Question

janet is designing a frame for a client. she wants to prove to her client that m∠age≅m∠che in her sketch. what is the missing justification in the proof? statement: ab || cd with transversal ef, m∠age≅m∠hgb, m∠hgb≅m∠che, m∠age≅m∠che. justification: given, vertical angles theorem, alternate interior angles theorem, transitive property, reflexive property, corresponding angles theorem, alternate exterior angles theorem

Explanation:

Step1: Recall angle - relationship theorems

We know that when two parallel lines are cut by a transversal, vertical angles are congruent, corresponding angles are congruent, alternate - interior angles are congruent and alternate - exterior angles are congruent. Also, the reflexive property states that an angle is congruent to itself, and the transitive property states that if $\angle A\cong\angle B$ and $\angle B\cong\angle C$, then $\angle A\cong\angle C$.

Step2: Analyze the given statements and justifications

We are given that $\overline{AB}\parallel\overline{CD}$ with transversal $\overline{EF}$, $\angle AGE\cong\angle HGB$ (by the Vertical Angles Theorem), $\angle HGB\cong\angle CHE$ (by the Corresponding Angles Theorem).

Step3: Determine the missing justification

To get from $\angle AGE\cong\angle HGB$ and $\angle HGB\cong\angle CHE$ to $\angle AGE\cong\angle CHE$, we use the Transitive Property of Congruence.

Answer:

Transitive Property