Question

complete the proof below by filling in the blanks. 23. given: $overline{de}perpoverline{ef}$, $angle dfe$ is complementary to $angle deg$. prove: $overline{df}paralleloverline{gh}$ statements reasons 1. 1. given 2. 2. given 3. $angle def$ is a right angle. 3. perpendicular lines intersect to form right angles. 4. 4. definition of a right angle 5. $mangle deg + mangle def + mangle feh=180$ 5. a straight angle measures 180. 6. $mangle deg + 90 + mangle feh=180$ 6. 7. 7. 8. $angle feh$ is complementary to $angle deg$. 8. 9. 9. if two angles are complementary to the same angle, then they are congruent. 10. 10.

Explanation:

Step1: State the first given

$\overline{DE}\perp\overline{EF}$

Step2: State the second given

$\angle DFE$ is complementary to $\angle DEG$

Step3: Define right - angle from perpendicular lines

Since $\overline{DE}\perp\overline{EF}$, $\angle DEF = 90^{\circ}$

Step4: Use straight - angle property

$\angle DEG+\angle DEF+\angle FEH = 180^{\circ}$ (as $\angle GEH$ is a straight angle)

Step5: Substitute the value of $\angle DEF$

Substitute $\angle DEF = 90^{\circ}$ into $\angle DEG+\angle DEF+\angle FEH = 180^{\circ}$, we get $\angle DEG + 90^{\circ}+\angle FEH=180^{\circ}$

Step6: Simplify the angle equation

Subtract 90 from both sides: $\angle DEG+\angle FEH = 90^{\circ}$, so $\angle FEH$ is complementary to $\angle DEG$

Step7: Use the congruence of complementary angles

Since $\angle DFE$ is complementary to $\angle DEG$ and $\angle FEH$ is complementary to $\angle DEG$, then $\angle DFE\cong\angle FEH$

Step8: Use the alternate - interior angles postulate

If $\angle DFE\cong\angle FEH$ (alternate - interior angles), then $\overline{DF}\parallel\overline{GH}$

Answer:

1. $\overline{DE}\perp\overline{EF}$
2. $\angle DFE$ is complementary to $\angle DEG$
3. $\angle DEF = 90^{\circ}$
4. $\angle DEG+\angle DEF+\angle FEH = 180^{\circ}$
5. Substitution ($\angle DEF = 90^{\circ}$)
6. $\angle DEG+\angle FEH = 90^{\circ}$
7. Definition of complementary angles
8. $\angle DFE\cong\angle FEH$
9. $\overline{DF}\parallel\overline{GH}$ (Alternate - interior angles are congruent, then lines are parallel)