The Exterior Angle of a Triangle

In a triangle, the exterior angle is equal to the sum of the two opposite interior angles. Given that one of the exterior angles is 140 degrees and one of the interior opposite angles is 75 degrees, we can find the other two angles of the triangle. Let's go through the process step-by-step.

Step-by-Step Solution

1. **Understanding the Relationship**: The exterior angle is equal to the sum of the two opposite interior angles. Given that the exterior angle is 140 degrees and one of the interior angles is 75 degrees, we can set up the equation:

Exterior angle Interior opposite angle 1 Interior opposite angle 2

Let the other interior angle be (x). Thus, we can write:

140o 75o (x)

2. **Finding the Other Interior Angle**: To find (x), we can rearrange the equation:

(x 140o - 75o 65osup>)

The Triangle's Angles

Now we have two of the interior angles: 75o and 65o. To find the third interior angle, we use the fact that the sum of the interior angles of a triangle is 180o:

Sum of interior angles 180o

Therefore, the third angle is:

180o - 75o - 65o 40o

Conclusion

Thus, the three angles of the triangle are:

75o, 65o, and 40o

Alternative Method

For clarity, let's solve another example with slightly different values. Suppose the exterior angle of a triangle is 120 degrees and one of its interior angles is 70 degrees. Let's find the measure of its other interior opposite angle:

Step-by-Step Solution

1. **Understanding the Relationship**: The exterior angle is equal to the sum of the two opposite interior angles. Given that the exterior angle is 120 degrees and one of the interior angles is 70 degrees, we can set up the equation:

Exterior angle Interior opposite angle 1 Interior opposite angle 2

Let the other interior angle be (y). Thus, we can write:

120o 70o (y)

2. **Finding the Other Interior Angle**: To find (y), we can rearrange the equation:

(y 120o - 70o 50o)

Conclusion

Thus, the other interior opposite angle is:

50o

Summary

Understanding the relationship between an exterior angle and its interior opposite angles can help solve various geometric problems. By using the properties of triangles and the given measures, we can accurately determine the unknown angles. Whether the triangle involves a 140-degree exterior angle or a 120-degree exterior angle, the process remains consistent, making it easier to solve such problems.

Additional Tips

To further solidify your understanding, consider the following additional points:

Always remember that the sum of the interior angles of a triangle is 180 degrees. Understand that an exterior angle is equal to the sum of the two opposite interior angles. Practice multiple examples to reinforce your knowledge. Utilize visual aids such as diagrams to better grasp the relationships.

These tips will help you solve similar problems efficiently and accurately.