Answer:
1. ∠C ≅ ∠B by definition of congruency
2. m∠Y by definition of obtuse angle
3. ∠C ≅ ∠A by definition of congruency
4. ∠N and ∠P are complementary angles by definition of complementary angles
Explanation:
1. The given parameters are;
Statement
Reason
ΔABC is a right triangle
Given
∠A is a right angle
Given
m∠B = 45°
Given
m∠A + m∠B + m∠C = 180°
Sum of interior angles of a triangle
m∠C = 180° - (m∠A + m∠B)
m∠C = 180° - (90° + 45°) = 45°
Substitution property
m∠C = 45° = m∠B
Substitution property
∠C ≅ ∠B
Definition of congruency
2. Statement
Reason
m∠X = 4·a + 2
Given
m∠Y = 21·a + 3
Given
∠X and ∠Y are linear pair
Given
m∠X + m∠Y = 180°
Sum of angles of linear pair
4·a + 2 + 21·a + 3 = 180°
Substitution property
25·a + 5 = 180°
a = (180 - 5)/25 = 7
m∠Y = 21 × 7 + 3 = 150°
Substitution property
m∠Y = 150° > 90
m∠Y is an obtuse angle
Definition of obtuse angle
3. Statement
Reason
∠A and ∠B are complementary angles
Given
∠A + ∠B = 180°
Definition of complementary angles
∠B and ∠C are complementary angles
Given
∠B + ∠C = 180°
Definition of complementary angles
∠B + ∠C = ∠A + ∠B
Transitive property
∠C = ∠A
Reverse of addition property of equality
∠C ≅ ∠A
Definition of congruency
4. Statement
Reason
ΔMNP is a right triangle
Given
∠M is a right angle
Given
∠M + ∠N + ∠P = 180°
Sum of interior angles of a triangle
∠N + ∠P = 180° - ∠M
Subtraction property of equality
∠N + ∠P = 180° - 90° = 90°
Substitution property of equality
∠N + ∠P = 90°
Substitution property of equality
∠N and ∠P are complementary angles
Definition of complementary angles.