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Congruence, Proof, and Constructions

Criteria for Triangle Congruence

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 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.

Side-Angle-Side Congruence Theorem

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

Proof

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

Two triangles with two congruent sides and one congruent angle
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.

1

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

2

Rotate So That Two Corresponding Sides Match
Rotate 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 onto B. Therefore, the rotation maps onto AB.
SAS rotation
As before, the image does not match ABC. Therefore, a third rigid motion is required.

3

Reflect So That Two More Corresponding Sides Match
Reflect across Because reflections preserve angles, is mapped onto AC. Additionally, it is given that Therefore, is mapped onto C, which gives that is mapped onto BC.
SAS reflection
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.

Trapezoid ABCD, (major base DC and minor base AB) with diagonals AC and BD. The diagonals intersect at E.

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.

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.

Highlighting 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.
Separating 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.
By the Side-Angle-Side (SAS) Congruence Theorem, it can be concluded that ADC and BCD are congruent.

ADCBCD

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.
Separating Triangles ABD and BAC
Below, the corresponding congruent parts between ABD and BAC are listed.
One more time, the Side-Angle-Side (SAS) Congruence Theorem can be used to conclude that triangles ABD and BAC are congruent.

ABDBAC

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.

Highlighting triangles ABE and DEC

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 and 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.

Proof

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.

1

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.

2

Rotate So That Two Corresponding Sides Match
Rotate 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 onto B. Therefore, the rotation maps onto AB.
As before, the image does not match ABC. Therefore, a third rigid motion is required.

3

Reflect So That All Corresponding Sides Match
Reflect across Because reflections preserve angles, and are mapped onto and respectively. Then, the point of intersection of the original rays 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.

Quadrilateral PQRS with diagonal SQ, PQ=6,QR=4.5,RS=2y+x,PS=5x-3, m angle R = 50, m angle P = 8z+2, angles PQS and RSQ are congruent, and angles PSQ and RQS are congruent
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.
Separating 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.
Consequently, PQS and RSQ are congruent because of the Angle-Side-Angle (ASA) Congruence Theorem. Therefore, the corresponding sides and angles are congruent.
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.
By solving Equation (I), the value of z can be found.
8z+2=50
Solve for z
8z=48
z=6
Next, solve Equation (III) to find the value of x.
5x3=4.5
Solve for x
5x=7.5
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
6=2y+1.5
Solve for y
4.5=2y
2.25=y
y=2.25
Finally, the required sum can be calculated by substituting the values found for x, y, and z.
x+y+z
1.5+2.25+6
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.

Two congruent triangles ABC and DEF

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

Proof

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

Two triangles ABC and DEF with congruent corresponding sides
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.

1

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

2

Rotate So That Two Corresponding Sides Match
Rotate 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 onto B. Consequently, is mapped onto AB.
Rotation of AE'F' about A
As before, the image does not match ABC. Therefore, a third rigid motion is required.

3

Reflect So That Two More Corresponding Sides Match

The points C and are on opposite sides of Now, consider Let G denote the point of intersection between and

It can be noted that and By the Converse Perpendicular Bisector Theorem, is a perpendicular bisector of Points along the perpendicular bisector are equidistant from the endpoints of the segment, so

Finally, can be mapped onto C by a reflection across by reflecting across Because reflections preserve angles, and are mapped onto and respectively.
Reflecting ABF'' across line AB
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.

Flowchart Summarizing the Previous paragraph
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, 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.

Diagram containing one rectangle, three isosceles triangles, two squares
If mJCK=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

Solution

To find the value of x, notice that x, mDCB, mJCK, and mJCH add up
Since R1 is a rectangle and S2 is a square, DCB and JCH are right angles. Therefore, these angles have a measure of each. Also, it is given that mJCK=3z+8. Use this information to solve for x.
x+mDCB+mJCK+mJCH=360
x+90+(3z+8)+90=360
Solve for z
x+3z+188=360
x=1723z
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.

Diagram containing one rectangle, three isosceles triangles, two squares

Proving that DLCK

Notice that KL is common to both triangles. By using the Segment Addition Postulate, the following pair of equations can be written.
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=CL+LK

Segment Addition Postulate

DL=CK
This equation implies that DL and CK are congruent.

DLCK

Proving that GLJK

Once more, the Segment Addition Postulate can be used to rewrite GL and JK.
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=MK+JM

Segment Addition Postulate

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

GLJK

Proving that DGCJ

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 DACB
S1 is a square DGDE
S2 is a square CHCJ
T1 is an isosceles triangle DEDA
T2 is an isosceles triangle CBCH

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

FlowChart Proving that segment DG is congruent to segment CJ

Proving that DGLCJK

Previously, the following three congruence statements were obtained.
The Side-Side-Side (SSS) Congruence Theorem allows to conclude that DGL is congruent to CJK.
Separating Triangles DGL and 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=1723z. Then solve for x.
x=1723z
x=1723(34)
Solve for x
x=172102
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 and 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.
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 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.

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.

1

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.

2

Rotate so that two corresponding sides match
Rotate 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 onto B. Therefore, the rotation maps onto CB.
As before, the image does not match ABC. Therefore, a third rigid motion is required.

3

Reflect so that two more corresponding sides match

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

Reflect across Because reflections preserve angles, and are mapped onto BA and CA, respectively. Then, the point of intersection of the original segments 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.

Traditional shaped boomerang with two overlapping triangles

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.
Separating triangles ABE and CBF
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.
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.

Two non-congruent Triangles meeting SSA

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.

Proof

Consider ABC and DEF, shown below.

By applying the Pythagorean Theorem in each triangle, the following equations can be written.
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.
Solve for b1
Since both b1 and b2 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 b1 and b2 are equal.
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.

ABCDEF

In the following chart, all the criteria for triangle congruence seen in the lesson are listed.

Chart with all the Criteria and a Pair of Triangles highlighting the respective parts
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