finish proof
given ABCD is an isoceles trapezoid with AD parallel to BC; DE=DC
prove ABED is a parallelogram
Where are points E and F?
ther is no e and f it a two column proof
statements reasons
1.AD parr.BC;DE=DC 1.given
2.AB=DC 2.def of isos trap
3.<B=<? 3.?
4.<DEC=? 4?
5.<DEC=<B 5.?
6.AB parr. DE 6.?
I should have just asked where point E is. There is no F mentioned; I was wring about that. The question makes sense if E lies along the extended line BCE
ABED is a parallelogram if BE = AD and is parallel to AD, AND if AB is parallel and equal to AB. That is what you have to prove.
Clearly CDE is an isosceles triangle appended to ABCD. Prove that the angle ADC is the same as DCE. This will prove that CE is an extension of BC.
To prove that ABED is a parallelogram, we need to show that opposite sides are parallel.
Given:
- ABCD is an isosceles trapezoid, which means that AB//CD and AD=BC.
- AD is parallel to BC.
- DE = DC.
We will use the properties of isosceles trapezoids to prove that ABED is a parallelogram.
Proof:
1. AB//CD (Given: ABCD is an isosceles trapezoid)
2. AD=BC (Given: ABCD is an isosceles trapezoid)
3. DE=DC (Given)
4. Draw lines AE and BD to connect the opposite vertices of the trapezoid.
5. In triangle ADE, DE=DC (Given)
6. In triangle ABE, AB//CD and AD=BC (Properties of isosceles trapezoids)
7. By the corresponding sides of congruent triangles, we have AE//BD (By corresponding angles and AD=BC, AD is parallel to BC)
8. AB//ED and AE//BD (By transitive property, AB//CD and AE//BD)
9. ABED is a quadrilateral with opposite sides parallel, so it is a parallelogram. (By definition of a parallelogram)
Therefore, ABED is a parallelogram.
In this proof, we used the properties of isosceles trapezoids to establish that ABED is a parallelogram.