The measure of one angle of a triangle is 105°. What do you know about the measures of the other two angles? Check all that apply.

Question

Both angles are acute.
The sum of the measures of the two angles is 75°.
The sum of the measures of the two angles is 155°.
One of the angles is a right angle.
The measure of neither angle is more than 75°.
The measures of the other angles must be 105° too.
The measure of one of the other angles equals 75°.
The measure of one of the other angles must be 45°.

Answer 1

In a triangle, the sum of the measures of all three angles is always 180°. Given that one angle measures 105°, we can determine the measures of the other two angles.

1. The sum of the measures of the other two angles:

   • If one angle is 105°, then the sum of the other two angles is 180° - 105° = 75°.

2. Since the sum of the two angles is 75°, we can analyze the possibilities:

   • Both angles cannot be acute since an acute angle is less than 90°. If both angles were acute, their maximum total would be just under 180°, which would not account for the 105° already present. Thus, at least one of the two angles must be obtuse (greater than 90°).

3. Since the sum of the other angles is 75°, we also know:

   • It is not possible for one of the angles to equal 75°, as that would leave no room for the third angle (which would then have to be 30°, making one angle obtuse, which contradicts the acute requirement).
   • It is also impossible for one of the other angles to be 105° as the angles in a triangle cannot be equal to the angle that has already been established as 105°.

4. The angle measures must be as follows:

   • The two angles could be a combination such that one is greater than 0 and the other adds up to 75°. For example, they could be 30° and 45° (which meet the sum requirement).

Therefore, based on the statements provided:

• The correct statements are:
  • The sum of the measures of the two angles is 75°.
  • The measure of neither angle is more than 75° (since the sum is 75°, each angle must be less than 75°).

The following options are incorrect:

• Both angles are acute (not true, one must be obtuse given 105°).
• The sum of the measures of the two angles is 155° (incorrect as established).
• One of the angles is a right angle (not possible).
• The measures of the other angles must be 105° too (not possible, one is already 105°).
• The measure of one of the other angles equals 75° (not possible as established).
• The measure of one of the other angles must be 45° (not guaranteed).

So, the only applicable statements are:

• The sum of the measures of the two angles is 75°.
• The measure of neither angle is more than 75°.