Question

1. select the correct proof from the options listed.

given: am || rz, ∠r ≅ ∠z
prove: ∠ram ≅ ∠zma
a.
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.

b.
statements

1. am || rz, ∠r ≅ ∠z
2. ∠r and ∠ram are supplementary.
3. ∠z and ∠zma are supplementary.
4. ∠ram ≅ ∠zma

reasons

1. given
2. if two parallel lines are cut by a transversal, then their alternate interior angles are supplementary.
3. if two parallel lines are cut by a transversal, then their alternate interior angles are supplementary.
4. if two angles are supplementary to congruent angles, then they are congruent.

c.
statements

1. am || rz, ∠r ≅ ∠z
2. ∠r and ∠ram are complementary.
3. ∠z and ∠zma are complementary.
4. ∠ram ≅ ∠zma

reasons

1. given
2. if two parallel lines are cut by a transversal, then their interior angles on the same side of the transversal are complementary.
3. if two parallel lines are cut by a transversal, then their interior angles on the same side of the transversal are complementary.
4. if two angles are complementary to congruent angles, then they are congruent.

d.
statements

1. am || rz, ∠r ≅ ∠z
2. ∠r and ∠ram are congruent.
3. ∠z and ∠zma are congruent.
4. ∠ram ≅ ∠zma

reasons

1. given
2. if two parallel lines are cut by a transversal, then their interior angles on the same side of the transversal are congruent.
3. substitution property
4. transitive property

e.
statements

1. am || rz, ∠r ≅ ∠z
2. ∠r and ∠ram are supplementary

reasons

1. given
2. if two parallel lines are cut by a transversal, then their interior angles on the same side of the transversal are supplementary.

Explanation:

Step1: Analyze option A

∠R and ∠RAM are not vertical - angles, so option A is incorrect.

Step2: Analyze option B

If two parallel lines are cut by a transversal, alternate - interior angles are congruent, not supplementary. So option B is incorrect.

Step3: Analyze option C

1. Given \(AM\parallel RZ\) and \(\angle R\cong\angle Z\).

• When two parallel lines \(AM\) and \(RZ\) are cut by a transversal, the interior angles on the same side of the transversal are supplementary. So \(\angle R+\angle RAM = 180^{\circ}\) and \(\angle Z+\angle ZMA=180^{\circ}\).
• Since \(\angle R\cong\angle Z\), if \(\angle R + \angle RAM=180^{\circ}\) and \(\angle Z+\angle ZMA = 180^{\circ}\), we can use the property: if two angles are supplementary to congruent angles, then they are congruent. So \(\angle RAM\cong\angle ZMA\). Option C is correct.

Step4: Analyze option D

If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary, not congruent. So option D is incorrect.

Step5: Analyze option E

The reasoning in step 2 of option E has the wrong angle - relationship. When two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary, but the way it is used in option E is incorrect.

Answer:

C