Question

complete the proof that ( mangle g + mangle k + mangle ghk = 180^circ ) without using the triangle angle sum theorem.

(image of a triangle ( gkh ) with a line ( hi ) and a line ( hj ) parallel to ( gk ))

statement	reason
2 ( angle g cong angle ihj )	converse of alternate interior angles theorem
3 ( angle k cong angle jhk )	alternate interior angles theorem
4 ( mangle ihk = mangle ihj + mangle jhk )
5 ( mangle ihk = mangle g + mangle k )	additive property of angle measure
6 ( mangle ihk + mangle ghk = 180^circ )
7 ( mangle g + mangle k + mangle ghk = 180^circ )

(reasons for steps 4 and 6 are to be filled, with options like additive property of angle measure, angles forming a linear pair sum to ( 180^circ ), etc.)

Explanation:

Step1: Analyze Statement 6

We know that \( \angle IHK \) and \( \angle GHK \) form a linear pair (they are adjacent angles on a straight line). By the definition of a linear pair, angles forming a linear pair sum to \( 180^\circ \). So the reason for \( m\angle IHK + m\angle GHK = 180^\circ \) is "Angles forming a linear pair sum to \( 180^\circ \)".

Step2: Analyze Statement 7

From Statement 5, we have \( m\angle IHK = m\angle G + m\angle K \). Substitute \( m\angle IHK \) in Statement 6 with \( m\angle G + m\angle K \). So \( (m\angle G + m\angle K) + m\angle GHK = 180^\circ \), which simplifies to \( m\angle G + m\angle K + m\angle GHK = 180^\circ \) by substitution (using the result from Statement 5 in Statement 6).

Answer:

For Statement 6, the reason is "Angles forming a linear pair sum to \( 180^\circ \)". For Statement 7, the reason is "Substitution Property (using \( m\angle IHK = m\angle G + m\angle K \) from Statement 5 into Statement 6)". The final proof shows \( m\angle G + m\angle K + m\angle GHK = 180^\circ \) is established.