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Rule

Congruent Supplements Theorem

If two angles are supplementary to the same angle, then they are congruent.

Based on the diagram above, the following relation holds true.

If $∠ 1$ and $∠ 2$ are supplementary and $∠ 3$ and $∠ 2$ are supplementary, then $∠ 1 ≅ ∠ 3 .$

Proof

Consider two pairs of supplementary angles:

∠ 1

and

∠ 2,

and

∠ 3

and

∠ 2 .

By the definition of supplementary angles, the sum of the angle measures in each of these pairs is

18 0^{\circ} .

m ∠ 1 + m ∠ 2 = 18 0^{\circ} m ∠ 3 + m ∠ 2 = 18 0^{\circ}

The right-hand sides of the equations are the same. Therefore, by the Transitive Property of Equality the left-hand sides must also be equal.

{m ∠ 1 + m ∠ 2 = 18 0^{\circ} m ∠ 3 + m ∠ 2 = 18 0^{\circ} ⇓ m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 2

Finally, by using the Subtraction Property of Equality

m ∠ 2

can be subtracted from each side of the obtained equation.

m ∠ 1 + m ∠ 2 = m ∠ 3 + m ∠ 2 ⇕ m ∠ 1 = m ∠ 3

The measures of

∠ 1

and

∠ 3

are the same, which indicates that these angles are congruent.

$∠ 1 ≅ ∠ 3$

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Proof