4 6 problem solving triangle congruence cpctc

The CPCTC theorem states that when two triangles are congruent, their corresponding parts are equal. The CPCTC is an abbreviation used for 'corresponding parts of congruent triangles are congruent'.

What is CPCTC?

The abbreviation CPCTC is for Corresponding Parts of Congruent Triangles are Congruent . The CPCTC theorem states that when two triangles are congruent , then every corresponding part of one triangle is congruent to the other. This means, when two or more triangles are congruent then their corresponding sides and angles are also congruent or equal in measurements. Let us understand the meaning of congruent triangles and corresponding parts in detail.

Congruent Triangles

Two triangles are said to be congruent if they have exactly the same size and the same shape. Two congruent triangles have three equal sides and equal angles with respect to each other.

Corresponding Parts

Corresponding sides mean the three sides in one triangle are in the same position or spot as in the other triangle. Corresponding angles mean the three angles in one triangle are in the same position or spot as in the other triangle.

In the given figure, △ABC ≅ △LMN. It means that the three pairs of sides and three pairs of angles of △ABC are equal to the three pairs of corresponding sides and three pairs of corresponding angles of △LMN.

In these two triangles ABC and LMN, let us identify the 6 parts: i.e. the three corresponding sides and the three corresponding angles. AB corresponds to LM, BC corresponds to MN, AC corresponds to LN. ∠A corresponds to ∠L, ∠B corresponds to ∠M, ∠C corresponds to ∠N. And if △ABC ≅ △LMN, then as per the CPCTC theorem, the corresponding sides and angles are equal, i.e. AB = LM, BC = MN, AC = LN, and ∠ A = ∠L, ∠B = ∠M, ∠C = ∠N.

CPCTC Triangle Congruence

CPCTC states that if two triangles are congruent by any criterion, then all the corresponding sides and angles are equal. Here, we are discussing 5 congruence criteria in triangles .

CPCTC Proof

To prove CPCTC, first, we need to prove that the two triangles are congruent with the help of any one of the triangle congruence criteria. For example, Consider triangles ABC and CDE in which BC = CD and AC = CD are given.

Follow the points to prove CPCTC

• BC = CD and AC = CD (Given)
• ∠ACB = ∠EDC ( Vertically opposite angles are equal)
• Thus, △ABC ≅ △EDC; By SAS (side-angle-side) criterion
• Now the two triangles are congruent, therefore, using CPCTC, AB = DE, ∠ABC = ∠EDC and ∠BAC = ∠DEC.

Important Notes

Given below are some important notes related to CPCTC. Have a look!

• Look for the congruent triangles keeping CPCTC in mind.
• Before using CPCTC, show that the two triangles are congruent.

Related Articles on CPCTC

Check out these interesting articles to know more about CPCTC and its related topics.

• Corresponding Angles
• Congruence in Triangles

CPCTC Examples

Example 1 : Observe the figure given below and find the length of LM using the CPCTC theorem, if it is given that △ EFG ≅ △LMN.

Solution: Given that △ EFG ≅ △LMN. So, we can apply the ASA congruence rule to it which states that if two corresponding angles and the included side are equal in two triangles, then the triangles will be congruent. Here, two angles are given which are 30 degrees and 102 degrees such that ∠EFG = ∠LMN and ∠FEG = ∠MLN. So, by applying the CPCTC theorem we can identify that FE and ML are the corresponding sides of two congruent triangles △ EFG and △LMN. Therefore, FE = ML. Hence, the length of side LM is 3 units.

Example 2 : Observe the figure given below in which PR = RS and QR is perpendicular to PS. Find y using the CPCTC theorem.

Solution: First let us prove that △PQR ≅ △SQR,

• PR = RS (given)
• QR = QR (common side)
• ∠QRP = ∠QRS (as QR is perpendicular to PS)
• Therefore, △PQR ≅ △SQR (SAS criterion)
• PQ = QS ( By CPCTC)

Now as PQ = QS

Therefore, 4y = 28

Answer ⇒ y = 7 units

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Practice Questions on CPCTC

Faqs on cpctc, is cpctc a theorem.

Yes. CPCTC is a theorem that says corresponding parts of congruent triangles are congruent. Corresponding means angles and sides that are in the same respective position in the two triangles.

What is CPCTC for Similar Triangles?

CPCTC for similar triangles is not true. So, we cannot apply the CPCTC theorem for similar triangles . Corresponding angles of the two similar triangles are equal, whereas, corresponding sides of the triangles are not equal, but proportional.

How do you Prove CPCTC?

After showing the proposed triangles are congruent, we can immediately say that the corresponding parts of congruent triangles are congruent. It can be justified by superimposing triangles on each other and then by observing the corresponding angles and side lengths.

What does CPCTC Stand for?

CPCTC stands for corresponding parts of congruent triangles are congruent. Sometimes, it is also called CPCT which means corresponding parts of congruent triangles.

What is an Example of CPCTC?

The theorem CPCTC tells that when two triangles are congruent then their corresponding sides and angles are also said to be congruent. For example, triangle ABC and triangle PQR are congruent triangles therefore according to the theorem the sides AB = PQ, BC = QR, and CA = RP. Also ∠A = ∠P, ∠B = ∠Q, and ∠C = ∠R.

How do you Prove CPCTC Using SSS Criterion?

In SSS triangle congruence all the three corresponding sides are equal. In other words, the two triangles are said to be congruent if all corresponding sides of one triangle are equal to the sides of another triangle. Thus, when two triangles are congruent then according to CPCTC all the corresponding angles are also equal.

How do you Prove CPCTC Using SAS Criterion?

In SAS triangle congruence the two corresponding sides and the included angle are equal. In other words, the two triangles are said to be congruent if two corresponding sides and the included angle are equal. Thus, when two triangles are congruent then according to CPCTC the other corresponding side and the other two corresponding angles are also equal.

Related Topics: More Lessons for Grade 9 Math Geometry

Examples, solutions, videos, worksheets, and activities to help Geometry students learn about CPCTC. CPCTC is an acronym for Corresponding Parts of Congruent Triangles are Congruent.

CPCTC Triangle Congruence

SSS, SAS, ASA Proofs with CPCTC

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Sides and angles of congruent triangles have the same measure.

Congruent Triangles

Two figures are congruent if they have exactly the same size and shape. If two triangles are congruent, they will have exactly the same three sides and exactly the same three angles. In other words, two triangles are congruent if you can turn, flip, and/or slide one so it fits exactly on the other.

\(\Delta ABC\) and \(\Delta DEF\) are congruent because

\(\begin{align*}\overline{AB}\cong DE &\qquad& \angle A\cong \angle D \\ \overline{BC}\cong EF\overline{AB}&\:and\:& \angle B\cong \angle E \\ \overline{AC}\cong \overline{DF} &\qquad& \angle C\cong \angle F \end{align*}\)

Notice that when two triangles are congruent their three pairs of corresponding angles and their three pairs of corresponding sides are congruent.

When referring to corresponding congruent parts of congruent triangles, you can use the phrase C orresponding P arts of C ongruent T riangles are C ongruent, or its abbreviation CPCTC .

What if you were given two triangles with all the angle measures and all the side lengths marked? How could you tell if the two triangles were congruent?

Example \(\PageIndex{1}\)

Determine if the triangles are congruent using the definition of congruent triangles.

We can see from the markings that \(\angle B\cong \angle C\), \(\angle A\cong \angle D\), and \(\angle AEB\cong \angle DEC\) because they are vertical angles. Also, we know that \(\overline{BA}\cong \overline{CD}\), \(\overline{EA}\cong \overline{ED}\), and \(\overline{BE}\cong \overline{CE}\). Because three pairs of sides and three pairs of angles are all congruent and they are corresponding parts, this means that the two triangles are congruent.

Example \(\PageIndex{2}\)

While there are congruent corresponding parts, there are only two pairs of congruent sides, the marked ones and the shared side. Without knowing whether or not the third pair of sides is congruent we cannot say if the triangles are congruent using the definition of congruent triangles. Note, this does not mean that the triangles are not congruent , it just means that we need more information in order to say they are congruent using the definition of congruent triangles (congruent triangles have three pairs of congruent angles and three pairs of congruent sides).

Example \(\PageIndex{3}\)

Are the two triangles below congruent?

To determine if the triangles are congruent, match up sides with the same number of tic marks: \(\overline{BC}\cong \overline{MN}\), \(\overline{AB}\cong \overline{LM}\), \(\overline{AC}\cong \overline{LN}\).

Next match up the angles with the same markings:

\(\angle A\cong \angle L\), \(\angle B\cong \angle M\), and \(\angle C\cong \angle N\).

Lastly, we need to make sure these are corresponding parts. To do this, check to see if the congruent angles are opposite congruent sides. Here, \(\angle A\) is opposite \(\overline{BC}\) and \(\angle L\) is opposite \(\overline{MN}\). Because \(\angle A\cong \angle L\) and \(\overline{BC}\cong \overline{MN}\), they are corresponding. Doing this check for the other sides and angles, we see that everything matches up and the two triangles are congruent.

Example \(\PageIndex{4}\)

If all three pairs of angles for two given triangles are congruent does that mean that the triangles are congruent?

Without knowing anything about the lengths of the sides you cannot tell whether or not two triangles are congruent. The two triangles described above might be congruent, but we would need more information to know for sure.

Example \(\PageIndex{5}\)

From the tic marks we can see that \(\overline{AB}\cong \overline{DE}\). We also know that \(\angle ACB\cong \angle ECD\) because they are vertical angles. However, this is not enough information to know whether or not the triangles are congruent.

The following illustrations show two parallel lines cut by a transversal (You may assume that even though the lines are not marked parallel, the two lines that look parallel, actually are). Are the triangles formed by them definitively congruent?

Based on the following details, are the triangles definitively congruent?

• Both triangles are right triangles in which one angle measures \(55^{\circ}\). All of their corresponding sides are congruent.
• Both triangles are equiangular triangles.
• Both triangles are equilateral triangles. All sides are 5 inches in length.
• Both triangles are obtuse triangles in which one angle measures \(35^{\circ}\). Two of their corresponding sides are congruent.
• Both triangles are obtuse triangles in which two of their angles measure \(40^{\circ}\) and \(20^{\circ}\). All of their corresponding sides are congruent.
• Both triangles are isosceles triangles in which one angle measures \(15^{\circ}\).
• Both triangles are isosceles triangles with two equal angles of \(55^{\circ}\). All corresponding sides are congruent.
• Both triangles are acute triangles in which two of their angles measure \(40^{\circ}\) and \(80^{\circ}\). All of their corresponding sides are congruent.
• Both triangles are acute triangles in which one angle measures \(60^{\circ}\). Two of their corresponding sides are congruent.
• Both triangles are equilateral triangles.

Review (Answers)

To see the Review answers, open this PDF file and look for section 4.3.

Additional Resources

Interactive Element

Video: Congruent and Similar Triangles

Activities: Congruent Triangles Discussion Questions

Study Aids: Triangle Congruence Study Guide

Practice: CPCTC

Real World: Congruent Triangles

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CPCTC: Definition, Postulates, Theorem, Proof, Examples

Created: December 30, 2023

Last updated: January 8, 2024

Welcome to Brighterly , where learning mathematics is an exciting adventure! Today, we invite you to delve into the captivating world of geometry as we explore the concept of CPCTC – Corresponding Parts of Congruent Triangles are Congruent. At Brighterly, we believe in making math accessible, engaging, and tailored for young minds.

In this article, we will embark on a journey through the definition, postulates, theorem, proof, and examples of CPCTC. By understanding this fundamental principle, our young mathematicians at Brighterly will develop a solid foundation in geometry, logical reasoning, and problem-solving skills.

Geometry, with its intricate shapes and spatial relationships, plays a vital role in numerous real-life applications, from designing buildings to creating stunning works of art. That’s why it’s crucial for our students to grasp the essence of CPCTC, which establishes the congruence of corresponding parts in congruent triangles.

What Is CPCTC in Geometry?

Geometry is a fascinating area of mathematics that involves shapes, sizes, and properties of space. One of the most interesting concepts in geometry is CPCTC. If you’re wondering, ‘What does CPCTC stand for?’, it’s an abbreviation that stands for Corresponding Parts of Congruent Triangles are Congruent. In simpler terms, if two triangles are congruent (exactly the same in size and shape), then their corresponding parts (sides and angles) are also congruent.

Understanding CPCTC can be a big step in your journey to mastering geometry. It is used in various real-life scenarios such as architecture, engineering, and even in art. So, let’s delve deeper and understand the concept of CPCTC.

CPCTC: Definition

CPCTC is a principle in geometry that affirms that if two triangles are congruent, then their corresponding parts – the sides and angles that match – are also congruent. This principle is widely used in proofs and problem-solving in geometry. Imagine two identical triangle-shaped cookies. The corresponding sides (the equal lengths) and angles (the equal corners) of these cookies would be exactly the same, and that’s precisely what CPCTC is all about!

Congruent Triangles

In geometry, congruent triangles are triangles that are identical in terms of size and shape. If you were to place one triangle on top of the other, they would match perfectly. The concept of congruent triangles is fundamental in understanding CPCTC. For two triangles to be congruent, they must satisfy one of the following conditions: Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), or Right angle-Hypotenuse-Side (RHS) congruence rules.

Corresponding Parts

The term ‘corresponding parts’ refers to the sides and angles in a triangle that occupy the same relative position. In congruent triangles, the corresponding parts are equal in measurement. It means that the first angle of one triangle is congruent to the first angle of the other, the second to the second, and so on. Similarly, the sides of the triangles that correspond are also equal in length. This is a crucial concept to understand when learning about CPCTC.

CPCTC Triangle Congruence

CPCTC is a powerful tool that helps us in proving the congruence of parts of two triangles, given that the triangles themselves are congruent. This concept is derived from the principles of triangle congruence mentioned above (SSS, SAS, ASA, AAS, RHS). Once we have established the congruence of the triangles using any of these rules, we can assert that their corresponding parts are congruent by CPCTC.

CPCTC Proof

The proof of CPCTC is based on the postulates and theorems of geometry. Here’s how it works:

Suppose we have two triangles, Triangle ABC and Triangle DEF. We know that they are congruent (using one of the congruence rules). By definition of congruence, the corresponding parts of these triangles are congruent. Therefore, we can say that ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F, AB ≅ DE, BC ≅ EF, and AC ≅ DF.

Solved Examples on CPCTC

Let’s apply CPCTC in practice with the following example:

Example: In two congruent triangles ABC and DEF, ∠A = 40°, ∠B = 80°, and ∠C = 60°. Find the measures of ∠D, ∠E, and ∠F.

Solution: Since the triangles are congruent, by CPCTC, the corresponding angles of the triangles will be congruent. Therefore, ∠D = ∠A = 40°, ∠E = ∠B = 80°, and ∠F = ∠C = 60°.

This example demonstrates how we can use CPCTC to solve problems involving congruent triangles.

Practice Problems on CPCTC

To further reinforce your understanding of CPCTC, here are some practice problems:

If triangle ABC is congruent to triangle DEF and AB = 7 cm, BC = 8 cm, and AC = 9 cm, find the lengths of DE, EF, and DF.

In two congruent triangles PQR and STU, ∠P = 35° and ∠Q = 55°. Determine the measures of ∠S and ∠T.

Remember, practice is the key to mastering any concept, especially when it comes to complex geometrical principles like CPCTC.

In conclusion, CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is a vital concept within the realm of geometry that holds immense value for our young mathematicians at Brighterly. By understanding CPCTC, our students unlock the ability to analyze the congruence of triangles and deduce the congruence of their corresponding parts.

Through this journey into the world of CPCTC, we empower our students to develop critical thinking skills, logical reasoning, and problem-solving abilities. By delving into the definition, postulates, theorem, and proof of CPCTC, our students gain a comprehensive understanding of the interconnections between congruent triangles and their corresponding parts.

At Brighterly, we believe in making mathematics enjoyable and accessible. By incorporating CPCTC into our curriculum, we foster a deep appreciation for the beauty and practical applications of geometry. From architecture to engineering, our young learners discover the significance of CPCTC in various real-world contexts.

Frequently Asked Questions on CPCTC

What does cpctc stand for.

CPCTC stands for “Corresponding Parts of Congruent Triangles are Congruent.” This acronym highlights the principle that when two triangles are congruent, their corresponding parts – angles and sides in the same positions – are also congruent. This concept serves as a powerful tool in geometry to establish relationships between congruent triangles.

How is CPCTC used in geometry?

CPCTC plays a crucial role in proving congruence and solving problems in geometry. By applying CPCTC, we can make assertions about the equality of corresponding parts when two triangles are known to be congruent. For example, if we have proven that two triangles are congruent using the SAS (Side-Angle-Side) criterion, we can confidently state that their corresponding sides and angles are congruent.

What are congruent triangles?

Congruent triangles are triangles that are identical in both shape and size. When two triangles are congruent, all corresponding angles and sides have equal measures. This means that if you were to superimpose one congruent triangle onto another, they would perfectly overlap.

How can I prove that two triangles are congruent?

There are several methods to prove the congruence of triangles. These include the Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Right angle-Hypotenuse-Side (RHS) congruence rules. To prove congruence, you need to show that the corresponding elements of the triangles satisfy one of these criteria.

Why is CPCTC important in geometry?

CPCTC is a fundamental concept in geometry that allows us to make logical connections between congruent triangles. It provides a systematic approach to deducing the equality of corresponding parts, which is vital for solving geometric problems and proving theorems. Understanding CPCTC enables students to analyze the relationships between congruent triangles and apply their knowledge to various geometric scenarios.

Can CPCTC be used in real-life applications?

Yes, CPCTC finds practical applications in real-life situations. For example, architects and engineers use CPCTC to ensure the accuracy and consistency of corresponding elements in structures. By applying the principles of CPCTC, they can ensure that congruent components of a building or object fit together seamlessly.

How can I practice applying CPCTC?

To practice applying CPCTC, you can work through a variety of geometry problems and proofs that involve triangle congruence. Start with basic examples and gradually progress to more complex scenarios. Engaging in hands-on activities, such as constructing congruent triangles and analyzing their corresponding parts, can also enhance your understanding of CPCTC.

• CPCTC – Wikipedia
• CPCTC – Regents Prep
• Congruent Triangles and CPCTC – GeoGebra

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CPCTC: Definition, Postulates, Theorem, Proof, Examples

What is cpctc in geometry, corresponding parts, cpctc triangle congruence, solved examples on cpctc, practice problems on cpctc, frequently asked questions on cpctc.

CPCTC in geometry simply stands for “Corresponding Parts of Congruent Triangles are Congruent.” CPCTC theorem states that if two or more triangles are congruent, then their corresponding angles and sides are congruent as well.

CPCTC: Definition

CPCTC meaning is “Corresponding Parts of Congruent Triangles are Congruent.” You can only use CPCTC after you know that two triangles are congruent. What is the CPCTC theorem? 

The CPCTC theorem states that if two or more triangles are congruent to each other, then the corresponding angles and the sides of the triangles are also congruent to each other. Let us understand the meaning of congruent triangles and corresponding parts in detail.

Congruent Triangles

A triangle has three sides and three angles. The two triangles need to be of the same size and shape to be congruent. Both the triangles under consideration should superimpose on each other. For two triangles to be congruent, all 3 sides and angles should be congruent . 

We also need to remember that when we state that two triangles are congruent, we name them such that the letters for corresponding vertices are in the same order. Take a look at the image given below:

In the figure above, we see that the letters A, B, and C for the vertices of ABC correspond to the letters D, E, and F for the vertices of DEF respectively. We can state the congruences of sides and angles.

Side congruences: $\overline{AB} \cong \overline{DE},\; \overline{BC}  \cong \overline{EF},\; \overline{AC}  \cong \overline{DF}$

Angle congruences: $\angle ABC \cong \angle QRP,\;  \angle BCA \cong \angle RPQ,\; \angle CAB \cong \angle PQR$

Now, we name the congruent triangles so that the letters for corresponding vertices match up in the triangle congruence statement.

$\Delta ABC \cong \Delta DEF$

The corresponding parts of two triangles refer to the parts of two triangles that are in the same relative position . When a triangle’s three sides correspond to those of another triangle, they are at the same location or space. Corresponding angles signify that the three angles of one triangle are located in the same area as those of the other triangle. Take a look at the figure given below.

Here, $\Delta MNO \cong \Delta XYZ$. It means that the 3 pairs of sides and 3 pairs of angles of $\Delta MNO$ are congruent to the 3 pairs of corresponding sides and 3 pairs of corresponding angles of $\Delta XYZ$.

Now, let’s identify the 3 corresponding sides and the 3 corresponding angles:

 MN corresponds to XY; NO corresponds to YZ; MO corresponds to XZ. 

$\angle M$ corresponds to $\angle X;\; \angle N$ corresponds to $\angle Y;\; \angle O$ corresponds to $\angle Z$.

Since $\Delta MNO \cong \Delta XYZ$, as per the CPCTC theorem, the corresponding sides and angles are equal, which means, 

$MN = XY,\; NO = YZ,\; MO = XZ,$ and

$\angle M = \angle X,\; \angle N = \angle Y,\; \angle O = \angle Z$.

According to the CPCTC, all of the corresponding sides and angles are congruent when two triangles are congruent by any criteria. Here, we’ll talk about five triangle congruence criteria or postulates.

Visually, we can represent the rules as follows:

CPCTC Proof

Once it is shown that two triangles are congruent using one of the above congruence methods, we also know that all corresponding parts of the congruent triangles are congruent (abbreviated CPCTC).

Let’s consider an example: 

Consider $\Delta DAB$ and $\Delta DCB$ in which $BA = BC$ and $AD = DC$ are given.

Now, let’s prove the CPCTC theorem.

 $BA = BC$                 (Given)

 $AD = DC$                (Given)

 $BD = BD$                 (Reflexive Property)

Thus, $Delta DAB \cong DCB$ by SSS criteria (Side-Side-Side) 

Since the two triangles are congruent, we can use the CPCTC property and write

$\angle DAB  \cong  \angle DCB,\; \angle ADB \cong \angle CDB$ and  $\angle ABD \cong  \angle CBD$

• There are five conditions to determine if two triangles are congruent. They are

            SSS, AAS, SAS, ASA, and HL criteria.          

• SSS, AAS, SAS, ASA, HL use corresponding parts to prove triangles congruent, while CPCTC uses congruent triangles to prove corresponding parts congruent.
• By CPCTC, if two triangles are congruent, then you know that all pairs of corresponding sides are congruent and all pairs of corresponding angles are congruent.
•  We use the symbol $“\cong”$ to denote congruence.
• If you know the two triangles are congruent, you need to match up the letters for their corresponding vertices, and then use those to state the congruences of angles and sides.

In this article, we learned in detail about the CPCTC definition, theorem, and proof. Let’s solve some examples and practice problems for better understanding.

Example 1: Given $\overline{YX} \cong \overline{YZ},\; \angle XYW \cong \angle ZYW$ , prove that $\overline{XW}  \cong  \overline{ZW}$ .

$YX \cong YZ$                    (Given)

$\angle XYW \cong \angle ZYW$         (Given)

$WY \cong WY$                 (Reflexive property)

Thus, $\Delta WXY \cong \Delta WZY$  by SAS criteria (Side-Angle-Side)

Thus, $XW \cong ZW$  (by CPCTC)

Example 2: Given: $\angle A \cong \angle C$ , BY bisects $\angle ABC$ . Prove that $\overline{AB} \cong \overline{CB}$ .

Solution: 

 $\angle A \cong \angle C$                        (Given)

 BY bisects $\angle ABC$          (Given)

$\angle ABY \cong \angle CBY$               (Definition of angle Bisector)

 $BY  \cong  BY$                     (Reflexive property/common side)

$\Delta AYB \cong \Delta CYB$  by AAS criteria  (Angle-Angle-Side)

Thus,  $\overline{AB}  \cong \overline{CB}$            (CPCTC)

Example 3: Observe the figure given below and find the measure of $\angle I$   and length of $\overline{VU}$   using the CPCTC theorem, if $\Delta HJI \cong \Delta TVU$ .

Solution: Given that $\Delta HJI \cong \Delta TVU$.

By applying the CPCTC theorem, 

        $\overline{HJ} = \overline{TV} = 25\; mm$

                 $\angle HJI \cong \angle VTU$

                  $\overline{HI}  \cong \overline{TU} = 50\; mm$. 

We can identify that $\overline{JI}$ and $\overline{VU}$ are the corresponding sides of two congruent triangles $\Delta HJI$ and $\Delta TVU$. 

Therefore, $\overline{JI} = \overline{VU} = 43$ units 

Thus, the length of $\overline{VU}$  is 43 units.

Also, by the same theorem,  $\angle I = \angle U$. 

Thus, $\angle I = 30^{\circ}$

Example 4: Observe the figure given below, in which $AC = CD$ and BC is perpendicular to AD. Find y by using the CPCTC theorem .

Solution: First let us prove that $\Delta ABC \cong \Delta BCD$,

$AC = CD$           (given)

$BC = BC$           (common side)

$\angle BCA = \angle BCD$ (as BC is perpendicular to AD)

Thus, $\Delta ABC \cong \Delta BCD$ (SAS criterion)

$AB = BD$ ( By CPCTC)

Now as $AB = BD$

Thus, $6y = 48$

$y = 8$ units

Example 5: Observe the figure given below in which $\angle PQR = \angle STU$ and $\angle PRQ  = \angle SUT$ , $\overline{QR} = \overline{TU}$ . Find the value of x and y by using the CPCTC theorem.

Solution: $\angle PQR = \angle STU$   (Given)

                $\angle PRQ  = \angle SUT$    (Given)

                       $\overline{QR} = \overline{TU}$       (Given)

Thus, $\Delta RPQ \cong  \Delta UST$  (ASA criterion)

As $\angle R = \angle U$, we get $x = 30^{\circ}$

The value of $x = 30^{\circ}$

As we know, sum of three angles $= 180^{\circ}$

$\angle R + \angle P + \angle Q  = 180^{\circ}$

$30^{\circ} + \angle P + 60^{\circ} =  180^{\circ}$

Thus, $\angle P = 180 \;-\; 90 = 90^{\circ}$

Now, by CPCTC, we know $\angle P = \angle S$ 

Thus, $\angle S = y = 90^{\circ}$

The value of $y =  90^{\circ}$

Attend this quiz & Test your knowledge.

What does CPCTC in math stand for?

If two triangles are congruent,it means that all of its corresponding _________ and _________ are congruent., cpctc theorem is used only when two triangles are ___., which criteria proves these two triangles congruent.

What additional information is required for the 2 triangles to be congruent by ASA?

What is reflexive property?

The reflexive property states that any line segment or angle or shape is congruent to itself.

What must be proven before using CPCTC?

To prove CPCTC, we first prove the triangles congruent using SSS, AAS, SAS, ASA, or HL. Next, we can use CPCTC, which states that the other parts of the triangle are also congruent.

When and why is CPCTC used?

CPCTC is commonly used at or near the end of a proof, which asks the student to show that the two sides or angles are congruent. It is frequently used to assess the similarity and congruence of two or more triangles.

What is the third angle theorem?

If two angles in one triangle are congruent to two angles in another triangle, then the third pair of angles must also be congruent. This is called the third angle theorem.

Is CPCTC true for similar triangles?

In similar triangles, corresponding angles are equal, whereas corresponding sides of the triangles are not equal but proportional. So, we cannot apply the CPCTC theorem for similar triangles.

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CPCTC – Definition, Proof, and Examples

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Definition of CPCTC

Proof of cpctc, corresponding sides, corresponding angles, other corresponding elements, application in geometric reasoning, construction of congruent figures, generalizability, historical significance, geometric proofs, engineering and architecture, surveying and mapping, computer-aided design (cad), robotics and kinematics, physics and trigonometry, navigation and global positioning systems (gps), art and design.

Within the realm of geometry , a remarkable principle known as Corresponding Parts of Congruent Triangles is Congruent (CPCTC) unveils a hidden symmetry at the heart of triangle congruence .

CPCTC is a powerful tool, illuminating the profound relationship between congruent triangles and their corresponding parts . By establishing that corresponding sides , angles , and other elements of congruent triangles are themselves congruent, CPCTC enables geometric reasoning and proof construction with elegance and precision.

In this article, we delve into the depths of CPCTC , unraveling its underlying principles, exploring its practical applications, and demonstrating how this powerful concept unlocks many geometric insights.

The Corresponding Parts of Congruent Triangles are Congruent (CPCTC) is a fundamental principle in geometry that states that if two triangles are congruent, their corresponding parts (sides, angles, and other elements) are also congruent.

In other words, when two triangles are proven to be congruent through methods such as Side-Angle-Side (SAS) , Side-Side-Side (SSS) , or Angle-Side-Angle (ASA) , the corresponding parts of these congruent triangles can be considered identical in terms of their measures or lengths.

This principle allows for applying c ongruence relationships to individual parts of triangles and facilitates geometric proofs and deductions . CPCTC is a powerful tool for establishing relationships and making conclusions about congruent triangles and their corresponding elements , contributing to the precision and rigor of geometric reasoning .

Below is the figure-1. We present a generic representation of CPCTC.

The proof of CPCTC ( Corresponding Parts of Congruent Triangles are Congruent ) relies on the establishment of triangle congruence using one of several congruence criteria, such as Side-Angle-Side (SAS) , Side-Side-Side (SSS) , or Angle-Side-Angle (ASA) . Let’s explore the proof of CPCTC in detail:

Proof of CPCTC :

• Given that triangle, ABC is congruent to triangle DEF (proven using a congruence criterion such as SAS, SSS, or ASA ).
• We want to prove: Corresponding parts of congruent triangles are congruent .
• CPCTC states that congruent triangles’ corresponding sides, angles, and other elements are congruent .
• Side AB corresponds to side DE .
• Side BC corresponds to side EF .
• Side AC corresponds to side DF .
• Angle A corresponds to angle D .
• Angle B corresponds to angle E .
• Angle C corresponds to angle F .
• Therefore, we have established that the corresponding sides and angles of congruent triangles ABC and DEF are congruent.
• Other corresponding elements, such as medians , altitudes , and diagonals , will also be congruent between the congruent triangles.
• Hence, we have successfully proven CPCTC , which states that corresponding parts of congruent triangles are congruent .

The proof of CPCTC relies on the foundation of triangle congruence , which can be established through different congruence criteria .

Once the congruence of the triangles is established, the proof applies the congruence relationships to the corresponding parts of the triangles , including sides , angles , and other elements.

Properties  

The property of Corresponding Parts of Congruent Triangles are Congruent (CPCTC) states that if two triangles are congruent, their corresponding parts (sides, angles, and other elements) are also congruent. Let’s explore the properties of CPCTC in detail:

When two triangles are congruent , CPCTC ensures their corresponding sides are congruent. This means that the lengths or measures of the sides in one triangle are equal to the lengths or measures of the corresponding sides in the other triangle. For example, if triangle ABC is congruent to triangle DEF , then AB = DE , BC = EF , and AC = DF .

CPCTC also guarantees that the corresponding angles of congruent triangles are congruent. This implies that the angles in one triangle are equal to the measures of the corresponding angles in the other triangle. For instance, if triangle ABC is congruent to triangle DEF , then ∠A = ∠D , ∠B = ∠E , and ∠C = ∠F .

CPCTC extends beyond sides and angles to include other corresponding elements of congruent triangles. These elements include medians , altitudes , diagonals , and other geometric properties . If two triangles are congruent, CPCTC ensures that their corresponding elements, such as medians AB’ and DE’ or altitudes CD and FG , are also congruent.

CPCTC is crucial in geometric reasoning and proof construction . By establishing the congruence of triangles using methods such as SAS , SSS , or ASA , CPCTC allows for applying congruence relationships to individual parts of triangles. This aids in proving various geometric theorems and solving problems involving congruent triangles .

CPCTC enables the construction of congruent figures using known congruent triangles. By knowing that corresponding parts of congruent triangles are congruent, we can create figures or shapes with specific congruence relationships based on the congruence of their component triangles .

The property of CPCTC is not limited to triangles but can be extended to other polygons . If two polygons are congruent , their corresponding parts will also be congruent, following the principles of CPCTC .

CPCTC is a powerful property in geometry that ensures that corresponding parts of congruent triangles (or polygons) are congruent. This property provides a reliable foundation for geometric reasoning , proof construction , and the exploration of congruence relationships among various elements of geometric figures .

The historical background of the principle “Corresponding Parts of Congruent Triangles are Congruent” (CPCTC) traces back to the development of Euclidean geometry and the study of triangles .

Euclidean geometry , named after the ancient Greek mathematician Euclid, formed the foundation of geometric reasoning and proof construction . Euclid’s work, particularly his seminal work “ Elements ,” established a rigorous axiomatic system for geometry and introduced various theorems and principles.

The concept of congruent triangles , where two triangles have the same shape and size, dates back to ancient times. However, the specific idea that corresponding parts of congruent triangles are congruent is attributed to the work of Euclid and his followers.

Euclid’s “ Elements ” (circa 300 BCE) served as a comprehensive compilation of the mathematical knowledge of the time. Book I of “ Elements ” focused on plane geometry, including studying triangles and their properties.

Within Book I , Euclid introduced the concept of congruence and the criteria for triangle congruence, such as Side-Angle-Side (SAS) , Side-Side-Side (SSS) , and Angle-Side-Angle (ASA) . These criteria provided a basis for proving that the two triangles were congruent.

While Euclid did not explicitly state the CPCTC principle as a separate theorem, it was implicit in his work. Euclid’s theorems and proofs demonstrated the congruence of corresponding parts by relying on the established criteria for triangle congruence .

Over time, as Euclidean geometry became widely studied and disseminated, the principle of CPCTC gained recognition as a powerful tool for establishing congruence relationships between corresponding parts of triangles.

Throughout history, mathematicians and geometers built upon Euclid’s work and expanded the field of geometry . Theorems and principles related to congruent triangles , including CPCTC , were refined, extended, and applied in various mathematical contexts.

Today, CPCTC is a fundamental principle in geometric reasoning and proof construction . It continues to be taught and utilized in geometry curricula worldwide, serving as a key concept for establishing congruence relationships and making deductions about the corresponding parts of congruent triangles .

The historical background of CPCTC showcases its development alongside the broader evolution of geometry , from Euclid’s foundational work to modern applications and advancements. The principle has endured over centuries , offering mathematicians and students a powerful tool for exploring the geometric properties of triangles .

Applications 

The principle of Corresponding Parts of Congruent Triangles is Congruent (CPCTC) finds applications in various fields where geometry is fundamental. Here are some notable applications of CPCTC :

CPCTC is a fundamental tool in geometric proofs , allowing mathematicians to establish congruence relationships between corresponding parts of triangles. It enables the construction of logical and rigorous arguments to prove geometric theorems , establish relationships between figures, and deduce properties based on congruence.

CPCTC is applied in engineering and architecture to ensure accuracy and consistency in the design and construction of structures. By utilizing congruent triangles , engineers and architects can ensure that corresponding parts of structural components, such as beams, trusses, or supports, are identical in size and shape, enhancing stability and structural integrity .

CPCTC plays a vital role in surveying and mapping applications. It allows surveyors to establish congruence between corresponding points, angles, or sides on the ground and in maps or plans. This aids in accurately measuring, positioning, and mapping land boundaries , infrastructure , and geographic features .

CPCTC is utilized in CAD software to ensure consistency and precision in geometric modeling and design. By applying CPCTC principles, CAD systems can automatically maintain congruence relationships between corresponding parts of objects, facilitating accurate transformations , scaling , and assembly of components in virtual design environments .

CPCTC ensures precise movements and alignments of robotic limbs or end effectors in robotics . By establishing congruence between corresponding parts of robotic components, engineers can achieve accurate positioning and kinematic control , enabling tasks such as pick-and-place operations , assembly , and robotic manipulations .

CPCTC finds applications in physics and trigonometry , where the congruence of triangles is utilized to analyze geometric properties and physical phenomena. Using CPCTC , physicists can establish congruence between corresponding parts of triangles in various contexts, such as projectile motion , wave propagation , or electrical circuit analysis .

CPCTC establishes congruence between corresponding elements in navigation systems and GPS technology in triangulation methods . By ensuring congruence between measured angles or distances, GPS systems can accurately determine positions on Earth’s surface and aid in navigation and tracking applications .

CPCTC is utilized in art and design to create visually pleasing and harmonious compositions. Artists and designers often apply the principles of CPCTC to establish congruent relationships between corresponding elements in their works, such as shapes , forms , or patterns , ensuring balance and visual coherence .

The applications of CPCTC extend beyond these examples and find relevance in various fields where geometric reasoning , congruence , and consistency play a crucial role. CPCTC enables accurate measurements, reliable proofs, and precise geometric relationships , enhancing the understanding and application of geometry in practical contexts .

Triangle ABC is congruent to triangle DEF . Prove: Angle ABC is congruent to angle DEF .

Since triangle ABC is congruent to triangle DEF , we can write it as:

Therefore, by CPCTC , we can conclude that angle ABC is congruent to angle DEF .

Triangle PQR is congruent to triangle STU . Prove: Side PQ is congruent to side ST .

Since triangle PQR is congruent to triangle STU , we can write it as:

By CPCTC , we can conclude that side PQ is congruent to side ST .

Triangle XYZ is congruent to triangle UVW . Prove: Side XY is congruent to side VW .

Since triangle XYZ is congruent to triangle UVW , we can write it as:

By CPCTC , we can conclude that side XY is congruent to side VW .

Triangle MNO is congruent to triangle PQR . Prove: Angle M is congruent to angle P .

Since triangle MNO is congruent to triangle PQR , we can write it as:

By CPCTC , we can conclude that angle. M is congruent to angle P .

Triangle DEF is congruent to triangle GHI . Prove: Side DE is congruent to side GH .

Since triangle DEF is congruent to triangle GHI , we can write it as:

By CPCTC , we can conclude that side DE is congruent to side GH .

Triangle JKL is congruent to triangle MNP . Prove: Angle K is congruent to angle N .

Since triangle JKL is congruent to triangle MNP , we can write it as:

By CPCTC , we can conclude that angle K is congruent to angle N .

Triangle ABC is congruent to triangle XYZ . Prove: Side AC is congruent to side XZ .

Since triangle ABC is congruent to triangle XYZ , we can write it as:

By CPCTC , we can conclude that side AC is congruent to side XZ.

Triangle RST is congruent to triangle UVW . Prove: Angle S is congruent to angle U .

Since triangle RST is congruent to triangle UVW , we can write it as:

By CPCTC , we can conclude that angle S is congruent to angle U .

All images were created with MATLAB.

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