Question

100 POINTS PLEASE HELP!!

Answer 1

Answer:

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°

Answer 2

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Solution (1a):

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:

• Given: ∠2 = 65°
• To find: ∠4

Using the 180° angle property:

• ∠2 + ∠4 = 180 (Straight line)
• => 65 + ∠4 = 180
• => ∠4 = 180 - 65
• => ∠4 = 115°

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

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Solution (1b):

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:

• Given: ∠2 = 65°
• To find: ∠7

Using the vertically opposite angles property:

• ∠2 = ∠3 = 65° (Vertically opposite angles)

Using the corresponding angles property:

• ∠3 = ∠7 = 65° (Corresponding angles)
• => ∠7 = 65° (Proved)

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

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Solution (1c):

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:

• Given: ∠2 = 65°
• To find: ∠3

Using the 180° angle property:

• ∠2 + ∠1 = 180
• => 65 + ∠1 = 180
• => ∠1 = 180 - 65
• => ∠1 = 115°

Using the 180° angle property:

• ∠1 + ∠3 = 180
• => 115 + ∠3 = 180
• => ∠3 = 180 - 115
• => ∠3 = 65° (Proved)

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

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