Answer:
m∠1 = 111°: m∠4 = 61°; m∠6 = 141°; m∠7 = 47°
m∠2 = 69°; m∠3 = 119°; m∠5 = 39°; m∠8 = 133°
Explanation:
The Polygon Exterior Angle Sum Theorem states the sum of the measures of the exterior angles of a polygon always equals 360°.
Step 1: Find x by setting sum of exterior angles equal to 360:
m∠1 + m∠4 + m∠6 + m∠7 = 360
(5x + 11) + (3x + 1) + (8x - 19) + (3x - 13) = 360
(5x + 3x + 8x + 3x) + (11 + 1 - 19 - 13) = 350
19x - 20 = 360
19x = 380
x = 20
Step 2: Check validity of answer by plugging in 20 for x in the equations representing the measures of angles 1, 4, 6, and 7 and checking that we get 360:
(5(20) + 11) + (3(20) + 1) + (8(20) - 19) + (3(20) - 13) = 360
(100 + 11) + (60 + 1) + (160 - 19) + (60 - 13) = 360
(111 + 61) + (141 + 47) = 360
172 + 188 = 360
360 = 360
Thus, x is indeed 20.
Step 3: Find the measures of angles 1, 4, 6, and 7 by plugging in 20 for x in the equations representing the measures of the angles.
Plugging in 20 for x in (5x + 11) to find m∠1:
m∠1 = 5(20) + 11
m∠1 = 100 + 11
m∠1 = 111°
Plugging in 20 for x in (3x + 1) to find m∠4:
m∠4 = 3(20) + 1
m∠4 = 60 + 1
m∠4 = 61°
Plugging in 20 for x in (8x - 19) to find m∠6:
m∠6 = 8(20) - 19
m∠6 = 160 - 19
m∠6 = 141°
Plugging in 20 for x in (3x - 13) to find m∠7:
m∠7 = 3(20) - 13
m∠7 = 60 - 13
m∠7 = 47°
In polygons, an interior angle and its corresponding exterior angle are always supplementary and thus the sum of their measures always equals 180°.
Step 4: Identify the interior angles and their corresponding exterior angles:
∠2 is the interior angle, and its corresponding exterior angle is ∠1.
∠3 is the interior angle, and its corresponding exterior angle is ∠4.
∠5 is the interior angle, and its corresponding exterior angle is ∠6.
∠8 is the interior angle, and its corresponding exterior angle is ∠7.
Step 3: Find the measures of angles 2, 3, 5, and 8 by subtracting the measures of angles 1, 4, 6, and 7 from 180:
Finding the measure of ∠2:
m∠1 + m∠2 = 180
m∠2 = 180 - m∠1
m∠2 = 180 - 111
m∠2 = 69°
Finding the measure of ∠3:
m∠4 + m∠3 = 180
m∠3 = 180 - m∠4
m∠3 = 180 - 61
m∠3 = 119°
Finding the measure of m∠5:
m∠6 + m∠5 = 180
m∠5 = 180 - m∠6
m∠5 = 180 - 141
m∠5 = 39°
Finding the measure of m∠8:
m∠7 + m∠8 = 180
m∠8 = 180 - m∠7
m∠8 = 180 - 47
m∠8 = 133°
Step 5: Check validity of answer.
We can find the sum of all the interior angles of a polygon using the formula 180(n-2), where
- n is the number of sides.
Since there are 4 sides, the sum of the interior angles of this polygon equals 180 as 180(4-2) = 360
We can check the validity of our answers for Step 3 by seeing if their sum is 360:
m∠2 + m∠3 + m∠5 + m∠8 = 360
(69 + 119) + (39 + 133) = 360
188 + 172 = 360
360 = 360
Thus, we've correctly found the measures of the interior angles.