Question

complete the proof that ( mangle v + mangle z + mangle vwz = 180^circ ) without using the triangle angle sum theorem.

(image of a triangle ( vzw ) with a line ( wy ) parallel to ( vz ) and a line ( wx ) extending from ( w ), and a table with statements and reasons for a proof)

statement | reason
--- | ---
1 ( overline{vz} parallel overline{wy} ) | given
2 ( angle v cong angle xwy ) | corresponding angles theorem
3 ( angle z cong angle ywz ) | alternate interior angles theorem
4 ( mangle xwz = mangle xwy + mangle ywz ) | additive property of angle measure
5 ( mangle xwz = mangle v + mangle z ) | substitution
6 ( mangle xwz + mangle vwz = 180^circ ) | angles forming a linear pair sum to ( 180^circ )
7 ( mangle v + mangle z + mangle vwz = 180^circ ) | (blank for reason)

Explanation:

Step1: Identify the substitution

From step 5, we know \( m\angle XWZ = m\angle V + m\angle Z \). From step 6, \( m\angle XWZ + m\angle VWZ = 180^\circ \).

Step2: Substitute \( m\angle XWZ \)

Replace \( m\angle XWZ \) in the equation from step 6 with \( m\angle V + m\angle Z \) (from step 5). This is the Substitution Property of Equality, where we substitute an equal quantity for another. So substituting gives \( (m\angle V + m\angle Z) + m\angle VWZ = 180^\circ \), which simplifies to \( m\angle V + m\angle Z + m\angle VWZ = 180^\circ \).

Answer:

Substitution (or Substitution Property of Equality)