Question

ab is parallel to de. complete the proof that when a transversal crosses parallel lines, corresponding angles are congruent. this proof uses the following theorem: when a transversal crosses parallel lines, alternate interior angles are congruent. statement reason 1 ac || df given. 2 m∠w + m∠x = 180° these angles form a linear pair. 3 m∠y + m∠z = 180° these angles form a linear pair. 4 m∠w + m∠x = substitution (2, 3). 5 m∠z = m∠y alternate interior angles of parallel lines are congruent. 6 m∠w + m∠x = substitution (4, 5). 7 m∠w = m∠z subtract m∠x from both sides of the equation (6).

Explanation:

Step1: Identify equal - value expressions

Since \(m\angle w + m\angle x=180^{\circ}\) and \(m\angle y + m\angle z = 180^{\circ}\), for step 4, by substitution, \(m\angle w + m\angle x=m\angle y + m\angle z\).

Step2: Substitute again

From step 5, we know \(m\angle z=m\angle y\). Substituting \(m\angle y\) with \(m\angle z\) in the equation from step 4, for step 6, we get \(m\angle w + m\angle x=m\angle z + m\angle x\).

Answer:

1. \(m\angle y + m\angle z\); 6. \(m\angle z + m\angle x\)