Question

∠x and ∠z are complementary and ∠x and ∠y are complementary. complete the proof that ∠y ≅ ∠z.

statement	reason
2. ∠x and ∠y are complementary	given
3. ( mangle x + mangle z = 90^circ )	definition of complementary angles
4. ( mangle x + mangle y = 90^circ )	definition of complementary angles
5. ( mangle x + mangle z = mangle x + mangle y )
6. ( mangle y = mangle z )	properties of addition, subtraction, multiplication, and division
7. ( angle y cong angle z )	definition of congruence

Explanation:

Step1: Identify the equality of right-hand sides

From steps 3 and 4, we have \( m\angle X + m\angle Z = 90^\circ \) and \( m\angle X + m\angle Y = 90^\circ \). So both \( m\angle X + m\angle Z \) and \( m\angle X + m\angle Y \) are equal to \( 90^\circ \), which means they are equal to each other by the Transitive Property of Equality (or Substitution Property, since both equal \( 90^\circ \)).

Step2: State the reason for step 5

The reason for \( m\angle X + m\angle Z = m\angle X + m\angle Y \) is that both expressions are equal to \( 90^\circ \) (from steps 3 and 4), so we can substitute (or use the transitive property) to say they are equal. In other words, if \( a = c \) and \( b = c \), then \( a = b \) (Transitive Property of Equality) or substituting \( c \) (which is \( 90^\circ \)) into both.

Answer:

Transitive Property of Equality (or Substitution Property, since both equal \( 90^\circ \))