Question

The proof that hg ≅ eg is shown. Given: g is the midpoint of kf, kh ∥ ef. Prove: hg ≅ eg. Triangles feg and khg are connected at point g. What is the missing reason in the proof?

1) Vert. ∠s are ≅
2) Given
3) Alt. int. ∠s are ≅
4) Def. of midpt.
5) CPCTC

Answer 1

In the provided geometric proof, the missing reason to prove that segments hg and eg are congruent is likely due to the use of alternate interior angles being congruent when a transversal crosses parallel lines.

Step-by-step explanation:

To prove that hg ⋅ eg, we are given that g is the midpoint of kf, which implies that hg and gf are congruent segments, and kh is parallel to ef. When we are given that a line segment is divided into two equal parts by its midpoint, that is an application of the definition of a midpoint (option 4). Also, if kh is parallel to ef and g is a transversal, then the alternate interior angles are congruent, which can be used to show that triangles feg and khg are similar by the Angle-Angle Similarity Postulate.

Once the triangles feg and khg are shown to be similar, we can then apply the CPCTC (Corresponding Parts of Congruent Triangles are Congruent, option 5) to establish the congruence of hg and eg. However, for the proof provided, the missing step seems to pertain to the relationship between the angles, so the correct missing reason is likely that 'Alternate interior angles are congruent' (option 3), assuming that the rest of the proof uses similarity and CPCTC to establish the final congruence of segments hg and eg.