Question

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 :

• A. \(54^\circ\)
• B. \(120^\circ\)
• C. \(108^\circ\)
• D. \(136^\circ\)

Hint:

Property: Interior angles on the same side of a transversal between parallel lines add up to \(180^\circ\). Ratio of angles = 2 : 3. Let angles be \(2x\) and \(3x\). Equation: \(2x + 3x = 180^\circ\). \(5x = 180^\circ \implies x = 36^\circ\). Angles are: \(2x = 2 \times 36^\circ = 72^\circ\) \(3x = 3 \times 36^\circ = 108^\circ\) Larger angle = \(108^\circ\).

Answer 1

The Correct Option is C

Concept: When a transversal intersects two parallel lines, specific relationships exist between the angles formed. Interior angles on the same side of the transversal are supplementary (their sum is \(180^\circ\)). These are also known as consecutive interior angles or same-side interior angles. Step 1: Understand the property of consecutive interior angles If two parallel lines are intersected by a transversal, then the sum of the interior angles on the same side of the transversal is \(180^\circ\). Let the two consecutive interior angles be \(A\) and \(B\). If the lines are parallel, then \(A + B = 180^\circ\). Step 2: Set up the angles based on the given ratio The two interior angles are in the ratio 2 : 3. Let the common factor for the ratio be \(x\). Then the measures of the two angles are \(2x\) and \(3x\). Step 3: Use the supplementary property to form an equation Since these are consecutive interior angles and the lines are parallel, their sum must be \(180^\circ\): \[ 2x + 3x = 180^\circ \] Step 4: Solve for \(x\) \[ 5x = 180^\circ \] \[ x = \frac{180^\circ}{5} \] \[ x = 36^\circ \] Step 5: Calculate the measures of the two angles The first angle is \(2x = 2 \times 36^\circ = 72^\circ\). The second angle is \(3x = 3 \times 36^\circ = 108^\circ\). Step 6: Identify the larger angle Comparing the two angles, \(72^\circ\) and \(108^\circ\), the larger angle is \(108^\circ\). (Check: \(72^\circ + 108^\circ = 180^\circ\), so they are supplementary). The measure of the larger angle is \(108^\circ\).