3. We know that in a triangle, the sum of lengths of any two sides is greater than the length of the third side. Is the sum of any angles of a triangle also greater than the third angle? If no, draw a rough sketch to show such a case.
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INTRODUCTION
No, the sum of any two angles in a triangle is always greater than the third angle.
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In a triangle, the sum of all three angles is always 180 degrees. So, we can say that if we take the third angle and subtract it from 180 degrees, the resulting value is the sum of the other two angles.
Let's consider a hypothetical scenario where one of the angles in a triangle is larger than 90 degrees. In this case, if we add the two smaller angles, their sum would be less than 90 degrees. However, the third angle should be greater than 90 degrees since the sum of all three angles must be 180 degrees. This violates the given condition of a triangle.
To illustrate this, let's draw a sketch. Imagine a triangle with one angle marked as 100 degrees, and the other two angles as 40 degrees each. If we add the two smaller angles (40 + 40), we get 80 degrees, which is less than the third angle of 100 degrees. This violates the condition that the sum of any two angles in a triangle must be greater than the third angle. Therefore, such a triangle cannot exist.
FINALLY
In summary, the sum of any two angles in a triangle will always be greater than the third angle, ensuring the validity of the triangular inequality.
No, the sum of any two angles of a triangle is always greater than the third angle. This is a fundamental property of triangles, known as the Triangle Inequality.
To illustrate this property, let's consider a hypothetical triangle ABC, where angle A measures 30 degrees, angle B measures 50 degrees, and angle C measures 100 degrees.
According to the Triangle Inequality, the sum of any two angles of the triangle must be greater than the remaining angle. However, if we add angles A and B (30 degrees + 50 degrees), we get a sum of 80 degrees, which is less than angle C (100 degrees).
Therefore, this triangle does not satisfy the Triangle Inequality, and therefore cannot exist in Euclidean geometry.
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