Urgent
pictures included

proof 6a
in parallelogram abcd, ae and cf are perpendicular to diagonal bd, and ef = 10 cm. if bd = 28 cm, then how long are de and bf? (diagram is not drawn to scale.)

fill in the missing blanks in the two tables below. you must support each statement in the proof with a theorem, property, or definition.

view the 2nd and 3rd picture for the questions

REASONS:

1. Given
We know for a fact that ABCD is a parallelogram since the problem statement says so and we can also observe from the figure that the opposite sides of the shape ABCD is parallel which adheres to the definition of a parallelogram.

2. Given
Similar to the previous item, the statement telling us that lines AE and CF are perpendicular to the diagonal BD can also be found on the problem statement. Therefore, we do not need any other reasoning to support this.

3. Definition of a parallelogram (Opposite sides are parallel)
As we have said in item #1, the basic definition of a parallelogram states that its opposite sides are parallel. In the figure, lines AD and BC are opposite to each other therefore, according to the definition of a parallelogram, they must be parallel.

4. In a pair of parallel lines cut by a transversal, alternate interior angles are congruent.
In the figure, the pair of parallel lines AD and BC are cut by a transversal DB. And, according to a theorem, the alternate interior angles that are formed by this cutting transversal should be congruent. In our case, these alternate interior angles are angles ADE and CBF.

5. Definition of perpendicular lines (Two lines intersect each other and form 90-degree angles)
Since we already know that AE and CF are both perpendicular to diagonal BD from item #2, then from the very definition of perpendicular lines, we'll know that the angles formed by the intersection of these lines would measure 90 degrees.

6. Definition of angle congruency (Angles with the same measure are congruent)
Through angle congruency, we just need to know that any pair of angles have the same measurement. For this item, we have already supported the statement that both AED and CFB measure 90 degrees in item #5. Thus, this will tell us that the angles are congruent.

7. Definition of a parallelogram (Opposite sides are congruent)
Just like how we proved item #3, we can also support the statement that AD is congruent to BC through the use of the definition of a parallelogram. For a figure to be a parallelogram, its opposite sides must also be equal in length. This is because it would be impossible for these sides to be parallel if it weren't.

8. ASA Postulate
We can prove that triangle AED is congruent to CFB by using the ASA postulate. This postulate states that if we successfully prove that two angles and the included side of a triangle is congruent to its corresponding parts in another triangle, then the two triangles are said to be congruent.

9. Definition of congruency
Since we have successfully proven that triangle AED is congruent to triangle CFB, then we can now deduce that the rest of their corresponding parts are also congruent. This would mean that DE is also congruent to BF.

10. Given
The statement here is also given in the problem statement. According to the problem, EF = 10cm and BD = 28 cm, and this is exactly what was reflected in statement #10. Thus, we do not need any other reasoning because this was the condition given to us.

11. The whole is equal to the sum of its parts.
For this item we can notice that the equation just illustrates that the measure of DE, EF, and FB, when combined, would be equal to the measure of BD. This would be true because in the figure, we can clearly see that these three lines are small parts that add up to form the line BD.

12. Substitution Property
In this item, 28 and 10 were just substituted for the variables BD and EF, respectively. This process is supported by the substitution property which states that anything equal to x can be substituted in x for any equations with the variable.

13. Additive Property
The process that was done in this statement could be justified by using the additive property of equality. The additive property states that any number added to both sides of the equation will not alter the result. In our case we just added -10 on both sides.

14. Definition of congruency
In this item, the statement just shows the equivalent of a previous statement found in item #9. Since we have already stated that DE is congruent to BF, then it will just follow that the measure of line DE is equal to FB because of the definition of congruency.

15. Substitution Property
Here we just applied the substitution property, replaced the variable FB with DE, and simplified the equation by combining like terms. This replacement is possible through the substitution property. Because we have previously stated that DE = FB, then we can just replace one of the variable with another one.

16. Multiplication and transitive property
For the statement in this item, both sides of the previous statement were just multiplied by 0.5. This resulted in the equation 9 = DE. After that, since FB = DE, then through transitive property, 9 = FB. This supports the equation in the item.