Find maximum number of acute angles which a convex, a quadrilateral, a pentagon and a hexagon can have. Observe the pattern and generalise the result for any polygon.

Solution:

Given, a convex quadrilateral, a pentagon and a hexagon.

We have to find the maximum number of acute angles the given polygons can have.

An acute angle is defined as an angle that measures less than 90 degrees i.e. measure between 0° to 90°.

We know that if an interior angle is acute, then the corresponding exterior angle is greater than 90°.

Consider a convex polygon that has four or more acute angles.

Since the polygon is convex, all the exterior angles are positive.

So, the sum of the exterior angle is at least the sum of the interior angles.

Now, the sum of the exterior angles is greater than 4 × 90° = 360°.

We know that the sum of exterior angles of a polygon must equal 360°.

This implies that the maximum number of acute angles in a convex polygon is 3.

Therefore, any convex polygon can have a maximum of three acute angles.

✦ Try This: Diagonals of a parallelogram ABCD intersect at point O. If ∠BOC = 90° and ∠BDC = 50°, find ∠OAB.

☛ Also Check: NCERT Solutions for Class 8 Maths

NCERT Exemplar Class 8 Maths Chapter 5 Problem 183

Find maximum number of acute angles which a convex, a quadrilateral, a pentagon and a hexagon can have. Observe the pattern and generalise the result for any polygon.

Summary:

The maximum number of acute angles which a convex, a quadrilateral, a pentagon and a hexagon can have is 3.

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