PROVING CONGRUENT TRIANGLES (ASA)

OBJECTIVE

• Review over the logic of proofs
• Review over tricks and suggestions
• Review over proving triangles with Side Angle Side
• Go over proving triangles with Angle Side Angle

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SO WHAT IS A PROOF?

Other than something that each student has been dreading all semester, a proof is:

“A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion.”

Or, in other words, a mathematical proof is a logical mathematical argument used to show the truth of a mathematical statement.

So, basically, we know something works, but that’s not enough, we need to prove that it works.

This is done because in the math, everyone is skeptical of everything (much like the science world).

So, to make sure something works in mathematics, we have to show that it’s legit.

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Example for SAS

Given: DA is congruent to CB

DA is parallel to CB

AC transverses DA and CB

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DA is congruent to CB

DA is parallel to CB

AC transverses DA and CB

Alternate Interior Angles

SAS Congruence

Given

Alright, so again, we need to start by putting what they gave us down.

So:

Reflexive Property

Okay, now, what do we know?

We know that AC is a transversal intersecting DA and CB

So, what do we know about angles when dealing with transversals that may help us out some?

Alternate Interior angles!

So we know, from alternate interior angles, that

We also know that we need a final congruent side to make this work.

However, due to the reflexive property, we know that AC is congruent to AC.

So:

And since we’ve proven a side, and angle, and a side are congruent, then we can conclude that:

Strategies to proofs

So, when it comes to proofs, there are a few things that we need to go over.

First, it’s important to know what you have that you can work with.

-This is where a journal of all of the theorems, postulates, and definitions can come in handy.

Second, come up with a plan

-Start with what they gave you

-Think about how you will get to the end of your proof. What are you trying to prove? What steps can you take to get there?

Suggestions:

Proofs can be pretty tough, but one thing that can help is instead of trying to prove whatever they are asking you to prove from the start (or what they gave you) to the end (or what they want you to prove), sometimes it helps to start at the end, and work backwards.

Write it out – explain, in your own words, how you know it works. Sometimes seeing it in your own writing can help you convert it to a formal proof.

Don’t stress – when you allow a proof to stress you out, it immediately becomes 10 times harder. If you need, take a break. Let your brain absorb what you need to prove it.

Talk to your teacher – emailing me for help in the next step can be beneficial as well. Mathematicians do this all of the time. They don’t email a teacher, per say, but they do get their peers to chime in and act as a second pair of eyes.

Example 2

Given: BC is congruent to CE

C is the midpoint of AD

BE intersects AD

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BC is congruent to CE

C is the midpoint of AD

BE intersects AD

Vertical Angles

SAS Congruence

Given

First, let’s try this out with the uno cards (only one).

And we want:

And what we have to work with is:

Definition of Midpoint

Example 3

Given: AC is a transversal between two parallel lines DB and EC

B is the midpoint between CA

DB is congruent to EC

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AC is a transversal between two parallel lines DB and EC

B is the midpoint between CA

DB is congruent to EC

Corresponding Angles

SAS Congruence

Given

Alright, now let’s try this out without the Uno cards

So, first, we need to put the Given down.

So:

Definition of Midpoint

Okay, now, what do we know?

We know that AC is a transversal intersecting DB and EC

So, what do we know about angles when dealing with transversals that may help us out some?

Corresponding angles!

So we know, from corresponding angles, that

We also know that since B is the midpoint between CA, it creates two congruent segments: BA and CB

And since we’ve proven a side, and angle, and a side are congruent, then we can conclude that:

Example 4

Given: CE is parallel to FB

AD transverses CE and FB

FB is congruent to CE

DE is congruent to AF

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CE is parallel to FB

AD transverses CE and FB

FB is congruent to CE

DE is congruent to AF

Alternate Exterior Angles

SAS Congruence

Given

Alright, so again, we need to start by putting what they gave us down.

So:

Now, the given gave us a ton of info

Most of the work is done here

We have two sides that are congruent, but to use SAS, we need to prove the angle between those two sides.

However , we do know that CE || FB, and AD transverses them

So what angle relationship can help us out here?

Alternate Exterior Angles!

So, because of alternate exterior angles, we know that

And since we’ve proven a side, and angle, and a side are congruent, then we can conclude that:

MOVING ON

So, of course, since mathematics is a very big subject, we need to look into other proofs that can be proved.

Now, we’ve gone over Side Side Side congruence proofs and Side Angle Side congruence proofs, so now we’re on to the bigger one.

Angle Side Angle Congruence proofs.

PROVING ANGLE SIDE ANGLE

So, Angle Side Angle Proofs are similar to Side Angle Side proofs, except we now have the addition of another angle, which means we need to be more careful how we proceed.

For Side Angle Side, we only had to worry about one angle.

Now we have two.

However, we’re still going to use the different angle relationships that we have been studying to prove them, just more of them.

So, let’s review over those again.

Corresponding Angles

Alright, let’s start off with the definition of corresponding angles.

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Corresponding angles – when two lines are crossed by a transversal, the angles in matching corners are called corresponding angles.

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So, something like this:

Vertical Angles

Vertical angles, by definition, are a pair of non-adjacent angles formed when two lines intersect.

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So, in plain English, basically when two lines cross, they create these two angles.

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These two angles are always congruent.

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ALTERNATE INTERIOR ANGLES

Alternate Interior Angles - If two parallel lines are transected by a third line (a transversal), the angles which are inside the parallel lines and on alternate sides of the third line are called alternate interior angles.

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Or, again, in plain English, if you have two parallel lines intersecting with a transversal, the angles on the inside are equal.

ALTERNATE EXTERIOR ANGLES

Similar to Alternate Interior Angles, we’ve already seen Alternate Exterior Angles.

So, as is tradition, let’s start with the definition.

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Alternate Exterior Angles -  Alternate Exterior Angles are a pair of angles on the outer side of each of those two lines but on opposite sides of the transversal.

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Again, in English, this means that when you have two parallel lines intersecting with a transversal, the angles on the outside are equal.

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Again, we’re looking at something like this:

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So, now that we’ve reviewed, we can get into it

Again, Angle Side Angle is a little more complicated than Side Angle Side.

Now that we’ve reviewed over what we know, we can use that to help prove some of these problems.

So, let’s jump in!

Example 1

Given: AB || CD

AD transverses AB and CD

BC transverses AB and CD

AB is congruent to CD

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AB || CD

AD transverses AB and CD

BC transverses AB and CD

AB is congruent to CD

Alternate Interior Angles

ASA Congruence

Given

Alright, let’s just try proving this one.

So, first, we need to put the Given down.

So:

Alternate Interior Angles

Okay, now, what do we know?

We know that AD is a transversal intersecting AB and CD

So, what do we know about angles when dealing with transversals that may help us out some?

Alternate Interior angles!

So we know, from alternate interior angles, that

And since we’ve proven an angle, a side, and an angle are congruent, then we can conclude that:

Example 2

Given: Angle BAE is congruent to Angle DEA

C is the midpoint of AE

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Angle BAE is congruent to Angle DEA

C is the midpoint of AE

Definition of Midpoint

ASA Congruence

Given

Alright, now let’s try this out without the Uno cards

So, first, we need to put the Given down.

So:

Vertical Angles

Okay, now, what do we know?

We know that C is the midpoint between AE, which means it creates two congruent sides, AC and CE

So:

And finally, we know that angle BCA is congruent to angle DCE because they are vertical angles.

So:

And since we’ve proven a side, and angle, and a side are congruent, then we can conclude that: