Question

1. name the definition, postulate, theorem, or property justifying each statement.

a. if ∠1 and ∠5 are vertical angles, then ∠1 ≅ ∠5.
b. if ∠p and ∠q are supplementary, then m∠p + m∠q = 180.
c. if m∠e = m∠f, then ∠e ≅ ∠f.
d. if \\( \overline{ab} \cong \overline{cd} \\) and \\( \overline{cd} \cong \overline{ef} \\), then \\( \overline{ab} \cong \overline{ef} \\).
e. if h is a point in the interior of ∠jkl, then m∠jkh + m∠hkl = m∠jkl.

Explanation:

• For part a: The Vertical Angles Theorem states that vertical angles are congruent. Since \(\angle1\) and \(\angle5\) are vertical angles, this theorem justifies \(\angle1\cong\angle5\).
• For part b: By the definition of supplementary angles, if two angles are supplementary, the sum of their measures is \(180^{\circ}\). So, if \(\angle P\) and \(\angle Q\) are supplementary, \(m\angle P + m\angle Q=180\) is justified by the definition of supplementary angles.
• For part c: If the measures of two angles are equal (\(m\angle E = m\angle F\)), then the angles are congruent (\(\angle E\cong\angle F\)) by the definition of congruent angles.
• For part d: The Transitive Property of Congruence states that if \(a\cong b\) and \(b\cong c\), then \(a\cong c\). Here, \(\overline{AB}\cong\overline{CD}\) and \(\overline{CD}\cong\overline{EF}\), so \(\overline{AB}\cong\overline{EF}\) is justified by the Transitive Property of Congruence.
• For part e: The Angle - Addition Postulate states that if a point \(H\) is in the interior of \(\angle JKL\), then \(m\angle JKH+m\angle HKL = m\angle JKL\).

Answer:

a. Vertical Angles Theorem
b. Definition of supplementary angles
c. Definition of congruent angles
d. Transitive Property of Congruence
e. Angle - Addition Postulate