Given: Angles A, B, And C Are Supplementary, { M \angle A = 48 $}$, And { M \angle B = 84 $}$.Prove: { M \angle A = M \angle C $}$.Statement: 1. { \angle A, \angle B, \angle C $}$ Are Supplementary. 2.
Introduction
In geometry, supplementary angles are two or more angles whose measures add up to 180 degrees. Given that angles A, B, and C are supplementary, and the measures of angles A and B are provided, we aim to prove that the measure of angle A is equal to the measure of angle C.
Given Information
- Angles A, B, and C are supplementary.
- The measure of angle A (m ∠A) is 48 degrees.
- The measure of angle B (m ∠B) is 84 degrees.
Objective
To prove that the measure of angle A (m ∠A) is equal to the measure of angle C (m ∠C).
Proof
Since angles A, B, and C are supplementary, their measures add up to 180 degrees. We can express this as an equation:
m ∠A + m ∠B + m ∠C = 180
Substituting the given values for m ∠A and m ∠B, we get:
48 + 84 + m ∠C = 180
Combine the constants:
132 + m ∠C = 180
Subtract 132 from both sides:
m ∠C = 48
This result shows that the measure of angle C is equal to the measure of angle A.
Conclusion
We have successfully proven that the measure of angle A (m ∠A) is equal to the measure of angle C (m ∠C). This result is a direct consequence of the supplementary nature of angles A, B, and C, and the given measures of angles A and B.
Supplementary Angles: Key Concepts
- Supplementary angles: Two or more angles whose measures add up to 180 degrees.
- Angle measure: The size or magnitude of an angle, usually expressed in degrees.
- Proof: A logical argument that demonstrates the truth of a statement or theorem.
Real-World Applications
Understanding supplementary angles has numerous real-world applications, including:
- Architecture: Designing buildings and structures that require precise angle measurements.
- Engineering: Calculating angles in mechanical systems, such as gears and pulleys.
- Navigation: Determining directions and angles in aviation, maritime, and land-based navigation.
Example Problems
- If angles X, Y, and Z are supplementary, and the measure of angle X is 60 degrees, find the measure of angle Z.
- In a triangle, angles A, B, and C are supplementary. If the measure of angle A is 120 degrees, find the measure of angle C.
Solutions
- Since angles X, Y, and Z are supplementary, their measures add up to 180 degrees. We can express this as an equation:
m ∠X + m ∠Y + m ∠Z = 180
Substituting the given value for m ∠X, we get:
60 + m ∠Y + m ∠Z = 180
Combine the constants:
60 + m ∠Y + m ∠Z = 180
Subtract 60 from both sides:
m ∠Y + m ∠Z = 120
Since angles Y and Z are supplementary, their measures add up to 180 degrees. We can express this as an equation:
m ∠Y + m ∠Z = 180
Substituting the result from the previous equation, we get:
120 = 180
This is a contradiction, so we must re-examine our assumptions. Let's assume that angles X, Y, and Z are not supplementary. Then, their measures do not add up to 180 degrees.
- Since angles A, B, and C are supplementary, their measures add up to 180 degrees. We can express this as an equation:
m ∠A + m ∠B + m ∠C = 180
Substituting the given value for m ∠A, we get:
120 + m ∠B + m ∠C = 180
Combine the constants:
120 + m ∠B + m ∠C = 180
Subtract 120 from both sides:
m ∠B + m ∠C = 60
Since angles B and C are supplementary, their measures add up to 180 degrees. We can express this as an equation:
m ∠B + m ∠C = 180
Substituting the result from the previous equation, we get:
60 = 180
This is a contradiction, so we must re-examine our assumptions. Let's assume that angles A, B, and C are not supplementary. Then, their measures do not add up to 180 degrees.
Final Thoughts
Q: What are supplementary angles?
A: Supplementary angles are two or more angles whose measures add up to 180 degrees.
Q: How do you find the measure of a supplementary angle?
A: To find the measure of a supplementary angle, you can use the following formula:
m ∠A + m ∠B + m ∠C = 180
where m ∠A, m ∠B, and m ∠C are the measures of the supplementary angles.
Q: What is the relationship between supplementary angles and the sum of their measures?
A: The sum of the measures of supplementary angles is always 180 degrees.
Q: Can you give an example of supplementary angles?
A: Yes, here's an example:
- Angle A: 60 degrees
- Angle B: 120 degrees
- Angle C: 0 degrees (since the sum of the measures is 180 degrees)
Q: How do you prove that the measure of angle A is equal to the measure of angle C?
A: To prove that the measure of angle A is equal to the measure of angle C, you can use the following steps:
- Write an equation representing the sum of the measures of the supplementary angles: m ∠A + m ∠B + m ∠C = 180
- Substitute the given values for m ∠A and m ∠B: 48 + 84 + m ∠C = 180
- Combine the constants: 132 + m ∠C = 180
- Subtract 132 from both sides: m ∠C = 48
Q: What are some real-world applications of supplementary angles?
A: Supplementary angles have numerous real-world applications, including:
- Architecture: Designing buildings and structures that require precise angle measurements.
- Engineering: Calculating angles in mechanical systems, such as gears and pulleys.
- Navigation: Determining directions and angles in aviation, maritime, and land-based navigation.
Q: Can you give some example problems involving supplementary angles?
A: Yes, here are some example problems:
- If angles X, Y, and Z are supplementary, and the measure of angle X is 60 degrees, find the measure of angle Z.
- In a triangle, angles A, B, and C are supplementary. If the measure of angle A is 120 degrees, find the measure of angle C.
Q: How do you solve problems involving supplementary angles?
A: To solve problems involving supplementary angles, you can use the following steps:
- Write an equation representing the sum of the measures of the supplementary angles.
- Substitute the given values for the measures of the angles.
- Combine the constants.
- Subtract the constants from both sides to isolate the measure of the unknown angle.
Q: What are some common mistakes to avoid when working with supplementary angles?
A: Some common mistakes to avoid when working with supplementary angles include:
- Failing to recognize that supplementary angles have a sum of 180 degrees.
- Not using the correct formula to find the measure of a supplementary angle.
- Not checking the units of the measures of the angles.
Q: How do you check your work when solving problems involving supplementary angles?
A: To check your work when solving problems involving supplementary angles, you can use the following steps:
- Verify that the sum of the measures of the supplementary angles is 180 degrees.
- Check that the units of the measures of the angles are consistent.
- Review the steps you took to solve the problem to ensure that you used the correct formula and procedures.