m∠3 = 33°
Step-by-step explanation:
Given that ∠1, ∠2, and ∠3 are adjacent angles, and that ∠1 and ∠2 are supplementary, while ∠2 and ∠3 are complementary angles.
We are also given the values for the following angles:
m∠1 = (8x + 3)°
m∠2 = (5x - 18)°
In order to find the measure of ∠3, we must first find the value of x using the details from the given prompt.
We know that ∠1 and ∠2 are supplements, which means that the sum of their measures add up to 180°. In other words:
m∠1 + m∠2 = 180°
Substitute the given values into the equation:
(8x + 3)° + (5x - 18)° = 180°
8x° + 3° + 5x° - 18° = 180°
Combine like terms:
13x° - 15° = 180°
Add 15° from both sides:
13x° - 15° + 15° = 180° + 15°
13x° = 195°
Divide both sides by 13 to solve for x:
x = 15°
Now that we have the value for x = 15°, we need to determine the measure of ∠3. Using the details from the given prompt that m∠2 and m∠3 are complementary angles (whose measures add up to 90°), then we can set up the following equation:
m∠2 + m∠3 = 90°
Substitute its value into m∠2° to find the m∠3°.
m∠2 + m∠3 = 90°
(5x - 18)° + m∠3 = 90°
5(15)° - 18° + m∠3 = 90°
75° - 18° + m∠3 = 90°
57° + m∠3 = 90°
Subtract 57° from both sides:
57° - 57° + m∠3 = 90° - 57°
m∠3 = 33°
Therefore, the measure of ∠3 is 33°.
Double-check:
In order to verify whether we have the correct measure for ∠3, we'll substitute the x = 15° into the given values for ∠1, ∠2, and the derived value for ∠3.
m∠1 + m∠2 = 180°
(8x + 3)° + (5x - 18)° = 180°
8(15)° + 3° + 5(15)° - 18° = 180°
120° + 3° + 75° - 18° = 180°
180° = 180° (True statement).
m∠2 + m∠3 = 90°
(5x - 18)° + 33° = 90°
5(15)° - 18° + 33° = 90°
75° - 18° + 33° = 90°
90° = 90° (True statement).
