If two interior angles on the same side of a transversal intersecting two parallel lines are in the ratio `2:3,` then the measure of the larger angle is

Options

A. `36^@`
B. `72^@`
C. `108^@`
D. `54^@`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have two interior angles on the same side of a transversal intersecting two parallel lines, and these angles are in the ratio of 2:3. We need to find the measure of the larger angle.

### Step 2: Define the angles
Let the two angles be represented as:
- Angle 1 = 2x
- Angle 2 = 3x

### Step 3: Use the property of interior angles
Since these angles are on the same side of the transversal, their sum is equal to 180 degrees:
\[ \text{Angle 1} + \text{Angle 2} = 180^\circ \]
Substituting the expressions for the angles:
\[ 2x + 3x = 180^\circ \]

### Step 4: Combine like terms
Combine the terms on the left:
\[ 5x = 180^\circ \]

### Step 5: Solve for x
To find the value of x, divide both sides by 5:
\[ x = \frac{180^\circ}{5} \]
Calculating this gives:
\[ x = 36^\circ \]

### Step 6: Find the measures of the angles
Now, substitute x back into the expressions for the angles:
- Angle 1 = 2x = 2(36) = 72 degrees
- Angle 2 = 3x = 3(36) = 108 degrees

### Step 7: Identify the larger angle
The larger angle is:
\[ \text{Angle 2} = 108^\circ \]

### Conclusion
Thus, the measure of the larger angle is **108 degrees**.

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