Question

what can be proven if given \\(\angle a\\) and \\(\angle b\\) are supplementary, and \\(\angle b\\) and \\(\angle c\\) are supplementary?

options:

• \\(\angle a \cong \angle b\\)
• \\(\angle a\\) and \\(\angle c\\) are supplementary
• \\(\angle a \cong \angle c\\)
• \\(\angle a\\) and \\(\angle b\\) are vertical

Explanation:

Define supplementary angles

Supplementary angles sum to \(180^\circ\).

$$ m\angle A + m\angle B = 180^\circ $$

$$ m\angle B + m\angle C = 180^\circ $$

Relate the equations

Equate the two expressions.
Using the Algebraic Proof knowledge point:

$$ m\angle A + m\angle B = m\angle B + m\angle C $$

Solve for the relationship

Subtract \(m\angle B\) from both sides.

$$ m\angle A = m\angle C $$

State the geometric conclusion

Equal measures mean congruent angles.

$$ \angle A \cong \angle C $$

Answer:

• \(\angle A \cong \angle B\)
• \(\angle A \cong \angle C\) (Correct answer)
• \(\angle A\) and \(\angle C\) are supplementary
• \(\angle A\) and \(\angle B\) are vertical