Question

find the measure of $\angle i$.

octagon labeled with vertices i, a, b, c, d, e, f, g, h in order

angle	measure
$\angle b$	$149^\circ$
$\angle c$	$144^\circ$
$\angle d$	$132^\circ$
$\angle e$	$133^\circ$
$\angle f$	$137^\circ$
$\angle g$	$134^\circ$
$\angle h$	$143^\circ$

Explanation:

Step1: Recall the formula for the sum of interior angles of a polygon.

The formula for the sum of interior angles of an \( n \)-sided polygon is \( S=(n - 2)\times180^\circ \). Here, the polygon is a nonagon (9 sides), so \( n = 9 \). Calculate the sum: \( S=(9 - 2)\times180^\circ=7\times180^\circ = 1260^\circ \).

Step2: Sum the given angles.

Given angles: \( \angle A = 144^\circ \), \( \angle B = 149^\circ \), \( \angle C = 144^\circ \), \( \angle D = 132^\circ \), \( \angle E = 133^\circ \), \( \angle F = 137^\circ \), \( \angle G = 134^\circ \), \( \angle H = 143^\circ \). Sum them: \( 144 + 149 + 144 + 132 + 133 + 137 + 134 + 143 \). Let's calculate step by step:
\( 144+149 = 293 \); \( 293+144 = 437 \); \( 437+132 = 569 \); \( 569+133 = 702 \); \( 702+137 = 839 \); \( 839+134 = 973 \); \( 973+143 = 1116 \).

Step3: Find \( \angle I \).

Let \( \angle I = x \). The sum of all angles is \( 1260^\circ \), so \( 1116 + x = 1260 \). Solve for \( x \): \( x = 1260 - 1116 = 144^\circ \).

Answer:

\( 144^\circ \)