Recall that two angles are complements of each other if their sum is 90degrees. Two angles are supplements of each other if their sum is 180degrees. Find the measure of each angle.
One angle is three times its supplement increased by 76degrees. Find the measures of the two supplementary angles.

To find the measures of the two supplementary angles, let's define the angles and set up an equation based on the problem statement.

### Step 1: Define the angles
- Let the measure of one angle be \( x \) degrees.
- Since the angles are supplementary, the measure of the other angle is \( 180^\circ - x \).

### Step 2: Set up the equation
According to the problem, one angle is three times its supplement increased by 76 degrees. This translates to the equation:
\[
x = 3(180^\circ - x) + 76^\circ
\]

### Step 3: Solve the equation
First, expand and simplify the equation:
\[
x = 540^\circ - 3x + 76^\circ
\]
\[
x = 616^\circ - 3x
\]

Next, add \( 3x \) to both sides to combine like terms:
\[
4x = 616^\circ
\]

Now, divide by 4 to solve for \( x \):
\[
x = \frac{616^\circ}{4} = 154^\circ
\]

### Step 4: Find the other angle
Since the angles are supplementary, the other angle is:
\[
180^\circ - 154^\circ = 26^\circ
\]

### Final Answer
The measures of the two supplementary angles are:
- One angle: \( 154^\circ \)
- The other angle: \( 26^\circ \)

Would you like details on any step? Here are five related questions to consider:

1. What are complementary angles, and how do they differ from supplementary angles?
2. How would the equation change if the problem involved complementary angles instead?
3. Can you provide a different scenario where you solve for angles using the same concept?
4. How does the relationship between angles affect the type of triangle formed?
5. What is the geometric significance of supplementary angles in quadrilaterals?

**Tip:** When dealing with angle problems, it's helpful to sketch a quick diagram to visualize the relationships between the angles.

Explore how to find the measures of supplementary angles using an example problem. Learn step-by-step how to set up and solve equations involving supplementary angles. Suitable for high school students studying geometry.

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