Question

statements 1. 2. ∠mpa is complementary to ∠ram. 3. 4. m∠map = 90° 5. m∠ram + m∠map + m∠nap = 180° 6. m∠ram + 90° + m∠nap = 180° 7. 8. 9. ∠mpa ≅ ∠nap 10. mp || rn reasons 1. given 2. given 3. perpendicular lines intersect to form right angles. 4. 5. 6. 7. subtraction property 8. definition of complementary angles 9. 10. a. (6) transitive property (7) m∠ram + m∠nap = 90° (9) complementary angles are equal. b. (7) m∠ram + m∠nap = 90° (9) definition of complementary angles (10) definition of parallel lines c. (3) m∠ram + m∠nap = 90° (7) m∠ram = 90° (9) vertical angles are equal. d. (1) (overline{ma}perpoverline{pa}) (9) if two angles are complementary to the same angle, then they are congruent. (10) if two lines form congruent alternate - interior angles with a transversal, then the lines are parallel. e. (3) ∠map is a right angle. (5) betweenness of rays (9) substitution property 24. select the correct proof from the options listed. given: (overline{am}paralleloverline{rz};angle rcongangle z) prove: ∠ram ≅ ∠zma a. statements 1. (overline{am}paralleloverline{rz};angle rcongangle 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: Analyze the first - part proof

We know that if two lines are perpendicular, the angle between them is a right - angle. So for the first proof, since perpendicular lines intersect to form right angles, we can state that $\overline{MA}\perp\overline{PA}$ (to complete the statement in step 3). Also, if two angles are complementary to the same angle, they are congruent, which is the reason for $\angle MPA\cong\angle NAP$ (step 9). And if two lines form congruent alternate interior angles with a transversal, then the lines are parallel (step 10).

Step2: Analyze the second - part proof

Given $\overline{AM}\parallel\overline{RZ}$ and $\angle R\cong\angle Z$. $\angle R$ and $\angle RAM$ are not vertical angles, and $\angle Z$ and $\angle ZMA$ are not vertical angles. The correct way is to use the property of alternate interior angles. Since $\overline{AM}\parallel\overline{RZ}$, $\angle R$ and $\angle RMA$ are alternate interior angles and $\angle Z$ and $\angle ZMA$ are alternate interior angles. And given $\angle R\cong\angle Z$, by substitution, $\angle RAM\cong\angle ZMA$.

For the first proof, the correct statements and reasons are as follows:
(3) $\overline{MA}\perp\overline{PA}$; (9) If two angles are complementary to the same angle, then they are congruent; (10) If two lines form congruent alternate interior angles with a transversal, then the lines are parallel.

Answer:

D. (1) $\overline{MA}\perp\overline{PA}$; (9) If two angles are complementary to the same angle, then they are congruent; (10) If two lines form congruent alternate interior angles with a transversal, then the lines are parallel.