BD are in the same region of the quadrilateral.
In a convex quadrilateral each angle measures less than 180°.
Concave quadrilateral: A quadrilateral is called a concave quadrilateral, if at least one line segment joining the vertices is not a part of the same region of the quadrilateral.
That is, any line segment that joins two interior points goes outside the figure. In a concave quadrilateral at least one angle is a reflex angle, i.e., an angle larger than 180°. In figure, ABCD is a concave quadrilateral because a line joining the vertices A and C is going outside the quadrilateral region.
Angle sum property of a quadrilateral
The sum of all angles of a quadrilateral is 360° or four right angles.
Draw a quadrilateral ABCD with one of its diagonals AC.
Diagonal AC divides the quadrilateral into two triangles, i.e., ΔADC and ΔABC.
Clearly ∠l + ∠2 = ∠A
and ∠3 + ∠4 = ∠C …(1)
We know that the sum of the angles of a triangle is 180°.
∴ In ΔABC, ∠1 + ∠3 + ∠B = 180°
In ΔADC, ∠2 + ∠4 + ∠D = 180°
Sum of the angles of a quadrilateral
= Sum of the angles of ΔABC and ΔADC
∴ (∠1 + ∠3 + ∠B) + (∠2 + ∠4 + ∠D) = 180° + 180°
or ∠1 + ∠3 + ∠B + ∠2 + ∠4 + ∠D = 360°
or (∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360°
or ∠A + ∠B + ∠C + ∠D = 360° (using 1)
Hence, the sum of the angles of a quadrilateral equals 360°.
Example 1: The angles of a quadrilateral are in the ratio of 1 : 2 : 1 : 2. Find the measure of each angle.
Solution: Let the first angle of a quadrilateral be x.
Here, second angle = 2x
third angle = x
fourth angle = 2x
Sum of all angles of a quadrilateral = 360°
∴ x + 2x + x + 2x = 360°
6x = 360°
x = 60°
∴ First angle = x = 60°
Second angle = 2x = 2 × 60° = 120°
Third angle = x = 60°
Fourth angle = 2x = 120°.