Question

complete the proof that ∠gkh ≅ ∠ikj.
diagram: two lines intersecting at point k, with points labeled g, h, j, i (lines g-i and h-j intersect at k)

statement	reason
2 ( mangle hkj + mangle ikj = 180^circ )
3 ( mangle gkh + mangle hkj = mangle hkj + mangle ikj )
4 ( mangle gkh = mangle ikj )
5 ( angle gkh cong angle ikj )

Explanation:

Step1: Identify the reason for \( m\angle GKH + m\angle HKJ = 180^\circ \)

These two angles form a linear pair (they are adjacent and their non - common sides form a straight line), so the reason is "Linear Pair Postulate" (angles in a linear pair are supplementary, i.e., their measures add up to \( 180^\circ \)).

Step2: Identify the reason for \( m\angle HKJ + m\angle IKJ = 180^\circ \)

Similarly, \( \angle HKJ \) and \( \angle IKJ \) form a linear pair. So the reason is also "Linear Pair Postulate".

Step3: Identify the reason for \( m\angle GKH + m\angle HKJ = m\angle HKJ + m\angle IKJ \)

We are setting the two equal expressions (from Step 1 and Step 2) equal to each other. This is the "Substitution Property of Equality" (if \( a = c \) and \( b = c \), then \( a = b \); here \( a=m\angle GKH + m\angle HKJ \), \( b = m\angle HKJ + m\angle IKJ \), and \( c = 180^\circ \)).

Step4: Identify the reason for \( m\angle GKH = m\angle IKJ \)

We can subtract \( m\angle HKJ \) from both sides of the equation \( m\angle GKH + m\angle HKJ = m\angle HKJ + m\angle IKJ \). This is the "Subtraction Property of Equality" (if \( a + c=b + c \), then \( a = b \)).

Step5: Identify the reason for \( \angle GKH\cong\angle IKJ \)

If two angles have equal measures, then they are congruent. This is the "Definition of Congruent Angles" (angles with equal measures are congruent).

Answer:

1. Reason: Linear Pair Postulate
2. Reason: Linear Pair Postulate
3. Reason: Substitution Property of Equality
4. Reason: Subtraction Property of Equality
5. Reason: Definition of Congruent Angles