Hi All, I'm a student of The American school and the only thing I have between me and my diploma is 3 proofs. I've gotten help from several people but we can't seem to figure out these proofs. Any help would be awesome!!

Given: In triangle ABC, (angle) B = 120 degrees

Prove: Angle A is not equal to 60 degrees

Plan: Use an indirect proof.

NOTE: Write the proof using the paragraph method.

Prove: If a diagonal of a parallelogram bisects an angle of the parallelogram, the parallelogram is a rhombus. (State your plan and give a proof.)

Given: ABCD is a parallelogram with (angle) 1 congruent with (angle) 2

To Prove: ABCD is a rhombus

Plan
:

Thanks to Soroban for help with this proof =)

Prove that the tangents to a circle at the endpoints of a diameter are parallel. State what is given, what is to be proved, and your plan of proof. Then write a two-column proof.

Thanks so much to anyone who can help me with one or all of these proofs. I'm so ready to be finished with High School LOL

2. Hello, proofsRkickingmybutt!

Here's the second one . . .

Prove: If a diagonal of a parallelogram bisects an angle of the parallelogram,
the parallelogram is a rhombus.

Given: $ABCD$ is a parallelogram with $\angle1 = \angle2$

To Prove: $ABCD$ is a rhombus.

We need to prove that two adjacent sides are equal.

Code:
          A * - - - - - - - - * B
/2 * 1            /
/     *           /
/        *        /
/           *     /
/            3 * 4/
D * - - - - - - - - * C

$1.\;\angle1 = \angle 2$. . . . . . . . . . $\text{Given}$

$2.\;AB \parallel DC,\:AD \parallel BC$. . $\text{d{e}f. parallelogram}$

$3.\;\angle1 = \angle 3,\:\angle 2 = \angle 4$. . . $\text{alt-int. angles}$

$4.\;\angle1 \,=\,\angle 4$. . . . . . . . . $\text{Transitivity}$

$5.\;\Delta ABC\text{ is isosceles}$. . . $\text{d{e}f. isosceles}$

$6.\;\therefore\:AB = BC$. . . . . . $\text{d{e}f. isosceles}$

. . . $Q.E.D.$

3. Oh Thank you SOOOOO much!!!

=D!!

4. Hello, proofsRkickingmybutt!

Prove that the tangents to a circle at the endpoints of a diameter are parallel.
State what is given, what is to be proved, and your plan of proof.
Then write a two-column proof.
Code:
                A
P - - - - * * * - - - - Q
*     |     *
*       |       *
*        |        *
|
*         |         *
*         *O        *
*         |         *
|
*        |        *
*       |       *
*     |     *
R - - - - * * * - - - - S
B

There is a Theorem that says:
. . If a line is tangent to a circle, the radius drawn to
. . the point of tangency is perpendicular to the tangent.

We have a circle with center $O$ and diameter $AB.$

Line $PQ$ is tangent to circle $O$ at $A.$
Line $RS$ is tangent to circle $O$ at $B.$

$1.\;OA \perp PQ,\:OB \perp RS$ . . . . . $\text{Theorem}$

$2.\;\angle OAP = 90^o,\:\angle OBS = 90^o$ . . $\text{d{e}f. perpendicular}$

$3.\;\angle OAP = \angle OBS$. . . . . . . . . $\text{All right angles are equal.}$

$4.\;\therefore\:PQ \parallel RS$. . . . . . . . . . . $\text{alt-int. angles}$

5. Originally Posted by proofsRkickingmybutt

Given: In triangle ABC, (angle) B = 120 degrees

Prove: Angle A is not equal to 60 degrees

Plan: Use an indirect proof.

NOTE: Write the proof using the paragraph method.
I'm guessing it's too trivial to say that if B is 120 degrees, and A is 60 degrees, then C must be equal to 180 (total number of degrees in a triangle) - 180 = 0... which is clearly nonsense... and therefore A is less than or at least, not equal to 60?

Edit: Unless you're supposed to prove the sum of the angles in a triangle = 180 degrees... but that's been done to death