Question

∠x and ∠z are complementary and ∠x and ∠y are complementary. complete the proof that ∠z ≅ ∠y.
statement | reason
1 ∠x and ∠z are complementary | given
2 ∠x and ∠y are complementary | given
3 m∠x + m∠z = 90° |
4 m∠x + m∠y = 90° | definition of complementary angles
5 m∠x + m∠z = m∠x + m∠y |
6 m∠z = m∠y | properties of addition, subtraction, multiplication, and division
7 ∠z ≅ ∠y | definition of congruence
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not feeling ready yet? these can help:
identify complementary, supplementary, vertical, adjacent, and co. | properties of equality
lesson: complementary angles | lesson: supplementary angles
lesson: vertical angles | lesson: adjacent angles

Explanation:

Step1: Analyze Statement 3

Statement 3 is \( m\angle X + m\angle Z = 90^\circ \). The reason for this is the definition of complementary angles (since \( \angle X \) and \( \angle Z \) are complementary, their measures sum to \( 90^\circ \)). So the reason for statement 3 is "Definition of complementary angles".

Step2: Analyze Statement 5

Statement 5 is \( m\angle X + m\angle Z = m\angle X + m\angle Y \). We know from statements 3 and 4 that \( m\angle X + m\angle Z = 90^\circ \) and \( m\angle X + m\angle Y = 90^\circ \). So we can set them equal because they both equal \( 90^\circ \). This is the "Transitive Property of Equality" (or "Substitution Property", since both equal the same value). So the reason for statement 5 is "Transitive Property of Equality (or Substitution, as both equal \( 90^\circ \))".

Answer:

• For Statement 3: Reason is "Definition of complementary angles"
• For Statement 5: Reason is "Transitive Property of Equality (or Substitution)"