Question

complete the proof below by filling in the blanks. 21. given: de ⊥ ef, ∠dfe is complementary to ∠deg. prove: df || gh statements reasons 1. 1. given 2. 2. given 3. ∠def is a right angle. 3. perpendicular lines intersect to form right angles. 4. 4. definition of a right angle 5. m∠deg + m∠def + m∠feh = 180 5. a straight angle measures 180. 6. m∠deg + 90 + m∠feh = 180 6. 7. 7. 8. ∠feh is complementary to ∠deg. 8. 9. 9. if two angles are complementary to the same angle, then they are congruent. 10. 10.

Explanation:

Step1: State given 1

$DE\perp EF$

Step2: State given 2

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

Step3: Define right - angle from perpendicular lines

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

Step4: Use straight - angle property

$\angle DEG+\angle DEF+\angle FEH = 180^{\circ}$ (A straight angle measures 180)

Step5: Substitute $\angle DEF = 90^{\circ}$

$\angle DEG + 90+\angle FEH=180^{\circ}$

Step6: Simplify the equation

Subtract 90 from both sides: $\angle FEH=90 - \angle DEG$, so $\angle FEH$ is complementary to $\angle DEG$ (Definition of complementary angles: two angles whose sum is 90 degrees)

Step7: Use the property of congruent complementary angles

Since $\angle DFE$ is complementary to $\angle DEG$ and $\angle FEH$ is complementary to $\angle DEG$, $\angle DFE\cong\angle FEH$ (If two angles are complementary to the same angle, then they are congruent)

Step8: Use the converse of alternate - interior angles theorem

$\angle DFE$ and $\angle FEH$ are alternate - interior angles. If alternate - interior angles are congruent, then the lines are parallel. So $DF\parallel GH$ (Converse of alternate - interior angles theorem: If two lines are cut by a transversal and the alternate - interior angles are congruent, then the two lines are parallel)

Answer:

1. $DE\perp EF$
2. $\angle DFE$ is complementary to $\angle DEG$
3. $\angle DEF$ is a right angle
4. Definition of a right angle
5. $\angle DEG+\angle DEF+\angle FEH = 180^{\circ}$
6. Substitution ($\angle DEF = 90^{\circ}$)
7. Simplification to show $\angle FEH$ is complementary to $\angle DEG$
8. $\angle DFE\cong\angle FEH$
9. If two angles are complementary to the same angle, then they are congruent
10. $DF\parallel GH$ (Converse of alternate - interior angles theorem)