Question

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

(image of a triangle ( ghk ) with a line ( hi ) and a parallel line ( hj ) to ( gk ), and a table with statements and reasons:

1. ( overline{hj} parallel overline{gk} ) (given)
2. ( angle g cong angle ihj ) (reason to be filled)
3. ( angle k cong angle jhk ) (alternate interior angles theorem)
4. ( mangle ihk = mangle ihj + mangle jhk ) (reason to be filled)
5. ( mangle ihk = mangle g + mangle k ) (substitution)
6. ( mangle ihk + mangle ghk = 180^circ ) (angles forming a linear pair sum to ( 180^circ ))
7. ( mangle g + mangle k + mangle ghk = 180^circ ) (substitution))

Explanation:

Step1: Identify the Reason for Step 2

Since \(\overline{HJ} \parallel \overline{GK}\) (given), \(\angle G\) and \(\angle IHJ\) are alternate interior angles. By the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, then alternate interior angles are congruent. So the reason for \(\angle G \cong \angle IHJ\) is the Alternate Interior Angles Theorem.

Step2: Identify the Reason for Step 4

The statement \(m\angle IHK = m\angle IHJ + m\angle JHK\) is based on the Angle Addition Postulate, which states that if a point lies in the interior of an angle, the sum of the two smaller angles formed is equal to the measure of the larger angle. So the reason for step 4 is the Angle Addition Postulate.

Answer:

• Step 2 Reason: Alternate Interior Angles Theorem
• Step 4 Reason: Angle Addition Postulate