- Identify that angles 3 and 5 are same-side interior angles, which are supplementary and sum to \(180^\circ\).
- Set up the equation \((3x + 4) + (2x + 11) = 180\) to solve for \(x\).
- Calculate \(x = 33\) and substitute it into the expression for \(m \angle 5\) to find \(m \angle 5 = 77^\circ\).
- Select the matching choice from the provided options. The final value is \(\boxed{77}\).
Explanation
- Analyze the relationship between the angles
The problem provides two expressions for the measures of two angles, \(m \angle 3 = (3x + 4)^\circ\) and \(m \angle 5 = (2x + 11)^\circ\). We are told that these are same-side interior angles. In geometry, when two parallel lines are intersected by a transversal, the same-side interior angles are supplementary, meaning their sum is \(180^\circ\). Therefore, we can set up the equation:
\((3x + 4) + (2x + 11) = 180\)
- Solve for x
To solve for \(x\), combine the terms in the equation:
\(5x + 15 = 180\)
Subtract 15 from both sides:
\(5x = 165\)
Divide by 5:
\(x = 33\)
- Calculate the measure of Angle 5
Now that we have the value of \(x\), we substitute it back into the expression for \(m \angle 5\):
\(m \angle 5 = 2(33) + 11 = 66 + 11 = 77^\circ\).
This matches one of the provided choices. The fill-in-the-blank equation in the problem statement intended to describe the sum of the angles, and the correct value for \(m \angle 5\) is 77.
Examples
In civil engineering and architecture, understanding the properties of angles formed by transversals is crucial for ensuring that structural elements like beams or pipes, which are meant to be parallel, are installed correctly. By measuring angles and using the supplementary property of same-side interior angles, engineers can verify if two components are perfectly parallel, ensuring the integrity and stability of the entire construction project. In this specific case, solving for the angle measure ensures that the angles meet the design specifications of \(180^\circ\) for proper alignment.