________________________________________________
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°.
________________________________________________
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°.
________________________________________________
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°.
________________________________________________