Question

1. select the correct proof from the options listed. given: am || rz; ∠r ≅ ∠z prove: ∠ram ≅ ∠zma statements 1. am || rz; ∠r ≅ ∠z 2. ∠r and ∠ram are vertical angles. 3. ∠z and ∠zma are vertical angles. 4. ∠ram ≅ ∠zma reasons 1. given 2. definition of vertical angles 3. substitution property 4. vertical angles are congruent.

Explanation:

Step1: State the given information

Given that $\overline{AM}\parallel\overline{RZ}$ and $\angle R\cong\angle Z$.

Step2: Identify vertical - angle pairs

$\angle R$ and $\angle RAM$ are vertical angles, and $\angle Z$ and $\angle ZMA$ are vertical angles. (By the definition of vertical angles)

Step3: Use the property of vertical angles

Vertical angles are congruent. So, $\angle R\cong\angle RAM$ and $\angle Z\cong\angle ZMA$

Step4: Use the substitution property

Since $\angle R\cong\angle Z$ (given), by substitution, $\angle RAM\cong\angle ZMA$

Answer:

The correct proof is:

1. Statements: $\overline{AM}\parallel\overline{RZ}$; $\angle R\cong\angle Z$

• Reasons: Given

1. Statements: $\angle R$ and $\angle RAM$ are vertical angles; $\angle Z$ and $\angle ZMA$ are vertical angles

• Reasons: Definition of vertical angles

1. Statements: $\angle R\cong\angle RAM$; $\angle Z\cong\angle ZMA$

• Reasons: Vertical angles are congruent

1. Statements: $\angle RAM\cong\angle ZMA$

• Reasons: Substitution Property