Question

$\overline{gk} \parallel \overline{hj}$. complete the proof that $m\angle g + m\angle k + m\angle ghk = 180\degree$ without using the triangle angle sum theorem.
image of a geometric figure with points g, k, h, j, i and a table with statements and reasons for a proof

Explanation:

Step1: Analyze ∠G and ∠IHJ

Since \( \overline{GK} \parallel \overline{HJ} \) and \( \overline{GH} \) is a transversal, \( \angle G \) and \( \angle IHJ \) are corresponding angles. By the Corresponding Angles Theorem, corresponding angles are congruent when lines are parallel. So the reason for \( \angle G \cong \angle IHJ \) is "Corresponding Angles Theorem".

Step2: Analyze \( m\angle IHK + m\angle GHK = 180^\circ \)

\( \angle IHK \) and \( \angle GHK \) form a linear pair (they are adjacent and their non - common sides form a straight line). By the Linear Pair Postulate, the sum of the measures of angles in a linear pair is \( 180^\circ \). So the reason for \( m\angle IHK + m\angle GHK = 180^\circ \) is "Linear Pair Postulate".

Step3: Analyze the last step

We know that \( m\angle IHK = m\angle G + m\angle K \) (from step 5) and \( m\angle IHK + m\angle GHK = 180^\circ \) (from step 6). By substituting \( m\angle IHK \) in the second equation with \( m\angle G + m\angle K \), we get \( m\angle G + m\angle K + m\angle GHK = 180^\circ \). The reason is "Substitution" (substituting \( m\angle IHK \) from step 5 into the equation from step 6).

Answer:

• For statement 2: Corresponding Angles Theorem
• For statement 6: Linear Pair Postulate
• For statement 7: Substitution (substituting \( m\angle IHK \) from step 5 into step 6's equation)