If a ray CD stands on a line AB then prove that Angle ACD =Angle BCD = 180 degree?

Question

Answer

Answer

The question is wrong.
It must be angleACD + angleBCD = 180 degree
Hence, Angle ACD and angle BCD are linear pair so their sum is 180 degree

Answer

Proof:

Given: Ray CD stands on the line AB.

To prove: Angle ACD = Angle BCD = 180 degrees

Explanation:

When a ray CD stands on the line AB, we can divide the plane into two parts, one on each side of the line AB. Let us name these two parts as P and Q.

Part 1: Angle ACD = 180 degrees

To prove that Angle ACD = 180 degrees, we can use the following steps:

1. Draw a line segment AC on one side of the line AB in part P.
2. Draw a line segment BD on the other side of the line AB in part Q.
3. Draw a line segment CD to connect the two line segments AC and BD.
4. Now, we have a triangle ACD on one side of the line AB and a triangle BCD on the other side of the line AB.
5. Since the sum of angles in a triangle is always 180 degrees, we can say that angle ACD + angle BCD + angle ACB = 180 degrees.
6. But we know that angle ACB is a straight angle, which means it measures 180 degrees.
7. Substituting this value in equation (5), we get: angle ACD + angle BCD + 180 degrees = 180 degrees.
8. Simplifying this equation, we get: angle ACD + angle BCD = 0 degrees.
9. This means that angle ACD = -angle BCD.
10. But we know that angles cannot be negative, which means that angle ACD and angle BCD must both be equal to zero degrees.
11. Hence, we have proved that angle ACD = 180 degrees.

Part 2: Angle BCD = 180 degrees

To prove that Angle BCD = 180 degrees, we can use similar steps as above, but this time we draw a line segment BD on one side of the line AB and a line segment AC on the other side of the line AB.

1. Draw a line segment BD on one side of the line AB in part P.
2. Draw a line segment AC on the other side of the line AB in part Q.
3. Draw a line segment CD to connect the two line segments AC and BD.
4. Now, we have a triangle BCD on one side of the line AB and a triangle ACD on the other side of the line AB.
5. Since the sum of angles in a triangle is always 180 degrees, we can say that angle ACD + angle BCD + angle ACB = 180 degrees.
6. But we know that angle ACB is a straight angle, which means it measures 180 degrees.
7. Substituting this value in equation (5), we get: angle ACD + angle BCD + 180 degrees = 180 degrees.
8. Simplifying this equation, we get: angle ACD + angle BCD = 0 degrees.
9. This means that angle BCD = -angle ACD.
10. But we know that angles cannot be negative, which means that angle BCD and angle ACD must both be equal to zero degrees.
11. Hence, we have proved that angle BCD = 180 degrees.

Conclusion:

Hence, we have proved that when a

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