Write a two column proof of the Corollary Triangle Sum Theorem. The acute angles of a right triangle are complementary.
m<A + m<B=90degrees
To write a two-column proof for the Corollary Triangle Sum Theorem, which states that the acute angles of a right triangle are complementary, we will need to use the properties and definitions related to angles and triangles. Here's an example of how the proof can be done:
| Statement | Reason |
|----------------|-------------------------------|
| 1. Given | Right triangle ABC |
| 2. m∠C = 90° | Definition of a right triangle |
| 3. ∠A + ∠B = m∠C | Triangle Angle Sum Theorem |
| 4. ∠A + ∠B = 90° | Substitution (from step 2) |
Explanation:
1. We are given a right triangle ABC. The right angle is denoted as ∠C in triangle ABC.
2. By definition, a right triangle has one angle measuring 90 degrees. Therefore, we can state that m∠C = 90°.
3. Using the Triangle Angle Sum Theorem, we know that the sum of the measures of the three angles in any triangle is always equal to 180 degrees. Substituting m∠C = 90° into this theorem, we get ∠A + ∠B = m∠C.
4. Finally, we substitute m∠C = 90° from step 2 into the equation from step 3. This gives us ∠A + ∠B = 90°, which shows that the acute angles of a right triangle are complementary.
Note: This proof is a straightforward application of the Triangle Angle Sum Theorem, which states that the sum of the measures of the three angles in a triangle is always 180 degrees. We use the given information and definitions to derive the conclusion that the acute angles of a right triangle (denoted as ∠A and ∠B) are complementary (summing to 90 degrees).