Two "angles are supplementary to each other. The larger of two supplementary angles exceeds the smaller by 20°. Then, the smaller angle is.

Options

A. `60^(@)`
B. `80^(@)`
C. `65^(@)`
D. `75^(@)`

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the Variables
Let the larger angle be \( X \) and the smaller angle be \( Y \).

**Hint:** Start by defining the variables for the angles you are working with.

### Step 2: Set Up the First Equation
According to the problem, the larger angle exceeds the smaller angle by 20 degrees. Therefore, we can write the equation:
\[ X = Y + 20 \]
This is our first equation.

**Hint:** Use the information given in the problem to create an equation that relates the two angles.

### Step 3: Set Up the Second Equation
Since the angles are supplementary, their sum is 180 degrees. Thus, we can write the second equation:
\[ X + Y = 180 \]

**Hint:** Remember that supplementary angles add up to 180 degrees, which will help you create another equation.

### Step 4: Substitute the First Equation into the Second Equation
Now, we will substitute the expression for \( X \) from the first equation into the second equation:
\[ (Y + 20) + Y = 180 \]

**Hint:** Substituting one equation into another can simplify the problem and help you solve for one variable.

### Step 5: Simplify the Equation
Combine like terms:
\[ 2Y + 20 = 180 \]

**Hint:** Look for like terms to combine, which will help you isolate the variable.

### Step 6: Solve for \( Y \)
Now, subtract 20 from both sides:
\[ 2Y = 160 \]
Next, divide by 2:
\[ Y = 80 \]

**Hint:** Isolate the variable by performing inverse operations, such as subtraction and division.

### Step 7: Conclusion
The smaller angle \( Y \) is 80 degrees.

**Hint:** After finding the value, make sure to state the answer clearly.

### Final Answer
The smaller angle is \( 80^\circ \).