Question

what is the value of a? (the figure is a heptagon with angles: 2a - 31°, a + 50°, 2a - 11°, 2a - 23°, 141°, 2a, 144°)

Explanation:

Step1: Find total interior angle sum

For a heptagon (7 sides), the sum of interior angles is calculated by the formula $(n-2)\times180^\circ$, where $n=7$.
$$(7-2)\times180^\circ = 5\times180^\circ = 900^\circ$$

Step2: Sum all given angles

Add up all the interior angle expressions and known values.
$$(2a-31^\circ)+(a+50^\circ)+(2a-11^\circ)+(2a-23^\circ)+141^\circ+2a+144^\circ$$
Combine like terms:
$$2a+a+2a+2a+2a + (-31+50-11-23+141+144)^\circ = 9a + 270^\circ$$

Step3: Set equal to total sum

Equate the summed angles to the total interior angle sum, then solve for $a$.
$$9a + 270^\circ = 900^\circ$$
$$9a = 900^\circ - 270^\circ$$
$$9a = 630^\circ$$
$$a = \frac{630^\circ}{9} = 70^\circ$$

Answer:

$70^\circ$