Question

given: $overline{de} perp overline{ef}$, $angle dfe$ is complementary to $angle deg$. prove: $overleftrightarrow{df} parallel overleftrightarrow{gh}$
diagram: horizontal lines with points d, f (top) and g, e, h (bottom); segments de, fe, with a right angle at e

1. click here to insert
2. click here to insert
3. $angle def$ is a right angle.
4. click here to insert
5. given
6. given
7. perpendicular lines intersect to form right angles.
8. definition of a right angle

Explanation:

1. Identify given info: \( DE \perp EF \), \( \angle DFE \) comp. to \( \angle DEG \).
2. Perpendicular lines: \( \angle DEF = 90^\circ \).
3. Triangle angle sum: \( \angle DFE + \angle EDF = 90^\circ \).
4. Given complementarity: \( \angle DFE + \angle DEG = 90^\circ \).
5. Set equal & simplify: \( \angle EDF = \angle DEG \).
6. Alternate interior angles: \( DF \parallel GH \).

Answer:

To prove \( \overleftrightarrow{DF} \parallel \overleftrightarrow{GH} \), we can follow these steps:

Step 1: Identify the given information

We are given that \( \overline{DE} \perp \overline{EF} \) and \( \angle DFE \) is complementary to \( \angle DEG \).

Step 2: Use the definition of perpendicular lines

Since \( \overline{DE} \perp \overline{EF} \), \( \angle DEF = 90^\circ \) (perpendicular lines intersect to form right angles).

Step 3: Use the definition of a right angle

By the definition of a right angle, \( \angle DFE + \angle EDF = 90^\circ \) (since the sum of the angles in a triangle is \( 180^\circ \), and \( \angle DEF = 90^\circ \), the remaining two angles must be complementary).

Step 4: Use the given complementary angles

We know \( \angle DFE \) is complementary to \( \angle DEG \), so \( \angle DFE + \angle DEG = 90^\circ \).

Step 5: Set the two complementary angle sums equal

From Step 3 and Step 4, we have:
\[
\angle DFE + \angle EDF = \angle DFE + \angle DEG
\]
Subtracting \( \angle DFE \) from both sides, we get:
\[
\angle EDF = \angle DEG
\]

Step 6: Use the alternate interior angles theorem

Since \( \angle EDF \) and \( \angle DEG \) are alternate interior angles and they are equal, by the alternate interior angles theorem, \( \overleftrightarrow{DF} \parallel \overleftrightarrow{GH} \).