Prove that the sum of the two acute angles of a right-angled triangle is 90 degrees.
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Step-by-Step Solution
Step 1: Define the triangle and its angles
Let's consider a right-angled triangle, which we'll call △ABC. In a right-angled triangle, one of the angles is always 90∘. Let's assume that ∠B is the right angle, so ∠B=90∘. The other two angles, ∠A and ∠C, are acute angles.
Step 2: Apply the Angle Sum Property
The Angle Sum Property of a triangle states that the sum of all interior angles in any triangle is always equal to 180∘. We can apply this property to our △ABC.
Step 3: Substitute the value of the right angle
Since we defined ∠B as the right angle, we know that ∠B=90∘. We can substitute this value into the Angle Sum Property equation.
Step 4: Solve for the sum of acute angles
To find the sum of the two acute angles, ∠A and ∠C, we subtract 90∘ from both sides of the equation. This shows that their sum is 90∘.