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## 4.4 CPCTC Using Corresponding Parts of Congruent Triangles are Congruent

Rooted in the definition of congruent triangles, the acronym CPCTC stands for Corresponding Part of Congruent Triangles are Congruent. Basically, you have to prove the triangles congruent with what is given and once the triangle congruence is established the remaining corresponding parts are congruent.

Anytime you are required to prove corresponding parts of congruent triangles congruent, you will be doing a triangle proof, meaning that you will need to prove the triangles congruent with ASA, AAS, SAS, SSS, or HL. The examples are solved with AAS and SAS to prove the other part of the triangle congruent using CPCTC.

## Example 1 Using CPCTC to Prove Two Segments Congruent

It is given that angle BAE is congruent to angles BCD and that segment AE is congruent to segment CD. With the given marked on the diagram you should recognize that there is one pair of corresponding angles and one pair of corresponding segments congruent. That leaves you with establishing ASA, AAS, or SAS in the two triangles.

Notice, triangles BAE and BCD share the vertex B. By the reflexive property of congruence, angle B is congruent to itself. Now triangles BAE and BCD are congruent because of AAS.

Since segment AB and CB are corresponding part of two triangles, they are congruent by CPCTC.

## Example 2 Using CPCTC to Prove Two Segments Congruent

I have not yet used this in my blog, it is a modified two column proof. There is still the statements and reasons, but the reasons are included in a set of parenthesis. The statements are numbered starting with 1.

1.            segment DA is congruent to segment DC (Given)

2.            angle BDA is congruent to angle BDC (Given)

3.            segment BD is congruent to segment BD (reflexive property of congruence)

4.            triangle ABD is congruent to triangle CBD (SAS)

5.            angle ABD is congruent to angle CBD (CPCTC)

As you can see, using CPCTC is just a matter of completing a triangle congruence proof first.

Until next time,

Mr. Pi

## Proving Lines Parallel with Triangle Congruence SSS SAS AAS and ASA

I got a great question via personal message on youtube,

“hello, i need help on this one proof i am doing. Do you have any video on how to prove lines are parallel by sss, sas, asa, or aas? please tell me if you do. thankyou”

I have not done any problems like this yet, but I put this image together to help the student of geometry. I hope it helps. Below, I write a paragraph proof.

To really understand this problem you have to remember the ways to prove lines parallel: the converse of the corresponding angles postulate, the converse of the alternate interior angles theorem and the converse of the same-side interior angles theorem. So, to prove that segment AB is congruent to Segment CD. The shortest way to complete this proof is to show that either angels MAB and MDC or angles MBA and MCD are congruent. To prove those angles congruent, triangles ABM and DMC must be proved congruent.

With the given information, M is the midpoint of AD and BC, segments AM and DM are congruent as well as segments BM and CM because of the definition of midpoint. Since segments AD and BC intersect at M and form vertical angles AMB and DMC, those angles are congruent by the vertical angles theorem. That gives a SAS congruence in triangles ABM and DMC, thus leaving them congruent. Because of corresponding parts of congruent triangles are congruent(CPCTC), angle A is congruent to angle D. Angles D and A are congruent alternating interior angles, so segments AB and CD are parallel by the converse of the alternate interior angles theorem. QED

I color coded the markings on the diagram with the proof. I like to start with a blank diagram and mark my corresponding congruent parts as I go. It helps me develop my proof. I hope this technique helps you. I encourage you to use different colored pencils to mark your diagram is my coloring coding helped you follow this proof on how to prove lines congruent with triangle congruence.

Regards,

Mr. Pi

## Triangle Congruence by AAS and ASA Part 2

If you have not read my previous post on triangle congruence, make sure you do. Anyway back to the angle-side-angle and angle-angle-side theorems. I will prove even more triangles congruent in this post! Like I wrote before, that is all I really remembering doing in high school geometry.

## Example 1 Triangle Congruence Proof with AAS

The first example is example 3 in the video. If you are wondering why I am writing about something I already did in a video, let me explain. It has to do with different learning styles. It is pretty important to understand your learning style. I will have to blog about learning styles soon. Anyway, through my blog, I try to reach a variety of learning styles. Thus, I create graphics, write and create videos to get my math point across. Anyway, on with example 3 from the video.

Given: angle S is congruent to angle Q and segment RP bisects angle SRQ

Prove: triangle SRT congruent to triangle QRP

When I look at what is given and the diagram, I notice the two triangles share a side which will be a corresponding congruent side, segment PR. Since, segment RP bisects angle SRQ, that means angles SRP and QRS are congruent corresponding parts by the definition of angle bisector. Those two facts taken with the given, provides enough proof to establish AAS in both triangles SRT and QRP.

## Example 2 Triangle congruence with AAS

Given: angles B and D are congruent and segment AB is parallel to segment CD

Prove: triangles ABC and CDA are congruent

As is true with any proof, you need to understand the given and how it will help you identify a pair of correspond angles or sides of a triangle congruent to use one of the methods to proving triangles congruent. From the given, we have a pair of corresponding angles congruent, D and B. From the diagram a pair of corresponding sides can be established. AC is a shared side and with the reflexive property of congruence it can stated AC is congruent to AC. Also in the given, it is stated that segments AB and DC are parallel. Whenever you hear the words parallel lines, you must remember the special angle pairs formed by two parallel lines and a transversal. In this case, angle BAC is congruent to angle DCA because of the alternate interior angles theorem. With all of that, it can be said that triangle ABC is congruent to triangle CDA by the AAS theorem.

## Example 3 Proving Triangles Congruent with AAS

Given: Segments XQ and TR are parallel and segment XR bisects QT

Prove: triangle XMQ congruent to triangle RMT

This proof requires the most work of all the proofs I have done on this blog. There are no corresponding parts given as congruent, which means we have to establish three pairs of corresponding parts congruent. The two keywords in the given are parallel and bisects. Angle X and angle R are alternate interior angles and are congruent because the two angles are formed by two parallel lines and a transversal. Angles XMQ and RMT are congruent because all vertical angles are congruent. I have two angles and need to prove 1 pair of corresponding sides congruent. Those sides will be segment TM and segment QM by the definition of segment bisector. Since the sides are the non-included sides, triangle XMQ is congruent to triangle RMT by the AAS theorem.

## 4.3 Triangle Congruence by AAS and ASA

In the video, it is not labeled example 1, but the first bit of information is critical to this lesson.

## Angle Side Angle Postulate

It two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

## Angle Angle Side Theorem

It two angles and a nonincluded side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the two triangles are congruent.

It would be good to remember that a postulate is something that is assumed to be true without question. Theorems are facts that have been proved true using definitions, postulates and other already proved theorems. We cannot use a theorem until we learn about it.

Example 2

In a recent post about proving The Converse of the Alternate Interior Angles Theorem, I created my first flow proof. It was a short proof, much like in this example using ASA. I color coded my markings, but on the vertical angles APX and BPY I did not get the correct shade of blue.

Example 3

As you can see, it is a fairly complicated diagram for a relatively easy proof.

Given: angle CAB is congruent to angle DAE, segment AS is congruent to segments AE and angle ABC and AED are right triangles

Prove: triangle ABC is congruent to AED

Planning this proof requires you to understand your given information and the prove statement. Since the diagram has an added dimension of difficulty because extra triangle in the middle, you need to focus on the two outer right triangles ABC and AED. These are the angles that need to be proved congruent.

It is given that angles CAB and DAE are congruent and are corresponding angles in each triangle. Also, the corresponding sides AB and AE are congruent. Finally, there is information relating to the corresponding angles ABC and AED. They are both right angles. Angles ABC and AED are the angle that gives Angle-Side-Angle in each triangle. Since all right angles are congruent, angles ABC and AED are congruent. Therefore, both triangles ABC and AED are congruent by ASA. QED

Example 4

Given: segments NM and NP are congruent and angles M and P are congruent

Prove: triangles NML and NPO congruent

There are many ways to prove the triangles congruent in this diagram, but I like to produce the most concise proof possible. Hopefully you can see the ASA relationship that will be able to be proved from this set up. Angles LNM and ONP can be proved congruent to establish the ASA congruence in these two triangles.

Since it is given that segments NM and NP are congruent and angles M and P are congruent, a ASA congruence can be established with triangles NML and NPO. Angle LNM and angle ONP are congruent because they are vertical are congruent. Finally, it can be stated that triangles NML and NPO congruent because of the Angle-Side-Angle Theorem. QED

As usual, I hope this has been helpful. Be sure to check back for more geometry.

Regards,

Mr. Pi

## 4.2 Triangle Congruence by SSS and SAS – Part 2

This post covers both triangle congruence using side-angle-side and side-side-side. If you have not checked out the first post on Triangle Congruence by SSS and SAS, follow the preceding link.

## Side Side Side Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

## SAS Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Example 1 (Additional Example 2 from the video below)

Given that segment AD is congruent to segment BC, what other information is needed to prove triangle ACD congruent to triangle BCD by SAS?

It can be established that segment DC is congruent to itself by the reflexive property of congruence.  So, there are two pairs of corresponding sides congruent in each triangle. You will need to prove that angle ADC is congruent to angle BCD to prove triangle ACD congruent to triangle BCD by SAS. Remember, the angle is the included side in the SAS postulate and angle ACD is the included angle of sides AD and DC, while angle BCD is the included of sides BC and DC.

Example 2 (Quick Check #1 from the video)

Given: segment HF is congruent to segment JH, segment FG is congruent to segment JK and H is the midpoint of segment GK

Prove: triangle FGH congruent to triangle JKH

Completing proofs in geometry requires some planning. It may not be planning that you write out as will in this post, but you should at least think about what you are given and how to prove the triangles congruent. In this problem, you are given two sets of corresponding sides as congruent. Also, you are given information in regards to the third side of each triangle. The previous sentence should indicate the use of the side-side-side postulate of triangle congruence.

Paragraph Proof

It is given that segment HF is congruent to segment JH, segment FG is congruent to segment JK and H is the midpoint of segment GK. Since H is the midpoint of segment GK, it can be concluded that segment GH is congruent to segment KH by the definition of midpoint. Now that all pairs of corresponding sides are congruent, it can be stated that triangle FGH congruent to triangle JKH by the Side-Side-Side Postulate. QED

Example 3 (Additional Example 3 from the video)

This example is a Side-Angle-Side proof.

Given: angle RSG is congruent to angle RSH and segment SG is congruent to SH

Prove: triangle RSG is congruent to triangle RSH

Planning this proof involves realizing that SAS is going to be used. Once you mark the given information on the diagram, it should be clear that you need prove side SR is congruent to itself to establish the SAS postulate in both triangles.

Paragraph Proof

It is given that angle RSG is congruent to angle RSH and segment SG is congruent to SH and triangle RSG is congruent to triangle RSH needs to be proved . Both triangles share the side SR, thus segment SR is congruent to segment SR by the reflexive property of congruence. From the given statement and the previous sentence, Side-Angle-Side has been established in each triangle so, triangle RSG is congruent to triangle RSH. QED

## Side Side Side Postulate

If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Example 1

The first example is proving two triangles congruent using the SSS postulate. You are given two pairs of corresponding sides congruent. Thus, you need to only prove one pair of corresponding sides is congruent. As you can see in the diagram, the two triangles share a side segment BD. You will be able to establish this side is congruent to itself with the reflexive property of congruent segments. Now that all three sides are congruent, the triangles are congruent by the SSS postulate.

Example 2

Given that segment AX is congruent to segment AY and M is the midpoint of segment XY, Prove that triangle AMX is congruent to triangle AMY. First one can establish that segment AM is congruent to itself by the reflective property of congruence. That gives two pairs of corresponding sides of each triangle congruent. The third pair of corresponding sides, MX and MY, is congruent, because M is the midpoint of segment XY which is made up the segments MX and MY. Since all three pairs of corresponding sides are congruent, the triangles, AMX and AMY are congruent by SSS.

## SAS Postulate

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Example 3 Using SAS

This is actually example 2 in the video, but it is example 3 in the blog! Given that segment RS is congruent to segment TK.

What other information is needed to prove triangle RSK is congruent to triangle TKS By Side-Angle-Side?

Since the triangles share side KS, it can be stated that segment KS is congruent to itself by the reflexive property of congruence. You now have two pairs of corresponding sides congruent. To prove the triangles congruent by SAS, angle RSK must be proved congruent to angle TKS.﻿

## Proving Triangles Congruent Using Definition of Congruent Triangles – Part 2

Part 1 of Congruent Figures 4.1

The third part of this series about congruent figures is proving triangles congruent. At this point, there is only one way to prove triangles congruent, with the definition of congruent triangles. Congruent triangles have 3 pairs of corresponding angles and 3 pairs of congruent angles. So, to prove two triangles congruent, we must establish that all 3 pairs of corresponding sides and angles are congruent.

Given: segment WX is congruent to YX, segment WZ is congruent to segment YZ, angle W is congruent to angle Y and segment XZ is perpendicular to segment WY

Prove: triangle WXZ is congruent to triangle YXZ

With the given information: segment WX is congruent to YX, segment WZ is congruent to segment YZ, angle W is congruent to angle Y and segment XZ is perpendicular to segment WY, prove triangle WXZ is congruent to triangle YXZ. First, segment XZ is congruent to segment XZ by the reflexive property of congruence. Angles XZW and XZY are right angles by the definition of right angles. Angle XZW and angle XZY are congruent because all right angles are congruent. Angle WXZ is congruent to angle YXZ by the third angle theorem. It can be established that triangle WXZ is congruent to triangle YXZ by the definition of congruent triangles.