Problem Analysis
The question states that every interior angle of a regular octagon is $135^\circ$. We need to find the measure of its exterior angle.
Octagon Exterior Angle Calculation
For any regular polygon, the interior angle and the exterior angle at a vertex are supplementary. This means they add up to $180^\circ$.
Given:
- Interior Angle = $135^\circ$
Using the relationship:
Interior Angle + Exterior Angle = $180^\circ$
Substituting the given value:
$ 135^\circ + \text{Exterior Angle} = 180^\circ $
To find the Exterior Angle, subtract the Interior Angle from $180^\circ$:
$ \text{Exterior Angle} = 180^\circ - 135^\circ $
$ \text{Exterior Angle} = 45^\circ $
Alternative Method: Using Polygon Properties
A regular octagon has 8 equal sides and 8 equal angles (n=8).
The sum of the exterior angles of any convex polygon is $360^\circ$. For a regular n-gon, each exterior angle is calculated as:
$ \text{Exterior Angle} = \frac{360^\circ}{n} $
For an octagon (n=8):
$ \text{Exterior Angle} = \frac{360^\circ}{8} $
$ \text{Exterior Angle} = 45^\circ $
Both methods confirm that the exterior angle of the regular octagon is $45^\circ$. The correct option is A.
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Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
