given: <1 and <2 and supplements, <3 and <4 are supplements, and <1 ≅ <4
prove: <2 ≅ <3

Two angles their sum is 180°
The supplement of an acute angle is an obtuse angle and vice versa
The supplement of a right angle is a right angle
If one angle is supplement to two other angles, then the two angles are equal in measure (if 50° is supplement to angles X and Y, then X and y must be 130°)

Supplement angles of congruent angles are congruent (If angles X and Y are supplement to angles A , B and angles A , B are congruent, then angles X and Y are congruent

∵ ∠1 and ∠2 are supplementary

∴ m∠1 + m∠2 = 180° ⇒ (1)

∵ ∠3 and ∠4 are supplementary

∴ m∠3 + m∠4 = 180° ⇒ (2)

We can equate the left hand sides of (1) and (2) because the right hand sides are equal

∴ m∠1 + m∠2 = m∠3 + m∠4 ⇒ (3)

∵ ∠1 ≅ ∠4

∴ m∠1 = m∠4

- Substitute m∠4 by m∠1 in (3)

∴ m∠1 + m∠2 = m∠3 + m∠1

- Subtract m∠1 from both sides

∴ m∠2 = m∠3

∴ ∠2 ≅ ∠3

By using the fact of supplement angles of congruent angles are congruent, we proved that ∠2 ≅ ∠3

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