Question

The angle measures of an octagon are in arithmetic sequence. Then what's the sum of its smallest and largest angles?
And if the difference between the smallest and largest angles is 70,what is the measure of smallest angle?
Than will any of the angles of this octagon be right angle?why?

Answer 1

Answer:

100°

No angle will be a right angle

Step-by-step explanation:

Let the smallest angle be a which is also the first term of the AP

If the common difference of the AP is d then

There will be 8 angles in octagon

The eighth angle will be $a+7d$

Given that

Difference between the smallest and largest angle is 70

Thus,

$a+7d-a=70$

$\implies 7d=70$

$\implies d=10$

Sum of all the angles

From sum of n terms of an AP

$\boxed{S=\frac{n}{2}[\text{First term}+\text{Last term}]}$

or, $S=\frac{8}{2}[a+a+7d]$

or, $S=4[2a+70]$

We know that the sum of all angles of a n sided polygon is given by

$(2n-4)\times 90^\circ$

Therefore,

$(2\times 8-4)\times 90=4(2a+70)$

$\implies 12\times 90=4(2a+70)$

$\implies 3\times 90=2a+70$

$\implies 2a=270-70$

$\implies 2a=200$

$\implies a=100^\circ$

Therefore, the smallest angle is 100°

No angle will be octagon because the measure of the smallest angle is 100°

Hope this answer is helpful.