# What is the CPCTC rate

## How to do a parallelogram proof

A great way to start a proof is to think through a game plan that sums up your main argument or a chain of logic. The following examples of parallelogram evidence show game plans followed by the resulting formal evidence.

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### Proof 1

Here is a game plan outlining how your thinking might go:

• Notice the congruent triangles. Always check for triangles that look congruent!

• Skip to the end of the proof and wonder if you could prove that QRVU if you knew a parallelogram that the triangles were congruent. With CPCTC (corresponding parts of congruent triangles are congruent), you could show that QRVU two pairs of congruent sides and that would make it a parallelogram. In order to . . .

• Figure out how you could show that the triangles are congruent. You already have segment QV congruent by the reflexive property and a pair of congruent angles (given), and you can get the other angles for the AAS (Angle-Angle-Side) using additions congruent angles. It does.

There are two other good ways to do this proof. If you noticed that given congruent angle, UQV and RVQIf, alternate interior angles are, you might be right to complete that segments UQ and VR are parallel. (This is a good thing to notice, so congratulations if you did.) You might then have had the good idea to try to prove the other pair of sides to be parallel and so you could use the first parallelogram detection method use. You can do this by proving the triangles to be congruent with CPCTC and then with alternate interior angles VQR and QVU, but assume, for the purposes of evidence, that you did not know this. It seems like you are at a dead end. Do not miss this frustration leave you. When evidence does, it is not uncommon for good ideas and good plans to lead to dead ends. When this happens, just go back to the drawing board. A third way to do the proof is to get the first pair of parallel lines and then show that they are also congruent - with congruent triangles and CPCTC - and then finish with the fifth parallelogram proof procedure.

Take a look at the formal proof:

Statement 1:

Reason for the statement 1: Given.

Statement 2:

Reason for the statement 2: Given.

Statement 3:

Reason for the statement 3: If two angles are complementary to two congruent angles, then they are congruent.

Statement 4:

Reason for the statement 4:Reflexive property.

Statement 5:

Reason for the statement 5: AAS (3, 1, 4)

Statement 6:

Reason for the statement 6: CPCTC (corresponding parts of congruent triangles are congruent).

Statement 7:

Reason for the statement 7: CPCTC.

Statement 8:

Reason for the statement 8:If both pairs of opposite sides of a square are congruent, then the square is a parallelogram.

### Proof 2

Here's another piece of evidence - with a pair of parallelograms. This problem gives you more practice with parallelogram proof methods, and because it is a bit longer than the first proof, it will give you a chance to think through a longer game plan.

Your game plan could go as follows:

• Look for congruent triangles. This diagram is the cake for including congruent triangles - it has six pairs of them! Don't spend a lot of time thinking about them - other than those who could help you - but at least make a quick mental note that they're there.

• Look at the givens. The specified congruent angles which parts

are a great indication that you should try to show these triangles to be congruent. You have these congruent angles and the congruent sides

of parallelogram HEJGYou just have to use a pair of congruent sides or angles SAS (Side-Angle-Side) or ASA (Angle-Side-Winkel).

• Think about the end of the proof.

So you should try the other option: prove the triangles congruent with ASA.

The second pair of angles you would need for ASA consists of anglesDHG and angles FJE.

• Let us consider parallelogram proof methods. You now have a pair of congruent sides DEFG. Use two of the parallelogram proof methods to find a pair of congruent sides. To complete either of these methods, you need to demonstrate one of the following:

• That the other pair of opposite sides is congruent

• The segment DG and the segment EF as well as congruent are parallel

Ask yourself which approach looks easier or faster.

This is packaging!

Now take a look at the formal proof:

Statement 1:

Reason for the statement 1: Given.

Statement 2:

Reason for the statement 2: Opposite sides of a parallelogram are congruent.

Statement 3:

Reason for the statement 3: Opposite sides of a parallelogram are parallel.

Statement 4:

Reason for the statement 4: If lines are parallel, then alternative exterior angles are congruent.

Statement 5:

Reason for the statement 5: Given.

Statement 6:

Reason for the statement 6: ASA (4, 2, 5).

Statement 7:

Reason for the statement 7: CPCTC.

Statement 8:

Reason for the statement 8: CPCTC.

Statement 9:

Reason for the statement 9: When alternative interior angles are congruent

then the lines are parallel.

Invoice 10:

Reason for the statement 10: When a pair of opposite sides of a square are parallel and congruent, the square becomes a parallelogram (lines 9 and 7).