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

CPCTCIt 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.

CPCTC

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.

Two column proof proving lines parallel with triangle congruenceTo 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

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



Triangle Congruence by SSS and SAS – How To Prove Triangles Congruent

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

Proving Triamgles Congruent with SSSThe 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

Proving Triagles Congruent with SSSGiven 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?

Using Sid-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.

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