Like the triangle inequality, can you form a rule that describes the two angles for which a triangle is possible? Can the sum of the two angles be used for framing this rule?
Yes, a rule can be formed using the sum of the two angles.
Rule: For a triangle to be possible with two given angles, the sum of those two angles must be less than 180°.
If the sum of two angles is greater than or equal to 180°, then a triangle with those angles cannot exist.
Reason: The sum of all three angles in a triangle must be exactly 180°.
If two angles already sum to 180° or more, there is no room for a third positive angle.
Explanation
The chapter develops this rule through exploration. The angle sum property states that the sum of all three angles in a triangle is 180°. Therefore, if two angles are given, the third angle would be 180° minus their sum. For this third angle to be positive, the sum of the two given angles must be less than 180°. If the sum equals 180°, the third angle would be 0°, and if the sum exceeds 180°, the third angle would be negative—both impossible for a triangle.
Solution Steps
A rule can be formed using the sum of the two angles.
For a triangle to be possible, the sum of two given angles must be less than 180°.
If the sum is greater than or equal to 180°, a triangle cannot exist.
The angle sum property states that all three angles in a triangle sum to 180°.
The third angle = 180° - (sum of two given angles).
For the third angle to be positive, the sum of two given angles must be less than 180°.
If sum = 180°, third angle = 0° (impossible for a triangle).
If sum > 180°, third angle would be negative (impossible).