Unlocking Proving Triangle Congruence through Congruence Theorems

Two triangles are congruent if their corresponding sides and angles are congruent. However, there could be cases where not all side lengths or angle measures are known. The good news is that congruence can still be verified depending on which parts are known.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

The primary goal of this lesson is to investigate exactly how much information about a pair of triangles has to be known in order to conclude that the triangles are congruent.

Explore

Investigating Congruence of Triangles Given Angles

Consider the following statement.

If two triangles have corresponding congruent angles, then they are congruent.

Using the following applet, investigate if that statement is true. To do so, try to map △ ABC onto △ JLK by applying different rigid motions to △ ABC.

Comparing different pairs of triangles with corresponding congruent angles

After exploring different cases, what can be said about the given statement?

Extra

How to Use the Applet

In the applet, rigid motions can be applied only on △ ABC.

To translate △ ABC, select its interior region and slide.
Point P acts as the center of rotation and can be moved by dragging it.
To rotate △ ABC about P, click on any vertex of △ ABC and drag it.
The given line acts as a line of reflection which can be moved by dragging it. Similarly, its inclination can be changed by dragging either of its two points.
To reflect △ ABC across the line of reflection, push the Reflect button.

Explore

Investigating Congruence of Triangles Given Two Sides and the Included Angle

In the previous exploration, it was seen that a pair of triangles can have corresponding congruent angles but not be congruent triangles. Therefore, relying only on the relationship of only angles is not a valid criterion.

Angle-Angle-Angle is not a valid criterion for proving triangle congruence.

Use segments AB and AC to construct two different triangles, one at a time, in such a way that the angle formed at A has the same measure in both triangles.

Applet to construct different triangles, Given two sides

Once the two triangles have been drawn, find the side lengths and angle measures of each triangle. Can any relationship between the triangles be found? Repeat the process a few times to see if the relationship remains true.

Discussion

Side-Angle-Side Congruence Theorem

The previous exploration suggests that two triangles are congruent whenever they have two pairs of corresponding congruent sides and the corresponding included angles are congruent. In fact, this conclusion is formalized in the Side-Angle-Side Congruence Theorem

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

Based on the diagram above, the theorem can be written as follows.

AB ≅ DE ∠ A ≅ ∠ D AC ≅ DF ⇒ △ ABC ≅ △ DEF

Proof

Side-Angle-Side Congruence Theorem

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

The primary purpose of the proof is finding a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways. One of the ways will be shown here.

Translate △ DEF So That Two Corresponding Vertices Match

Translate △ DEF so that D is mapped onto A. If this translation maps △ DEF onto △ ABC, the proof is complete.

Since the image of the translation does not match △ ABC, at least one more transformation is needed.

Rotate △ AE'F' So That Two Corresponding Sides Match

Rotate △ AE'F' counterclockwise about A so that a pair of corresponding sides match. If the image of this transformation is △ ABC, the proof is complete. Note that this rotation maps E' onto B. Therefore, the rotation maps AE' onto AB.

As before, the image does not match △ ABC. Therefore, a third rigid motion is required.

Reflect △ ABF'' So That Two More Corresponding Sides Match

Reflect △ ABF'' across AB. Because reflections preserve angles, AF'' is mapped onto AC. Additionally, it is given that AC=AF''. Therefore, F'' is mapped onto C, which gives that BF'' is mapped onto BC.

This time the image matches △ ABC.

Consequently, after a sequence of rigid motions, △ DEF can be mapped onto △ ABC. This means that △ DEF and △ ABC are congruent triangles. The proof is complete.

Example

Identifying Congruent Triangles

In the following diagram, triangles ADE and BCE are congruent, and ∠ ADC is congruent to ∠ BCD.

How many more pairs of congruent triangles are there in the diagram? Name each congruent triangle pair.

Answer

There are two more pairs of congruent triangles. △ ADC &≅ △ BCD △ ABD &≅ △ BAC

Hint

Remember, if two triangles are congruent, then their corresponding sides and angles are congruent.

Solution

Start by highlighting the given pair of congruent triangles, △ ADE and △ BCE.

Since these triangles are congruent, their corresponding parts are congruent. This implies that AD is congruent to BC.

First Pair

Because △ ADE and △ BCE are parts of △ ADC and △ BCD respectively, consider triangles ADC and BCD.

Notice that CD is a common side for triangles ADC and BCD. Because of the Reflexive Property of Congruence, CD is congruent to itself. Next, list the corresponding congruent parts between these two triangles. cl AD ≅ BC & Side ∠ADC ≅ ∠BCD & Angle DC ≅ CD & Side By the Side-Angle-Side (SAS) Congruence Theorem, it can be concluded that △ ADC and △ BCD are congruent.

△ ADC ≅ △ BCD

Second Pair

Next, consider triangles ABD and BAC. Because △ ADE and △ BCE are congruent, ∠ ADB and ∠ BCA are congruent. Additionally, since △ ADC and △ BCD are congruent, CA is congruent to DB.

Below, the corresponding congruent parts between △ ABD and △ BAC are listed. cl AD ≅ BC &Side ∠ADB ≅ ∠BCA &Angle DB ≅ CA &Side One more time, the Side-Angle-Side (SAS) Congruence Theorem can be used to conclude that triangles ABD and BAC are congruent.

△ ABD ≅ △ BAC

Third Pair

The last two triangles to consider are triangles ABE and DEC. Unlike the first two pairs, these dimensions seem to be quite different. Therefore, it can be concluded that they are not congruent.

Consequently, in the initial diagram, there are two more pairs of congruent triangles in addition to the given one.

Explore

Investigating Congruence of Triangles Given Two Angles and the Included Side

Use segment AB and the rays AX and BY to construct two different triangles, one at a time, in such a way that the following conditions are met.

The angle formed at A has the same measure in both triangles.
The angle formed at B has the same measure in both triangles.

Applet to construct different triangles, Given one side and two angles

Once the two triangles are drawn, find the side lengths and angle measures of each triangle. Is there any relationship between the triangles? Repeat the process a few times to see if the relationship remains true.

Discussion

Angle-Side-Angle Congruence Theorem

The following statement could be seen in the previous applet. When two triangles have two pairs of corresponding congruent angles, and the included corresponding sides are congruent, the triangles are then congruent. That leads to the second criteria for triangle congruence.

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

Based on the diagram above, the theorem can be written as follows.

∠ A ≅ ∠ D AB ≅ DE ∠ B ≅ ∠ E ⇒ △ ABC ≅ △ DEF

Proof

Angle-Side-Angle Congruent Theorem

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

The goal of the proof is to find a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways. One of the ways will be shown here.

Translate △ DEF So That Two Corresponding Vertices Match

Translate △ DEF so that D is mapped onto A. If this translation maps △ DEF onto △ ABC, the proof is complete.

Translation that maps vertex D of triangle DEF onto vertex A of triangle ABC

Since the image of the translation does not match △ ABC, at least one more transformation is needed.

Rotate △ AE'F' So That Two Corresponding Sides Match

Rotation that maps vertex E' of AE'F' onto B of ABC

As before, the image does not match △ ABC. Therefore, a third rigid motion is required.

Reflect △ ABF'' So That All Corresponding Sides Match

Reflect △ ABF'' across AB. Because reflections preserve angles, AF'' and BF'' are mapped onto AC and BC, respectively. Then, the point of intersection of the original rays F'' is mapped onto the point of intersection of the image rays C.

This time the image matches △ ABC.

Consequently, after a sequence of rigid motions △ DEF can be mapped onto △ ABC. This means that △ DEF and △ ABC are congruent triangles.

Example

Write Equations Based on Congruent Triangles

Consider the following diagram.

What is the value of x+y+z?

Hint

Take note that QS is a common side for two triangles. Use the fact that if two triangles are congruent, their corresponding sides and angles are congruent.

Solution

Notice that QS is a common side for triangles PQS and RSQ.

By the Reflexive Property of Congruence, QS is congruent to itself. Additionally, ∠ PQS and ∠ RSQ are congruent, as are ∠ PSQ and ∠ RQS. cl ∠ PQS ≅ ∠ RSQ & Angle QS ≅ QS & Side ∠ PSQ ≅ ∠ RQS & Angle Consequently, △ PQS and △ RSQ are congruent because of the Angle-Side-Angle (ASA) Congruence Theorem. Therefore, the corresponding sides and angles are congruent. △ PQS ≅ △ RSQ ⇒ { ∠ P &≅ ∠ R PQ &≅ RS PS &≅ RQ . By definition, congruent angles have the same measure, and congruent segments have the same length. Therefore, the congruence statements on the right-hand side support the formation of the following three equations. 8z+2=50 & (I) 6=2y+x & (II) 5x-3=4.5 & (III) By solving Equation (I), the value of z can be found.

8z+2=50

▼

Solve for z

SubEqn

LHS-2=RHS-2

8z = 48

DivEqn

.LHS /8.=.RHS /8.

z = 6

Next, solve Equation (III) to find the value of x.

5x-3=4.5

▼

Solve for x

AddEqn

LHS+3=RHS+3

5x = 7.5

DivEqn

.LHS /5.=.RHS /5.

x = 7.5/5

UseCalc

Use a calculator

x = 1.5

Then, the value of y can be found by substituting x=1.5 into the Equation (II) and solving the resulting equation for y.

6=2y+x

Substitute

x= 1.5

6=2y+ 1.5

▼

Solve for y

SubEqn

LHS-1.5=RHS-1.5

4.5 = 2y

DivEqn

.LHS /2.=.RHS /2.

4.5/2 = y

UseCalc

Use a calculator

2.25 = y

RearrangeEqn

Rearrange equation

y = 2.25

Finally, the required sum can be calculated by substituting the values found for x, y, and z.

x+y+z

SubstituteValues

Substitute values

1.5+ 2.25+ 6

AddTerms

Add terms

9.75

Explore

Congruence of Triangles Given Three Pairs of Congruent Sides

At the beginning of the lesson, it was shown that the Angle-Angle-Angle is not a valid criterion for determining triangle congruence. Next, using the following applet, it will be investigated if the Side-Side-Side is a valid criterion. Use segments AB, AC, and BC to construct two different triangles. Construct the triangles one at a time.

Applet to construct different triangles, Given three sides

Once the two triangles are drawn, find the angle measures of each triangle. Is there any relationship between the triangles? Repeat the process a couple of times to see if the relationship holds true.

Discussion

Side-Side-Side Congruence Theorem

As seen in the previous exploration, the Side-Side-Side is a valid criterion for checking triangle congruence.

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

Based on the diagram above, the theorem can be written as follows.

AB ≅ DE BC ≅ EF AC ≅ DF ⇒ △ ABC ≅ △ DEF

Proof

Side-Side-Side Congruence Theorem

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

The primary purpose of this proof is finding a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways. One of them will be shown here.

Translate △ DEF So That Two Corresponding Vertices Match

Translate △ DEF so that D is mapped onto A. If this translation maps △ DEF onto △ ABC, the proof is complete.

Since the image of the translation does not match △ ABC, at least one more transformation is needed.

Rotate △ AE'F' So That Two Corresponding Sides Match

Rotate △ AE'F' counterclockwise about A so that a pair of corresponding sides matches. If the image of this transformation is △ ABC, the proof is complete. Note that this rotation maps E' onto B. Consequently, AE' is mapped onto AB.

As before, the image does not match △ ABC. Therefore, a third rigid motion is required.

Reflect △ ABF'' So That Two More Corresponding Sides Match

The points C and F'' are on opposite sides of AB. Now, consider CF'. Let G denote the point of intersection between AB and CF''.

It can be noted that AC = AF'' and BC = BF''. By the Converse Perpendicular Bisector Theorem, AB is a perpendicular bisector of CF''. Points along the perpendicular bisector are equidistant from the endpoints of the segment, so CG = GF''.

Finally, F'' can be mapped onto C by a reflection across AB by reflecting △ ABF'' across AB. Because reflections preserve angles, AF'' and BF'' are mapped onto AC and BC, respectively.

This time the image matches △ ABC.

Consequently, the application of a sequence of rigid motions allows △ DEF to be mapped onto △ ABC. This means that △ DEF and △ ABC are congruent triangles. The proof is complete.

Given three random segments, it is not always possible to construct a triangle. But, when possible, this triangle will be unique. This fact implies that the angle measures of that triangle are also unique.

Therefore, any other triangle with the same side lengths will also have the same angle measures. Consequently, the two triangles are congruent. Notice that, in contrast, having the same angle measures does not force the side lengths to be unique.

Example

Using Triangle Congruence Conditions to Solve Problems

In the following diagram, R_1 is a rectangle, S_1 and S_2 are squares, T_1, T_2, and △ MKL are isosceles triangles, DK is congruent to CL, and GM is congruent to JM.

If m∠ JCK = 3z+8, what is the value of x?

Hint

Using the Segment Addition Postulate and the Side-Side-Side (SSS) Congruence Theorem, prove that △ DGL is congruent to △ CJK. Then, find the measure of ∠ JCK. Use the fact that m∠ JCK + m∠ DCB + m∠ JCH + x = 360^(∘).

Solution

To find the value of x, notice that x, m∠ DCB, m∠ JCK, and m∠ JCH add up 360^(∘). x + m∠ DCB + m∠ JCK + m∠ JCH = 360^(∘) Since R_1 is a rectangle and S_2 is a square, ∠ DCB and ∠ JCH are right angles. Therefore, these angles have a measure of 90^(∘) each. Also, it is given that m∠ JCK = 3z+8. Use this information to solve for x.

x + m∠ DCB + m∠ JCK + m∠ JCH = 360

SubstituteValues

Substitute values

x + 90 + ( 3z+8) + 90 = 360

▼

Solve for z

AddTerms

Add terms

x + 3z + 188 = 360

SubEqn

LHS-188-3z=RHS-188-3z

x = 172-3z

An expression of x was found in terms of z. Therefore, to find the value of x, the value of z should first be known.

Finding the Value of z

If △ DGL and △ CJK can be proven to be congruent, that would provide the needed information to find the value of z. Therefore, focus on those two triangles.

Proving that DL ≅ CK

Notice that KL is common to both triangles. By using the Segment Addition Postulate, the following pair of equations can be written. DL=DK+KL & (I) CK=CL+LK & (II) Since DK is congruent to CL, these segments have the same length, that is, DK = CL. Simlarly, KL=LK. By substituting these expressions into Equation (I), a relation between DL and CK will be obtained.

DL=DK+KL

▼

Substitute values and simplify

SubstituteII

DK= CL, KL= LK

DL= CL+ LK

Segment Addition Postulate

DL=CK

This equation implies that DL and CK are congruent.

DL ≅ CK

Proving that GL ≅ JK

Once more, the Segment Addition Postulate can be used to rewrite GL and JK. GL = GM+ML & (I) JK = JM+MK & (II) Because △ MKL is isosceles, MK is congruent to ML. Therefore, ML=MK. Keep in mind that it is given that GM and JM are congruent. That means GM=JM. These two expressions can be substituted into Equation (I).

GL = GM+ML

▼

Substitute values and simplify

SubstituteII

ML= MK, GM= JM

GL = MK + JM

Segment Addition Postulate

GL = JK

Based on the equation just obtained, it can be concluded that GL and JK are congruent.

GL ≅ JK

Proving that DG ≅ CJ

Since R_1 is a rectangle, S_1 and S_2 are squares, and T_1 and T_2 are isosceles triangles, the following consequences can be drawn.

Given	Consequence
R_1 is a rectangle	DA≅CB
S_1 is a square	DG≅DE
S_2 is a square	CH≅CJ
T_1 is an isosceles triangle	DE≅DA
T_2 is an isosceles triangle	CB≅CH

Next, organize the information in the right-hand column in a flow chart and use the Transitive Property of Congruence to prove that DG ≅ CJ.

Proving that △ DGL ≅ △ CJK

Previously, the following three congruence statements were obtained. cl DL ≅ CK & Side GL ≅ JK & Side DG ≅ CJ & Side The Side-Side-Side (SSS) Congruence Theorem allows to conclude that △ DGL is congruent to △ CJK.

Since corresponding parts of congruent triangles are congruent, it can be concluded that ∠ DGL is congruent to ∠ CJK. Therefore, z=34.

Finding the Value of x

Finally, to find the value of x, substitute z=34 into the equation x = 172-3z. Then solve for x.

x = 172-3z

Substitute

z= 34

x = 172-3( 34)

▼

Solve for x

Multiply

x = 172-102

SubTerms

Subtract terms

x = 70

Explore

Congruence of Triangles Given Two Angles and a Nonincluded Side

Notice that the ASA criterion requires the congruent sides to be included between the two pairs of corresponding congruent angles. Using the following applet, investigate what happens when the congruent sides are not the included sides.

Use segment AB and the rays AX and BY to construct two different triangles, one at a time, in such a way that these conditions are met:

The angle formed at A has the same measure in both triangles.
The angle formed at C, the intersection of the rays, has the same measure in both triangles.

Once the two triangles are drawn, find the side lengths and angle measures of each triangle. Is there any relationship between the triangles? Repeat the process a couple of times to see if the relationship holds true.

Discussion

Angle-Angle-Side Congruence Theorem

As seen in the previous exploration, the Angle-Angle-Side condition is a valid criterion for triangle congruence.

If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

Based on the diagram above, the theorem can be written as follows.

∠A ≅ ∠D ∠B ≅ ∠E BC ≅ EF ⇒ △ ABC ≅ △ DEF

Proof

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

The primary purpose of the proof is finding a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways. One of the ways will be shown here.

Translate △ DEF So That Two Corresponding Vertices Match

Translate △ DEF so that F is mapped onto C. If this translation maps △ DEF onto △ ABC, the proof is complete.

Since the image of the translation does not match △ ABC, at least one more transformation is needed.

Rotate △ CD'E' So That Two Corresponding Sides Match

Rotate △ CD'E' clockwise about C so that a pair of corresponding sides match. If the image of this transformation is △ ABC, the proof is complete. Note that this rotation maps E' onto B. Therefore, the rotation maps CE' onto CB.

As before, the image does not match △ ABC. Therefore, a third rigid motion is required.

Reflect △ ABF'' So That Two More Corresponding Sides Match

It is given that two angles of △ ABC are congruent to two angles of △ BCD''. Hence, by the Third Angle Theorem, ∠ BCD'' is congruent to ∠ BCA.

Reflect △ CBD'' across BC. Because reflections preserve angles, BD'' and CD'' are mapped onto BA and CA, respectively. Then, the point of intersection of the original segments D'' is mapped onto the point of intersection of the image segments A.

This time the image matches △ ABC.

Consequently, after a sequence of rigid motions, △ DEF can be mapped onto △ ABC. This means that △ DEF and △ ABC are congruent triangles. The proof is complete.

Example

Proving Congruence in Triangles

Dylan bought a new boomerang to play with his friends next summer. In the drawing printed on the boomerang, ∠ A and ∠ C are congruent, and BF and BE are congruent.

Show that AE is congruent to CF.

Answer

See solution.

Hint

Separate triangles ABE and CBF and notice they have a common angle. Then, use the Angle-Angle-Side (AAS) Congruence Theorem.

Solution

Start by separating △ ABE and △ CBF from the design.

Notice that ∠ B is common to both triangles. By the Reflexive Property of Congruence, ∠ B is congruent to itself. Also, it is given that ∠ A is congruent to ∠ C, and BF is congruent to BE. cl ∠ A ≅ ∠ C & Angle ∠ B ≅ ∠ B & Angle BF ≅ BE & Side Applying the Angle-Angle-Side (AAS) Congruence Theorem, it is obtained that △ ABE is congruent to △ CBF. Consequently, their corresponding parts are congruent, which means that AE is congruent to CF.

Explore

Investigating Side-Side-Angle

With the help of the following applet, investigate if the Side-Side-Angle is a valid criterion for determining triangle congruence.

Use segments AB and AC to construct two different triangles in such a way that the angle formed at B has the same measure in both triangles.

Applet to construct different triangles, Given two sides and one non included angle

Once the two triangles are drawn, find the side lengths and angle measures of each triangle. Are the triangles congruent in all cases?

Closure

Side-Side-Angle Congruence Theorem for Right Triangles

With the previous applet, it can be checked that, in general, the Side-Side-Angle is not a valid criterion to determine triangle congruence. For instance, the following triangles meet the conditions of this criterion, and they are not congruent.

However, this criteria is valid in the particular case that both triangles are right triangles.

Rule

Hypotenuse Leg Theorem

If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.

Based on the diagram, the following relations hold true.

{ c m∠A = 90^(∘) m∠D = 90^(∘) AB ≅ DE BC ≅ EF . ⇒ △ ABC ≅ △ DEF

Proof

Consider △ ABC and △ DEF, shown below.

By applying the Pythagorean Theorem in each triangle, the following equations can be written. c^2 = a^2 + b_1^2 & (I) c^2 = a^2 + b_2^2 & (II) The expression on the right hand-side of the first equation can be substituted into the second equation. Then a relation between b_1 and b_2 can be found.

c^2 = a^2 + b_1^2 & (I) c^2 = a^2 + b_2^2 & (II)

Substitute

(II): c^2= a^2+b_1^2

c^2= a^2 + b_1^2 a^2 + b_1^2= a^2 + b_2^2

▼

Solve for b_1

SubEqn

(II): LHS-a^2=RHS-a^2

c^2= a^2 + b_1^2 b_1^2= b_2^2

SqrtEqn

(II): sqrt(LHS)=sqrt(RHS)

c^2= a^2 + b_1^2 sqrt(b_1^2) =sqrt(b_2^2)

SqrtToAbs

(II): sqrt(a^2)=|a|

c^2= a^2 + b_1^2 |b_1| = |b_2|

Since both b_1 and b_2 represent side lengths, they are positive numbers. Moreover, the absolute value of a positive number is the number itself. Therefore, the second equation implies that b_1 and b_2 are equal. |b_1| &= |b_2| b_1 &> 0 b_2 &> 0 ⇒ b_1=b_2 Consequently, the three sides of △ ABC are congruent to the corresponding three sides of △ DEF.