Question
The measure of one interior angle of a triangle is 45°. What can you conclude about the other two interior angles of the triangle?
One of the other angles must be a right angle.
The sum of the measures of the other two angles is 135°.
The other two angles must be acute angles.
One of the other angles must have a measure greater than 90°.
One of the other angles must be a right angle.
The sum of the measures of the other two angles is 135°.
The other two angles must be acute angles.
One of the other angles must have a measure greater than 90°.
Answers
GPT-4o mini
In a triangle, the sum of the measures of the interior angles is always 180°. Given that one interior angle is 45°, we can determine the sum of the measures of the other two angles as follows:
\[
180° - 45° = 135°
\]
This means that the sum of the measures of the other two angles is 135°. Now, let's analyze the statements one by one:
1. **One of the other angles must be a right angle.** - This is not necessarily true. While it's possible for one of the other angles to be a right angle (which would make the other angle 90°), it is not a requirement.
2. **The sum of the measures of the other two angles is 135°.** - This statement is true, as we calculated.
3. **The other two angles must be acute angles.** - This is not necessarily true. The other two angles could be acute angles, but they also could be one acute angle and one obtuse angle (e.g., 45° and 90°).
4. **One of the other angles must have a measure greater than 90°.** - This is not necessarily true, since both angles could also be less than 90° and still sum up to 135° (e.g., 67.5° and 67.5°).
From this analysis, the only conclusion that can be drawn with certainty is:
**The sum of the measures of the other two angles is 135°.**
\[
180° - 45° = 135°
\]
This means that the sum of the measures of the other two angles is 135°. Now, let's analyze the statements one by one:
1. **One of the other angles must be a right angle.** - This is not necessarily true. While it's possible for one of the other angles to be a right angle (which would make the other angle 90°), it is not a requirement.
2. **The sum of the measures of the other two angles is 135°.** - This statement is true, as we calculated.
3. **The other two angles must be acute angles.** - This is not necessarily true. The other two angles could be acute angles, but they also could be one acute angle and one obtuse angle (e.g., 45° and 90°).
4. **One of the other angles must have a measure greater than 90°.** - This is not necessarily true, since both angles could also be less than 90° and still sum up to 135° (e.g., 67.5° and 67.5°).
From this analysis, the only conclusion that can be drawn with certainty is:
**The sum of the measures of the other two angles is 135°.**