Question

*50 points​**urgent**plz help!!**

In the figure shown, the angles formed when two parallel lines are cut by a transversal are labeled 1 through 8.

Question:

A. Prove that any two vertical angles are congruent.

B. One of the two angles from Part A has a corresponding alternate interior angle. Use the Corresponding Angles Postulate, which says that corresponding angles are congruent, to prove that the angle from Part A and its alternate interior angle are congruent.

C. How can you use the proof from Part A and the proof from Part B to prove that alternate exterior angles are congruent?

Answer 1

See explanation

Explanation:

A. Prove that angles 1 and 3 are vertical.

1.
`\angle 1, \angle 2` - linear pair of angles

2.
`m\angle 1+m\angle 2 =180^(\circ)` - linear pair of angles are supplementary

3.
`\angle 2, \angle 3` - linear pair of angles

4.
`m\angle 2+m\angle 3 =180^(\circ)` - linear pair of angles are supplementary

5.
`m\angle 1+m\angle 3 =m\angle 2+m\angle 3` - substitution property of equality

6.
`m\angle 1 =m\angle 3` - subtraction property of equality

7.
`\angle 1\cong \angle 3` - definition of congruent angles.

So, any two vertical angles are congruent.

B. Prove that angle 3 and angle 5 are congruent.

1.
`\angle 1\cong \angle 5` - Corresponding Angles Postulate

2.
`m\angle 1=m\angle 5` - definition of congruent angles

3.
`\angle 1\cong \angle 3` - vertical angles

4.
`m\angle 1=m\angle 3` - definition of congruent angles

5.
`m\angle 3=m\angle 5` - substitution property of equality

6.
`\angle 3\cong \angle 5` - definition of congruent angles

So, alternate interior angles are congruent

C. Prove that angle 1 and angle 7 are congruent.

1.
`\angle 1\cong \angle 5` - Corresponding Angles Postulate

2.
`m\angle 1=m\angle 5` - definition of congruent angles

3.
`\angle 5\cong \angle 7` - vertical angles

4.
`m\angle 5=m\angle 7` - definition of congruent angles

5.
`m\angle 1=m\angle 7` - substitution property of equality

6.
`\angle 1\cong \angle 7` - definition of congruent angles

So, alternate exterior angles are congruent