- Identify the relationship between supplementary angles: \(L + M = 180^{\circ}\).
- Use the given difference to set up a system of linear equations: \(L = M + 25^{\circ}\).
- Substitute the value of \(L\) to solve for \(M\): \(2M + 25^{\circ} = 180^{\circ} \implies M = 77.5^{\circ}\).
- Calculate the final value for angle \(L\): \(L = 77.5^{\circ} + 25^{\circ} = 102.5^{\circ}\).
- Final Answer: \(\boxed{102.5^{\circ}}\).
Explanation
- Define the system of equations based on the given information.
We are given two angles, \(L\) and \(M\), which are supplementary. By definition, the sum of supplementary angles is \(180^{\circ}\). This gives us the equation:
\[L + M = 180^{\circ}\]
The problem also states that angle \(L\) is \(25^{\circ}\) larger than angle \(M\). We can represent this relationship as:
\[L = M + 25^{\circ}\]
Now we have a system of two linear equations with two variables:
1) \(L + M = 180^{\circ}\)
2) \(L = M + 25^{\circ}\)
- Solve the system of equations.
To find the measure of angle \(L\), we can substitute the expression for \(L\) from the second equation into the first equation:
\[(M + 25^{\circ}) + M = 180^{\circ}\]
Now, simplify and solve for \(M\):
\[2M + 25^{\circ} = 180^{\circ}\]
\[2M = 180^{\circ} - 25^{\circ}\]
\[2M = 155^{\circ}\]
\[M = \frac{155^{\circ}}{2} = 77.5^{\circ}\]
Now that we have the measure of angle \(M\), we can find the measure of angle \(L\) using the relationship \(L = M + 25^{\circ}\):
\[L = 77.5^{\circ} + 25^{\circ} = 102.5^{\circ}\]
- State the final answer.
Comparing our calculated value of \(L = 102.5^{\circ}\) with the given options:
A \(32.5^{\circ}\)
B \(57.5^{\circ}\)
C \(77.5^{\circ}\)
D \(102.5^{\circ}\)
The calculated measure of angle \(L\) is \(102.5^{\circ}\), which corresponds to option D. The final answer is \(\boxed{102.5^{\circ}}\).
Examples
Supplementary angles are fundamental in construction and design. For example, if a roof rafter is set at an angle to the horizontal beam, the interior angle formed between the rafter and the beam is supplementary to the exterior angle of the roof. Understanding how to solve for these angles allows builders to ensure the roof is pitched correctly, leading to a stable and aesthetically pleasing structure. For instance, if the exterior angle is \(102.5^{\circ}\), the interior angle would be \(180^{\circ} - 102.5^{\circ} = 77.5^{\circ}\).