Final answer:
The students' questions are addressed using geometric principles such as the Triangle Sum Theorem and the Linear Pair Postulate, demonstrating how to prove angle congruences and relationships in triangles.
Step-by-step explanation:
To address the geometry problems presented by the student, we shall use fundamental geometric principles and theorems.
1. Proving ∠B ≅ ∠C in a right triangle with one 45° angle
Triangle Sum Theorem: The sum of angles in a triangle is 180°.
In ΔABC, m∠A + m∠B + m∠C = 180°.
Since ∠A is a right angle, m∠A = 90° and we are given m∠B = 45°.
So, m∠C = 180° - 90° - 45° = 45°. Therefore, ∠B ≅ ∠C.
2. Proving ∠Y is an obtuse angle
Linear Pair Postulate: Angles forming a linear pair sum to 180°.
m∠X + m∠Y = 180° because ∠X and ∠Y are a linear pair.
Substitute the given expressions: (4a + 2) + (21a + 3) = 180°.
Solve for a, then calculate m∠Y. If m∠Y > 90°, then ∠Y is obtuse.
3. Proving ∠A and ∠C are equal
Complementary Angles: Two angles are complementary if their measures sum to 90°.
Since ∠A and ∠B are complementary: m∠A + m∠B = 90°.
Since ∠B and ∠C are also complementary: m∠B + m∠C = 90°.
Therefore, m∠A must equal m∠C, which means ∠A ≅ ∠C.
4. Proving ∠N and ∠P are complementary angles in ΔMNP
Triangle Sum Theorem: The sum of angles in a triangle is 180°.
In ΔMNP, m∠M + m∠N + m∠P = 180°.
Since ∠M is a right angle, m∠M = 90°.
Thus, m∠N + m∠P = 90°, indicating ∠N and ∠P are complementary.