I need help with this proof:
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?
Given: ABCD is Parallelogram, AE and CF are perpendicular to diagonal BD, EF = 10 cm. BD = 28
Prove: How long are DE and BF
1 ABCD is a parallelogram 1 Given
2 AE and CF are perpendicular to diagonal BD. 2 Given
3 AD is parallel to BC 3
4 Angle ADE is congruent to angle CBF 4
5 Angle AED and angle CFB are right angles. 5
6 Angle AED is congruent to angle CFB 6
7 AD is congruent to BC 7
8 Triangle AED is congruent to triangle CFB 8
9 DE is congruent to BF 9
10 EF = 10 and BD = 28 10
11 BD = DE + EF + FB 11
12 28 = DE + 10 + FB 12
13 18 = DE + FB 13
14 DE = FB 14
15 18 = 2(DE) 15
16 9 = DE = FB 16
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You get nothing if you don't make any question on it.
Are all the statements given in the problem and you need to supply the missing reasons?
1. ABCD is a parallelogram : Given
2. AE and CF are perpendicular to diagonal BD: Given
3. AD is parallel to BC: Property of parallelogram
4. Angle ADE is congruent to angle CBF : Alternate interior angles.
5. Angle AED and angle CFB are right angles: given
6. Angle AED is congruent to angle CFB: both are 90 degree from 5
7. AD is congruent to BC: Sides of parallelogram
8. Triangle AED is congruent to triangle CFB: conclusion from4,5,6,7
9. DE is congruent to BF : Result from8
10. EF = 10 and BD = 28: Given
11. BD = DE + EF + FB: From10
12. 28 = DE + 10 + FB: Replace the given information
13. 18 = DE + FB: Minus 10 both sides
14. DE = FB: From 9
15. 18 = 2(DE)
16. 9 = DE = FB : Divide both sides by 2and from14.
3. Definition of parallelogram.
4. Theorem of alternate interior angles of parallel lines
5. Definition of perpendicular lines
6. Theorem: All right angles are congruent
7. Theorem: opposite sides of a parallelogram are congruent
11. Segment addition postulate
12. Substitution property of equality
13. Subtraction property if equality
14. Definition of congruent segments
16. Division prop of equality & transitive prop of equality
Can someone help me with the attached proof? The statements are given, I am having problems with the Reasons 5, 6 and 7. Thanks
Reason 2: A property of proportions.
In my opinion, there are two statements missing here.
Statement 2a. XR + RQ = XQ Reason 2a. Segment addition postulate
Statement 2b. YS + SQ = SQ Reason 2b. Segment addition postulate
Reason 3. Substitution property of equality
Reason 4. If two triangles have 2 pairs of corresponding sides with proportional lengths and a pair of corresponding congruent included angles, then the triangles are similar.
Reason 6. Definition of similar triangles.
Reason 7. If two lines are cut by a transversal, such that corresponding angles are congruent, then the lines are parallel.