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.

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.

Since we defined B as the right angle, we know that B = 90^°. We can substitute this value into the Angle Sum Property equation.

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^°.

Prove that the sum of the two acute angles of a right-angled triangle is 90 degrees.

Answer: The sum of the two acute angles of a right-angled triangle is $90^\circ$ .

Step-by-step solution

Step 1: Define the triangle and its angles

Let's consider a right-angled triangle, which we'll call $\triangle ABC$ . In a right-angled triangle, one of the angles is always $90^\circ$ . Let's assume that $\angle B$ is the right angle, so $\angle B = 90^\circ$ . The other two angles, $\angle A$ and $\angle C$ , 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 $180^\circ$ . We can apply this property to our $\triangle ABC$ .

Step 3: Substitute the value of the right angle

Since we defined $\angle B$ as the right angle, we know that $\angle B = 90^\circ$ . 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, $\angle A$ and $\angle C$ , we subtract $90^\circ$ from both sides of the equation. This shows that their sum is $90^\circ$ .

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