Two supplementary angle differ by `20^(@)`. The smaller of the two measures

Options

A. `60^(@)`
B. `80^(@)`
C. `100^(@)`
D. `120^(@)`

Answer

To solve the problem of finding the smaller of two supplementary angles that differ by 20 degrees, we can follow these steps:

### Step 1: Understand the Definition of Supplementary Angles
Supplementary angles are two angles whose sum is 180 degrees.

### Step 2: Set Up the Equations
Let the two angles be \( x \) and \( y \). According to the problem, we have two equations:
1. \( x + y = 180 \) (since they are supplementary)
2. \( x - y = 20 \) (since they differ by 20 degrees)

### Step 3: Solve the Equations
We can solve these equations simultaneously.

First, we can add the two equations:
\[ (x + y) + (x - y) = 180 + 20 \]
This simplifies to:
\[ 2x = 200 \]
Now, divide both sides by 2:
\[ x = 100 \]

### Step 4: Find the Value of \( y \)
Now that we have \( x \), we can substitute it back into one of the original equations to find \( y \). We can use the first equation:
\[ 100 + y = 180 \]
Subtract 100 from both sides:
\[ y = 180 - 100 \]
\[ y = 80 \]

### Step 5: Identify the Smaller Angle
Now we have both angles:
- \( x = 100 \) degrees
- \( y = 80 \) degrees

The smaller angle is \( y \), which is \( 80 \) degrees.

### Final Answer
The smaller of the two supplementary angles is \( 80 \) degrees.

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