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)
Answers
The sum of all interior angles of a quadrilateral is 360º. This property holds true for both convex and concave quadrilaterals.
GIVEN
A convex quadrilateral and a concave quadrilateral both named as ABCD, (with interior ∠B > 180º in case of concave quadrilateral)
TO PROVE
Sum of interior angles
∠A +∠B + ∠C + ∠D = 360º
CONSTRUCTION
Draw the diagonal BD in both quadrilateral.
PROOF(for both types of quadrilateral)
Theorem
Sum of interior angles of a triangle is 180º.
In ΔABD
∠ABD + ∠BDA + ∠DAB = 180º -----(1)
In ΔCBD
∠CBD + ∠BDC + ∠DCB = 180º -----(2)
Adding (1) and (2), we get
(∠ABD + ∠CBD) + (∠BDA + ∠BDC) + ∠DAB + ∠DCB = 180º + 180º
or ∠ABC + ∠CDA + ∠DAB + ∠BCD = 360º
or ∠DAB + ∠ABC + ∠BCD + ∠CDA = 360º
or ∠A + ∠B + ∠C + ∠D = 360º
HENCE PROVED
Answer:
360°
The sum of the measures of the angles of a convex quadrilateral is 360° as a convex quadrilateral is made of two triangles. In above convex quadrilateral, it made of two triangles.
Therefore, the sum of all the interior angles of this quadrilateral will be same as the sum of all the interior angles of these two triangles i.e., 180º + 180º = 360º This property also holds true for a quadrilateral which is not convex. This is because any quadrilateral can be divided into two triangles.
Step-by-step explanation:
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