Unlocking Proving Triangle Congruence through Congruence Theorems

△ D E F

so that

D

is mapped onto

A .

If this translation maps

△ D E F

onto

△ A B C,

the proof is complete.

Since the image of the translation does not match

△ A B C,

at least one more transformation is needed.

Rotate

△ A E^{'} F^{'}

So That Two Corresponding Sides Match

△ A E^{'} F^{'}

counterclockwise about

A

so that a pair of corresponding sides match. If the image of this transformation is

△ A B C,

the proof is complete. Note that this rotation maps

E^{'}

onto

B .

Therefore, the rotation maps

\overline{A E^{'}}

onto

\overline{A B} .

As before, the image does not match

△ A B C .

Therefore, a third rigid motion is required.

Reflect

△ A B F^{''}

So That Two More Corresponding Sides Match

Reflect

△ A B F^{''}

across

A B .

Because reflections preserve angles,

\overline{A F^{''}}

is mapped onto

\overline{A C} .

Additionally, it is given that

A C = A F^{''} .

Therefore,

F^{''}

is mapped onto

C,

which gives that

B F^{''}

is mapped onto

B C .

This time the image matches

△ A B C .

Consequently, after a sequence of rigid motions,

△ D E F

can be mapped onto

△ A B C .

This means that

△ D E F

and

△ A B C

are congruent triangles. The proof is complete.

Example

Identifying Congruent Triangles

In the following diagram, triangles $A D E$ and $B C E$ are congruent, and $∠ A D C$ is congruent to $∠ B C D .$

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.

△ A D C △ A B D ≅ △ B C D ≅ △ B A C

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, $△ A D E$ and $△ B C E .$

Since these triangles are congruent, their corresponding parts are congruent. This implies that $\overline{A D}$ is congruent to $\overline{B C} .$

First Pair

Because

△ A D E

and

△ B C E

are parts of

△ A D C

and

△ B C D

respectively, consider triangles

A D C

and

B C D .

Notice that

\overline{C D}

is a common side for triangles

A D C

and

B C D .

Because of the Reflexive Property of Congruence,

\overline{C D}

is congruent to itself. Next, list the corresponding congruent parts between these two triangles.

\overline{A D} ≅ \overline{B C} ∠ A D C ≅ ∠ B C D \overline{D C} ≅ \overline{C D} S ide A ngle S ide

By the Side-Angle-Side (SAS) Congruence Theorem, it can be concluded that

△ A D C

and

△ B C D

are congruent.

$△ A D C ≅ △ B C D$

Second Pair

Next, consider triangles

A B D

and

B A C .

Because

△ A D E

and

△ B C E

are congruent,

∠ A D B

and

∠ B C A

are congruent. Additionally, since

△ A D C

and

△ B C D

are congruent,

\overline{C A}

is congruent to

\overline{D B} .

Below, the corresponding congruent parts between

△ A B D

and

△ B A C

are listed.

\overline{A D} ≅ \overline{B C} ∠ A D B ≅ ∠ B C A \overline{D B} ≅ \overline{C A} S ide A ngle S ide

One more time, the Side-Angle-Side (SAS) Congruence Theorem can be used to conclude that triangles

A B D

and

B A C

are congruent.

$△ A B D ≅ △ B A C$

Third Pair

The last two triangles to consider are triangles $A B E$ and $D E C .$ 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 $\overline{A B}$ and the rays $A X$ and $B Y$ 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.

Triangle ABC is congruent to triangle DEF

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

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ ∠ A ≅ ∠ D \overline{A B} ≅ \overline{D E} ∠ B ≅ ∠ E \Rightarrow △ A B C ≅ △ D E F

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

△ D E F

So That Two Corresponding Vertices Match

△ D E F

so that

D

is mapped onto

A .

If this translation maps

△ D E F

onto

△ A B C,

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

△ A B C,

at least one more transformation is needed.

Rotate

△ A E^{'} F^{'}

So That Two Corresponding Sides Match

△ A E^{'} F^{'}

counterclockwise about

A

so that a pair of corresponding sides match. If the image of this transformation is

△ A B C,

the proof is complete. Note that this rotation maps

E^{'}

onto

B .

Therefore, the rotation maps

\overline{A E^{'}}

onto

\overline{A B} .

As before, the image does not match

△ A B C .

Therefore, a third rigid motion is required.

Reflect

△ A B F^{''}

So That All Corresponding Sides Match

Reflect

△ A B F^{''}

across

A B .

Because reflections preserve angles,

A F^{''}

and

B F^{''}

are mapped onto

A C

and

B C,

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

△ A B C .

Consequently, after a sequence of rigid motions

△ D E F

can be mapped onto

△ A B C .

This means that

△ D E F

and

△ A B C

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 $\overline{Q S}$ 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

\overline{Q S}

is a common side for triangles

P Q S

and

R S Q .

By the Reflexive Property of Congruence,

\overline{Q S}

is congruent to itself. Additionally,

∠ P Q S

and

∠ R S Q

are congruent, as are

∠ P S Q

and

∠ R Q S .

∠ P Q S ≅ ∠ R S Q \overline{Q S} ≅ \overline{Q S} ∠ P S Q ≅ ∠ R Q S A ngle S ide A ngle

Consequently,

△ P Q S

and

△ R S Q

are congruent because of the Angle-Side-Angle (ASA) Congruence Theorem. Therefore, the corresponding sides and angles are congruent.

△ P Q S ≅ △ R S Q \Rightarrow ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ ∠ P \overline{P Q} \overline{P S} ≅ ∠ R ≅ \overline{R S} ≅ \overline{R Q}

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.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ 8 z + 2 = 50 6 = 2 y + x 5 x - 3 = 4.5 (I) (II) (III)

By solving Equation (I), the value of

z

can be found.

8 z + 2 = 50

Solve for

z

$LHS - 2 = RHS - 2$

8 z = 48

DivEqn

$LHS / 8 = RHS / 8$

z = 6

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

x .

5 x - 3 = 4.5

Solve for

x

AddEqn

$LHS + 3 = RHS + 3$

5 x = 7.5

DivEqn

$LHS / 5 = RHS / 5$

x = \frac{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 = 2 y + x

Substitute

$x = 1.5$

6 = 2 y + 1.5

Solve for

y

$LHS - 1.5 = RHS - 1.5$

4.5 = 2 y

DivEqn

$LHS / 2 = RHS / 2$

\frac{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

\overline{A B},

\overline{A C},

and

\overline{B C}

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.

⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ \overline{A B} ≅ \overline{D E} \overline{B C} ≅ \overline{E F} \overline{A C} ≅ \overline{D F} \Rightarrow △ A B C ≅ △ D E F

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

△ D E F

So That Two Corresponding Vertices Match

△ D E F

so that

D

is mapped onto

A .

If this translation maps

△ D E F

onto

△ A B C,

the proof is complete.

Since the image of the translation does not match

△ A B C,

at least one more transformation is needed.

Rotate

△ A E^{'} F^{'}

So That Two Corresponding Sides Match

△ A E^{'} F^{'}

counterclockwise about

A

so that a pair of corresponding sides matches. If the image of this transformation is

△ A B C,

the proof is complete. Note that this rotation maps

E^{'}

onto

B .

Consequently,

\overline{A E^{'}}

is mapped onto

\overline{A B} .

As before, the image does not match

△ A B C .

Therefore, a third rigid motion is required.

Reflect

△ A B F^{''}

So That Two More Corresponding Sides Match

The points $C$ and $F^{''}$ are on opposite sides of $A B .$ Now, consider $\overline{C F^{'}} .$ Let $G$ denote the point of intersection between $A B$ and $\overline{C F^{''}} .$

It can be noted that $A C = A F^{''}$ and $B C = B F^{''} .$ By the Converse Perpendicular Bisector Theorem, $A B$ is a perpendicular bisector of $\overline{C F^{''}} .$ Points along the perpendicular bisector are equidistant from the endpoints of the segment, so $C G = G F^{''} .$

Finally,

F^{''}

can be mapped onto

C

by a reflection across

A B

by reflecting

△ A B F^{''}

across

A B .

Because reflections preserve angles,

A F^{''}

and

B F^{''}

are mapped onto

A C

and

B C,

respectively.

This time the image matches

△ A B C .

Consequently, the application of a sequence of rigid motions allows

△ D E F

to be mapped onto

△ A B C .

This means that

△ D E F

and

△ A B C

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.

A flowchart demonstrating the uniqueness of a triangle formed by three segments.

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 $△ M K L$ are isosceles triangles, $\overline{D K}$ is congruent to $\overline{C L},$ and $\overline{G M}$ is congruent to $\overline{J M} .$

Diagram containing one rectangle, three isosceles triangles, two squares

m ∠ J C K = 3 z + 8,

what is the value of

x ?

Hint

Using the Segment Addition Postulate and the Side-Side-Side (SSS) Congruence Theorem, prove that $△ D G L$ is congruent to $△ C J K .$ Then, find the measure of $∠ J C K .$ Use the fact that $m ∠ J C K + m ∠ D C B + m ∠ J C H + x = 36 0^{\circ} .$

Solution

To find the value of

x,

notice that

x,

m ∠ D C B,

m ∠ J C K,

and

m ∠ J C H

add up

36 0^{\circ} .

x + m ∠ D C B + m ∠ J C K + m ∠ J C H = 36 0^{\circ}

Since

R_{1}

is a rectangle and

S_{2}

is a square,

∠ D C B

and

∠ J C H

are right angles. Therefore, these angles have a measure of

9 0^{\circ}

each. Also, it is given that

m ∠ J C K = 3 z + 8 .

Use this information to solve for

x .

x + m ∠ D C B + m ∠ J C K + m ∠ J C H = 360

SubstituteValues

Substitute values

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

Solve for

z

AddTerms

Add terms

x + 3 z + 188 = 360

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

x = 172 - 3 z

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 $△ D G L$ and $△ C J K$ 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 $\overline{D L} ≅ \overline{C K}$

Notice that

\overline{K L}

is common to both triangles. By using the Segment Addition Postulate, the following pair of equations can be written.

{D L = D K + K L C K = C L + L K (I) (II)

Since

\overline{D K}

is congruent to

\overline{C L},

these segments have the same length, that is,

D K = C L .

Simlarly,

K L = L K .

By substituting these expressions into Equation (I), a relation between

\overline{D L}

and

\overline{C K}

will be obtained.

D L = D K + K L

Substitute values and simplify

SubstituteII

$D K = C L$ , $K L = L K$

D L = C L + L K

Segment Addition Postulate

D L = C K

This equation implies that

\overline{D L}

and

\overline{C K}

are congruent.

$\overline{D L} ≅ \overline{C K}$

Proving that $\overline{G L} ≅ \overline{J K}$

Once more, the Segment Addition Postulate can be used to rewrite

G L

and

J K .

{G L = G M + M L J K = J M + M K (I) (II)

Because

△ M K L

is isosceles,

\overline{M K}

is congruent to

\overline{M L} .

Therefore,

M L = M K .

Keep in mind that it is given that

\overline{G M}

and

\overline{J M}

are congruent. That means

G M = J M .

These two expressions can be substituted into Equation (I).

G L = G M + M L

Substitute values and simplify

SubstituteII

$M L = M K$ , $G M = J M$

G L = M K + J M

Segment Addition Postulate

G L = J K

Based on the equation just obtained, it can be concluded that

\overline{G L}

and

\overline{J K}

are congruent.

$\overline{G L} ≅ \overline{J K}$

Proving that $\overline{D G} ≅ \overline{C J}$

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	$\overline{D A} ≅ \overline{C B}$
$S_{1}$ is a square	$\overline{D G} ≅ \overline{D E}$
$S_{2}$ is a square	$\overline{C H} ≅ \overline{C J}$
$T_{1}$ is an isosceles triangle	$\overline{D E} ≅ \overline{D A}$
$T_{2}$ is an isosceles triangle	$\overline{C B} ≅ \overline{C H}$

Next, organize the information in the right-hand column in a flow chart and use the Transitive Property of Congruence to prove that $\overline{D G} ≅ \overline{C J} .$

Proving that $△ D G L ≅ △ C J K$

Previously, the following three congruence statements were obtained.

\overline{D L} ≅ \overline{C K} \overline{G L} ≅ \overline{J K} \overline{D G} ≅ \overline{C J} S ide S ide S ide

The Side-Side-Side (SSS) Congruence Theorem allows to conclude that

△ D G L

is congruent to

△ C J K .

Since corresponding parts of congruent triangles are congruent, it can be concluded that

∠ D G L

is congruent to

∠ C J K .

Therefore,

z = 34 .

Finding the Value of $x$

Finally, to find the value of

x,

substitute

z = 34

into the equation

x = 172 - 3 z .

Then solve for

x .

x = 172 - 3 z

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 $\overline{A B}$ and the rays $A X$ and $B Y$ 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 \overline{B C} ≅ \overline{E F} \Rightarrow △ A B C ≅ △ D E F

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

△ D E F

So That Two Corresponding Vertices Match

△ D E F

so that

F

is mapped onto

C .

If this translation maps

△ D E F

onto

△ A B C,

the proof is complete.

Since the image of the translation does not match

△ A B C,

at least one more transformation is needed.

Rotate

△ C D^{'} E^{'}

So That Two Corresponding Sides Match

△ C D^{'} E^{'}

clockwise about

C

so that a pair of corresponding sides match. If the image of this transformation is

△ A B C,

the proof is complete. Note that this rotation maps

E^{'}

onto

B .

Therefore, the rotation maps

\overline{C E^{'}}

onto

\overline{C B} .

As before, the image does not match

△ A B C .

Therefore, a third rigid motion is required.

Reflect

△ A B F^{''}

So That Two More Corresponding Sides Match

It is given that two angles of $△ A B C$ are congruent to two angles of $△ B C D^{''} .$ Hence, by the Third Angle Theorem, $∠ B C D^{''}$ is congruent to $∠ B C A .$

Triangles ABC and CBD'' with a common side CB

Reflect

△ C B D^{''}

across

B C .

Because reflections preserve angles,

\overline{B D^{''}}

and

\overline{C D^{''}}

are mapped onto

\overline{B A}

and

\overline{C A},

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

△ A B C .

Consequently, after a sequence of rigid motions,

△ D E F

can be mapped onto

△ A B C .

This means that

△ D E F

and

△ A B C

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 $\overline{B F}$ and $\overline{B E}$ are congruent.

Traditional shaped boomerang with two overlapping triangles

Show that $\overline{A E}$ is congruent to $\overline{C F} .$

Answer

See solution.

Hint

Separate triangles $A B E$ and $C B F$ and notice they have a common angle. Then, use the Angle-Angle-Side (AAS) Congruence Theorem.

Solution

Start by separating

△ A B E

and

△ C B F

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

\overline{B F}

is congruent to

\overline{B E} .

∠ A ≅ ∠ C ∠ B ≅ ∠ B \overline{B F} ≅ \overline{B E} A ngle A ngle S ide

Applying the Angle-Angle-Side (AAS) Congruence Theorem, it is obtained that

△ A B E

is congruent to

△ C B F .

Consequently, their corresponding parts are congruent, which means that

\overline{A E}

is congruent to

\overline{C F} .

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

\overline{A B}

and

\overline{A C}

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.

A pair of non-congruent triangles, ABC and PQR, satisfy the side-side-angle criterion.

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 = 9 0^{\circ} m ∠ D = 9 0^{\circ} \overline{A B} ≅ \overline{D E} \overline{B C} ≅ \overline{E F} \Rightarrow △ A B C ≅ △ D E F

Proof

Consider $△ A B C$ and $△ D E F,$ shown below.

By applying the Pythagorean Theorem in each triangle, the following equations can be written.

{c^{2} = a^{2} + b_{1}^{2} c^{2} = a^{2} + b_{2}^{2} (I) (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} c^{2} = a^{2} + b_{2}^{2} (I) (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}