100 POINTS PLEASE HELP!! 1. Given: Angle 2 is 65 degrees. Please note that L || M.

(a) What is the angle measurement of Angle 4? Explain the angle relationship used and show your work.
(b) What is the angle measurement of Angle 7? Explain the angle relationship used and show your work.
(c) What is the angle measurement of Angle 3? Explain the angle relationship used and show your work.

a)  ∠2 and ∠4 are a linear pair

∠4 = 115°

b)  ∠2 and ∠7 are alternate exterior angles

∠7 = 65°

c) ∠2 and ∠3 are vertical angles

∠3 = 65°

Step-by-step explanation:

Linear pair : a pair of adjacent angles formed when two lines intersect. The two angles of a linear pair are always supplementary (two angles whose measures add up to 180°)

Alternate exterior angles : when two parallel lines are cut by a transversal (a line that intersects two or more other, often parallel, lines), the resulting alternate exterior angles are congruent.

Vertical angles : a pair of opposite angles formed by intersecting lines. Vertical angles are always congruent.

a)  ∠2 and ∠4 are a linear pair

⇒ ∠2 +∠4 = 180

⇒ 65 + ∠4 = 180

⇒ ∠4 = 180 - 65

⇒ ∠4 = 115°

b)  ∠2 and ∠7 are alternate exterior angles

⇒ ∠2 ≅ ∠7

⇒ ∠7 = 65°

c) ∠2 and ∠3 are vertical angles

⇒ ∠2 ≅ ∠3

⇒ ∠3 = 65°

The measure of ∠4 is 115° because if we use the 180° angle property, the sum of ∠2 and ∠4 must be 180, as it forms a straight line. Seek below to know how to find ∠4 using the 180° angle property.

Reviewing the known clues:

Using the 180° angle property:

We can conclude that the measure of ∠4 is 115°.

There are some other ways to solve for ∠7. However, the easiest property to use here is the Exterior angle property. If we use that property, we can say that ∠7 is 65°, because ∠2 is 65°. For proof, let's use an alternate way to prove this.

Reviewing the known clues:

Using the vertically opposite angles property:

Using the corresponding angles property:

We can conclude that the measure of ∠7 is 65°.

As said in the above solution, ∠2 = ∠3 is equivalent due to vertically opposite angles. To prove this, we will use the 180° angle property. Seek below for proof.

Reviewing the known clues:

Using the 180° angle property:

Using the 180° angle property:

We can conclude that the measure of ∠3 is 65°.