To prove that in a triangle, if one angle is 90 degrees, then the sum of the other two angles is 90 degrees, we can use the properties of triangles and the concept of supplementary angles.
First, let’s recall that a triangle consists of three angles. The sum of all three angles in any triangle is always equal to 180 degrees. This fundamental property of triangles is stated as:
Angle A + Angle B + Angle C = 180 degrees
Now, if we assume one of these angles, say Angle A, is a right angle, we have:
Angle A = 90 degrees
Substituting this value into the equation above gives us:
90 degrees + Angle B + Angle C = 180 degrees
To find the sum of Angle B and Angle C, we can isolate these angles in the equation:
Angle B + Angle C = 180 degrees – 90 degrees
This simplifies to:
Angle B + Angle C = 90 degrees
Thus, we have demonstrated that if one angle of a triangle is 90 degrees (i.e., the triangle is a right triangle), then the other two angles must indeed add up to 90 degrees. This conclusion is supported by both the properties of triangles and the definition of supplementary angles, confirming that the sum of the angles in any triangle will always be 180 degrees.
In summary, we have proven that if one angle of a triangle is 90 degrees, the remaining two angles will necessarily sum to 90 degrees.