What conjecture can you make about the two acute angles of a right triangle? Explain your reasoning. Complete a flow proof for the Triangle Sum Theorem.
The correct answer and explanation is :
Conjecture about the Two Acute Angles of a Right Triangle:
Conjecture: The two acute angles in a right triangle are complementary.
Explanation:
In any triangle, the sum of the three interior angles is always 180 degrees. This is known as the Triangle Sum Theorem. In a right triangle, one of the angles is a right angle, meaning it measures exactly 90 degrees. That leaves 90 degrees to be shared between the two remaining angles — which must both be acute (less than 90 degrees) because no triangle can have more than one right or obtuse angle.
Therefore, if the right angle takes up 90°, and the total sum must be 180°, then:
Acute Angle A + Acute Angle B = 180° – 90° = 90°
This proves that the two acute angles in a right triangle must add up to 90 degrees, meaning they are complementary by definition.
Flow Proof for the Triangle Sum Theorem:
| Statement | Reason |
|---|---|
| 1. Draw triangle ABC | Given |
| 2. Draw line DE parallel to base AC through point B | Through a point outside a line, there is exactly one parallel |
| 3. ∠1 ≅ ∠4 and ∠3 ≅ ∠5 | Alternate interior angles (parallel lines) |
| 4. ∠4 + ∠2 + ∠5 = 180° | Straight angle on line DE |
| 5. ∠1 + ∠2 + ∠3 = 180° | Substitution using congruent angles (∠1 ≅ ∠4, ∠3 ≅ ∠5) |
| 6. The sum of interior angles in triangle ABC is 180° | Definition of triangle interior angle sum |
Conclusion:
Using the Triangle Sum Theorem and the definition of a right triangle, we can logically deduce that the remaining two angles in a right triangle are always acute and must add up to 90 degrees. Thus, the conjecture is true: the two acute angles in a right triangle are complementary. This property is fundamental in geometry and is often used when solving for missing angles in triangles.
