9. Criteria for Triangle Congruence
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Chapter 1
9.
Criteria for Triangle Congruence
In geometry, triangles play a pivotal role. Understanding when two triangles are congruent (meaning they have the exact same size and shape) is crucial. The key to this understanding lies in the congruence theorems. These theorems provide criteria that, when met, guarantee that two triangles are congruent. For instance, if you know certain sides or angles of one triangle match those of another, you can confidently state they are congruent. This knowledge is invaluable for architects, engineers, and anyone looking to ensure precision in their designs or constructions.
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| | 16 Theory slides |
| | 10 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Criteria for Triangle Congruence
Slide of 16
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.
Explore
Investigating Congruence of Triangles Given Angles
Consider the following statement.
Extra
How to Use the AppletIn 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.
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≅∠DAC≅DF⇒△ABC≅△DEF
Proof
Side-Angle-Side Congruence TheoremThis proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate △DEF So That Two Corresponding Vertices Match
expand_more 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.
2
Rotate △AE′F′ So That Two Corresponding Sides Match
expand_more 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.
3
Reflect △ABF′′ So That Two More Corresponding Sides Match
expand_more 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.
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△ABD≅△BCD≅△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.
AD≅BC∠ADC≅∠BCDDC≅CDSideAngleSide
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.
AD≅BC∠ADB≅∠BCADB≅CASideAngleSide
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.
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≅∠DAB≅DE∠B≅∠E ⇒ △ABC≅△DEF
Proof
Angle-Side-Angle Congruent TheoremThis proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate △DEF So That Two Corresponding Vertices Match
expand_more 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.
2
Rotate △AE′F′ So That Two Corresponding Sides Match
expand_more 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.
3
Reflect △ABF′′ So That All Corresponding Sides Match
expand_more 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.
Example
Write Equations Based on Congruent Triangles
Consider the following diagram.
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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.
Then, the value of y can be found by substituting x=1.5 into the Equation (II) and solving the resulting equation for y.
Finally, the required sum can be calculated by substituting the values found for x, y, and z.
∠PQS≅∠RSQQS≅QS∠PSQ≅∠RQSAngleSideAngle
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⇒⎩⎪⎪⎨⎪⎪⎧∠PPQPS≅∠R≅RS≅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=506=2y+x5x−3=4.5(I)(II)(III)
By solving Equation (I), the value of z can be found.
Next, solve Equation (III) to find the value of x.
5x−3=4.5
x=1.5
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
24.5=y
UseCalc
Use a calculator
2.25=y
RearrangeEqn
Rearrange equation
y=2.25
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.
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≅DEBC≅EFAC≅DF ⇒ △ABC≅△DEF
Proof
Side-Side-Side Congruence TheoremThis proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate △DEF So That Two Corresponding Vertices Match
expand_more 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.
2
Rotate △AE′F′ So That Two Corresponding Sides Match
expand_more 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.
3
Reflect △ABF′′ So That Two More Corresponding Sides Match
expand_more 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′′.
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.
Example
Using Triangle Congruence Conditions to Solve Problems
In the following diagram, R1 is a rectangle, S1 and S2 are squares, T1, T2, and △MKL are isosceles triangles, DK is congruent to CL, and GM is congruent to JM.
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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∘.
This equation implies that DL and CK are congruent.
Based on the equation just obtained, it can be concluded that GL and JK are congruent.
Since corresponding parts of congruent triangles are congruent, it can be concluded that ∠DGL is congruent to ∠CJK. Therefore, z=34.
x+m∠DCB+m∠JCK+m∠JCH=360∘
Since R1 is a rectangle and S2 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.
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+KLCK=CL+LK(I)(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
DL=CK
DL≅CK
Proving that GL≅JK
Once more, the Segment Addition Postulate can be used to rewrite GL and JK.{GL=GM+MLJK=JM+MK(I)(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
GL=JK
GL≅JK
Proving that DG≅CJ
Since R1 is a rectangle, S1 and S2 are squares, and T1 and T2 are isosceles triangles, the following consequences can be drawn.
| Given | Consequence |
|---|---|
| R1 is a rectangle | DA≅CB |
| S1 is a square | DG≅DE |
| S2 is a square | CH≅CJ |
| T1 is an isosceles triangle | DE≅DA |
| T2 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.DL≅CKGL≅JKDG≅CJSideSideSide
The Side-Side-Side (SSS) Congruence Theorem allows to conclude that △DGL is congruent to △CJK.
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.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.
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≅∠EBC≅EF ⇒ △ABC≅△DEF
Proof
This proof will be developed based on the given diagram, but it is valid for any pair of triangles.
1
Translate △DEF So That Two Corresponding Vertices Match
expand_more 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.
2
Rotate △CD′E′ So That Two Corresponding Sides Match
expand_more 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.
3
Reflect △ABF′′ So That Two More Corresponding Sides Match
expand_more 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.
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.
∠A≅∠C∠B≅∠BBF≅BEAngleAngleSide
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.
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.
⎩⎪⎪⎪⎨⎪⎪⎪⎧m∠A=90∘m∠D=90∘AB≅DEBC≅EF⇒△ABC≅△DEF
Proof
Consider △ABC and △DEF, shown below.
{c2=a2+b12c2=a2+b22(I)(II)
The expression on the right hand-side of the first equation can be substituted into the second equation. Then a relation between b1 and b2 can be found.
{c2=a2+b12c2=a2+b22(I)(II)
Substitute
(II): c2=a2+b12
{c2=a2+b12a2+b12=a2+b22
{c2=a2+b12∣b1∣=∣b2∣
∣b1∣b1b2=∣b2∣>0>0⇒b1=b2
Consequently, the three sides of △ABC are congruent to the corresponding three sides of △DEF. 
Therefore, by the Side-Side-Side Congruence Theorem the triangles are congruent.
△ABC≅△DEF
In the following chart, all the criteria for triangle congruence seen in the lesson are listed.
Criteria for Triangle Congruence
Exercises
Ignacio says that △ABD is congruent to △CDB by the SSS Congruence Theorem. Paulina says that the triangles are congruent by the HL Congruence Theorem. Who, if either, is correct?
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Let's start by explaining the theorems.
- HL Congruence Theorem: When the hypotenuse and a pair of legs in two right triangles are congruent, the triangles are congruent.
- SSS Congruence Theorem: When three pairs of sides in two triangles are congruent, the triangles are congruent.
From the diagram, we can see that the triangles share their hypotenuse. Therefore, by the Reflexive Property of Congruence, we can claim that the hypotenuses are congruent. Let's add this to the diagram.
Next, by separating the two triangles, it becomes easier to see both Ignacio's and Paulina's points of view.
As we can see, congruence can be proved by both theorems. Therefore, both Ignacio and Paulina are correct.
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Are the following triangles congruent? Justify your conclusion with every triangle congruence theorem that is applicable.
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Examining the diagram, we can identify two right triangles. Let's introduce some variables to label the unknown sides. Then we can write two equations to find the missing lengths.
Let's solve for the unknown hypotenuse c.
Let's also solve for the unknown leg b.
The unknown leg is 6 units long. Since the two triangles have three pairs of congruent corresponding sides, we know that they are congruent by the Side-Side-Side Congruence Theorem. We could also use the Hypotenuse-Leg Congruence Theorem.
From the diagram, we notice that the triangles have two pairs of congruent angles and one pair of corresponding congruent non-included sides. With this information, we can claim congruence by the Angle-Angle-Side Congruence Theorem.
Examining the diagram, we see that the triangles have a pair of congruent angles and a pair of congruent sides. Additionally, the triangles also share one of their sides. By the Reflexive Property of Congruence, we know that this side is congruent as well. Let's mark this in the diagram.
However, this information does not match any triangle congruence theorem. Therefore, we cannot prove congruence.
Just like in the previous exercise, we have two triangles that share a side. This means this side is congruent according to the Reflexive Property of Congruence. Let's add this to the diagram.
Now we can claim congruence by the Side-Angle-Side Congruence Theorem.
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Which triangles are congruent to △A? Select all that apply.
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Looking at △ A, we see a couple of things.
- Two angles are marked. One of them is a right angle.
- Both of the triangle's legs are marked.
Possible Theorems
Given the information about △ A, we cannot use the Side-Side-Side Congruence Theorem or the Hypotenuse-Leg Congruence Theorem. We are left with three theorems which can be used to prove triangle congruence.
- Side-Angle-Side Congruence Theorem (SAS)
- Angle-Side-Angle Congruence Theorem (ASA)
- Angle-Angle-Side Congruence Theorem (AAS)
SAS
In order to use SAS, two pairs of sides and their included angles must be congruent. Examining the triangles, we see that only △ D fulfills this criteria.
ASA and AAS
To use ASA or AAS, we need to know two angles and one side, either the included side (ASA) or a non-included side (AAS). Examining the diagram, we see that △ E fulfills the ASA criteria.
Conclusion
The triangles that are congruent to △ A are △ D and △ E.
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Are the triangles congruent? If so, justify your answer with a triangle congruence statement.
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Examining the diagram, we see three pairs of congruent sides. &AC≅DE &BC≅BE &AB≅DB Since we have three pairs of congruent sides, the triangles are congruent by the Side-Side-Side Congruence Theorem.
Notice that we have not been given any information about the triangle's angles. Therefore, we can only attempt to use the Side-Side-Side Congruence Theorem. From the diagram we can identify a pair of congruent sides. Also, the triangles share QS as a side. Therefore, according to the Reflexive Property of Congruence, this side of the triangles are congruent as well. Let us add that to the diagram.
To prove congruence by the SSS Congruence Theorem, we also need the remaining pair of sides to be congruent. Even though it appears as though PS and RS are congruent, we do not have a way to confirm that they are. Therefore, we cannot claim congruence.
If the hypotenuse and a pair of legs in two right triangles are congruent, then by the HL Congruence Theorem, the triangles are congruent. From the diagram, we can see that the hypotenuses are congruent, and we can also identify a pair of congruent legs. BC ≅ EF AB ≅ DE Since the triangles are both right triangles, we can claim congruence by the HL Congruence Theorem.
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Are the triangles congruent? If so, justify your answer with a triangle congruence statement.
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Examining the triangles, we can identify two pairs of congruent angles and one pair of non-included congruent sides. Angle:& ∠ E ≅ ∠ A Angle:& ∠ D ≅ ∠ C Side:& DF≅CB As we can see, we have enough information to claim congruence by the AAS Congruence Theorem.
Examining the diagram, we can identify a pair of congruent angles. We can also see that ∠ ACD is a straight angle, and because ∠ BCD is a right angle, ∠ BCA must be a right angle as well. Additionally, BC is a shared side, which means we can claim that this side is congruent according to the Reflexive Property of Congruence.
Let's list everything we know. Angle:& ∠ BAC≅ ∠ BDC Angle:& ∠ BCA ≅ ∠ BCD Side:& BC≅BC As we can see, we can claim congruence by the AAS Congruence Theorem.
From the diagram, we can see that the triangles have two pairs of congruent sides and a pair of congruent non-included angles. Angle:& ∠ F ≅ ∠ C Side:& EF≅AC Side:& ED≅AB There is no congruence theorem that uses two sides and a non-included angle. Therefore, there is not enough information to claim congruence.
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Criteria for Triangle Congruence
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