Question

DB is perpendicular to AE at C, AB is congruent to DE, and C is the midpoint of AE. Prove DE is parallel to AB

Answer 1

1.
`\overline{DB}` is perpendicular to
`\overline{AE}` at
`C` (given)

2.
`\angle ACB` and
`\angle DCB` are right angles (perpendicular lines form right angles)

3.
`\triangle ACB` and
`\triangle DCE` are right triangles (a triangle with a right angle is a right triangle)

4.
`C` is the midpoint of
`\overline{AE}` (given)

5.
`\overline{AC} \cong \overline{CE}` (definition of midpoint)

6.
`\overline{AB} \cong \overline{DE}` (given)

7.
`\triangle ACB \cong \triangle ECB` (HL)

8.
`\angle BAC \cong \angle DEC` (CPCTC)

9.
`\overline{DE} \parallel \overline{AB}` (converse of alternate interior angles theorem)