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Triangles

Proving Congruent Triangles

Concept

Identifying Corresponding Parts in Triangles

To identify if two triangles are congruent, their corresponding parts can be compared. The triangles are congruent if all the angles and sides are congruent with their counterpart.

The congruent angles are the ones with the same number of arcs. In this case, they are
For the sides, hatch marks are used to show that they are congruent. The following pairs of sides are congruent.
fullscreen
Exercise

Determine mE.

Show Solution
Solution
To find the measure of angle E, we could use that the sum of a triangle's interior angles is . However, this requires that we know mD. Notice that angle A and angle D are corresponding and congruent.
AD
Thus, mA and mD are equal.
We can now write and solve the following equation, using that the sum of mD, mE, and mF is
The desired angle measure is

Method

Showing Congruence in Triangles using Rigid Motion

It can be shown that two triangles are congruent through rigid motions.

Every side and angle in ABC triangle has a corresponding congruent part in DEF. Therefore, by applying rigid motions to one triangle, it's possible to map it onto the other, showing congruence. The triangle DEF can be translated so that the F maps to C. The image, can then be rotated so that  maps to A. This is possible because ACDF.
Transform

The final step is now to map the point  to B. This will map all angles and sides of to the corresponding ones in ABC. This is done with a reflection in the line Thus, there is a rigid motion that maps DEF to ABC — the triangles are congruent. In some cases, it's not necessary to know that all angles and sides are congruent to show that the triangles are.

Rule

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

Side-Angle-Side Congruence Theorem
Consider the triangles ABC and DEF, where
If either of these can be mapped onto the other using rigid motion, then they are congruent. As AB is congruent with DE, there is a rigid motion that maps one of these onto the other. This can be performed for one of the triangles, which leads to the two congruent sides overlapping.
Transform

The triangle can now be reflected in the line If the image of then falls onto C, the triangles will completely overlap. As the angles CAB and are congruent, the ray will be mapped onto the ray This, combined with
means that will indeed be mapped onto C when gets reflected in

Thus, there is a rigid motion that maps DEF onto ABC. Consequently, ABC and DEF are indeed congruent.

Rule

Side-Side-Side Congruence Theorem

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.

Proof

Side-Side-Side Congruence Theorem
Consider the triangles ABC and DEF, where
If there exists a rigid motion that maps one of these onto the other, then they are congruent. As the sides AB and DE are congruent, there is a rigid motion that maps one of these onto the other. Performing this transformation for one of the triangles leads to the two congruent sides overlapping.
Transform

The line can now be drawn, dividing the angle C into ∠1 and ∠3, and into ∠2 and ∠4.

Notice that the is an isosceles triangle, leading to ∠1 and ∠2 being congruent. Similarly, ∠3 and ∠4 are congruent as is also an isosceles triangle. This leads to
m∠1+m∠3=m∠2+m∠4,
which by construction means that
Thus, C and are congruent.
The triangles have two sides, and their included angle, that are congruent. Thus, by the SAS Congruence Theorem, the triangles are indeed congruent.

Rule

Angle-Side-Angle Congruence Theorem

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

Angle-Side-Angle Congruence Theorem
Consider the triangles ABC and DEF, where
If either of these can be mapped onto the other using rigid motion, then they are congruent. As AB is congruent with DE, there is a rigid motion that maps one of these onto the other. This can be performed for one of the triangles, which leads to the two congruent sides overlapping.
Transform

The triangle can now be reflected in the line If the image of falls onto C, the triangles will completely overlap. As the angles CAB and are congruent, the ray will be mapped onto the ray Similarly, will be mapped onto


Thus, the intersection of and which is will be mapped onto the intersection of and which is C.

There is a rigid motion that maps DEF onto ABC. Consequently, ABC and DEF are indeed congruent.


fullscreen
Exercise

Show that the triangles below are congruent.

Show Solution
Solution

Two triangles are congruent only if their corresponding angles and sides are congruent. However, using one of the congruence theorems, SAS, SSS, or ASA, it's only necessary to know three congruent parts to prove complete congruence. Let's study the triangles to see if we can identify three of these parts.

It is given that AD and ABDE. This is one side and one angle. Since SSS and SAS both require more than one congruent side, they cannot be used. To use ASA, we'll need to prove that B is congruent to E.


Since the measure of B is given, it is necessary to show that The angles D and F are known,  and Since the sum of the angles in a triangle is we can write an equation that can be solved for mE.

Solve for mE
Thus, the measure of Therefore, two angles and their included side are congruent in the triangles.

Using the ASA congruence theorem, we have now proven that the triangles are congruent.

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