Question

Drag each reason to the correct location on the flowchart. Not all reasons will be used.

Given: AO OC and BO OD
Prove: ABCD and AD||BC
B
Complete the flowchart proof.
converse of alternate
interior angles theorem
AO OC
BO OD
=
given
vertical angles SAS
theorem
ZAOD ZCOB
ZAOB ZCOD
AAODACOB
AAOB ACOD
alternate interior
angles theorem
ZDAC ZBCA
ZBAC ZDCA
ASA
CPCTC
AB|| CD
AD || BC
He

Answer 1

Step-by-step explanation:

Ok the first box is the vertical angle theorem because AOD and COB and the other two are opposite angles formed by the same lines which means that they are vertical angles and the theorem proves that they are equal.

The next one is SAS because as you can see in the diagram, because we know the vertical angles are equal we now know that the triangles are congruent because there are two sides and an angle between them on both triangles that are equal.

The next one is CPCTC which stands for Corresponding Parts of Congruent Triangles Congruent this basically means that because the triangles are congruent, the sides that match up and angles that match up in the triangles are equal.

After that you would use Converse Alternate Interior Angle Theorem because in the previous step you already proved that the alternate interior angles in this structure are equal through congruent triangles and now you are saying that because the are equal the lines must be parallel because alt interior angles are only equal when lines are parallel. You have to use the converse because you are kind of going backward. If you used the regular one you would be using parallel lines to state that the angles are equal. Let me know if you have any questions and good luck!

Answer 2

To prove ABCD and AD||BC, we can use a combination of given information and theorems such as the converse of the alternate interior angles theorem, AA similarity theorem, and SAS theorem.

Step-by-step explanation:

To prove ABCD and AD||BC, we can use a combination of given information and theorems. Here is the step-by-step process:

1. Given: AO OC and BO OD
2. Reason: Given
3. Given: Triangle BAO and BCO are similar
4. Reason: Similar triangles
5. Using the converse of the alternate interior angles theorem, we can conclude that angle ZAOD = angle ZCOB and angle ZAOB = angle ZCOD
6. Reason: Converse of alternate interior angles theorem
7. Using the AA similarity theorem, we can conclude that triangle AOB is similar to triangle COD
8. Reason: AA similarity theorem
9. Using the SAS theorem, we can conclude that triangle DAC is congruent to triangle BCA
10. Reason: SAS theorem
11. Using the CPCTC (Corresponding Parts of Congruent Triangles are Congruent) theorem, we can conclude that AB || CD and AD || BC
12. Reason: CPCTC theorem