Proving Congruent Triangles

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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 AD,  BE,  and  CF. \angle A \cong \angle D, \ \ \angle B \cong \angle E,\ \text{ and } \ \angle C \cong \angle F. For the sides, hatch marks are used to show that they are congruent. The following pairs of sides are congruent.

ACDF,ABDE,BCEF \overline{AC} \cong \overline{DF}, \quad \overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}
Exercise

Determine mE.m\angle E.

Solution

To find the measure of angle E,E, we could use that the sum of a triangle's interior angles is 180180^\circ. However, this requires that we know mD.m\angle D. Notice that angle AA and angle DD are corresponding and congruent. AD \angle A \cong \angle D Thus, mAm\angle A and mDm\angle D are equal.

We can now write and solve the following equation, using that the sum of mD,m\angle D, mE,m\angle E, and mFm\angle F is 180.180^\circ.
mD+mE+mF=180m\angle D + m\angle E + m\angle F = 180^\circ
29+mE+90=180{\color{#0000FF}{29^\circ}} + m\angle E + {\color{#009600}{90^\circ}} = 180^\circ
mE+119=180m\angle E + 119^\circ = 180^\circ
mE=61m\angle E = 61^\circ
The desired angle measure is 61.61^\circ.
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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 ABC triangle has a corresponding congruent part in DEF.\triangle DEF. Therefore, by applying rigid motions to one triangle, it's possible to map it onto the other, showing congruence. The triangle DEF\triangle DEF can be translated so that the FF maps to C.C. The image, DEC,\triangle D'E'C, can then be rotated so that DD' maps to A.A. This is possible because ACDF.\overline{AC} \cong \overline{DF}.
Transform

The final step is now to map the point EE'' to B.B. This will map all angles and sides of AEC\triangle AE''C to the corresponding ones in ABC.\triangle ABC. This is done with a reflection in the line AC.\overleftrightarrow{AC}. Thus, there is a rigid motion that maps DEF\triangle DEF to ABC\triangle 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 in a triangle are congruent to corresponding parts in another triangle, then the triangles are congruent.

This can be proven using rigid motions.

Proof

Side-Angle-Side Congruence Theorem

Consider the triangles ABC\triangle ABC and DEF,\triangle DEF, where ABDE,AD, and  ACDF. \overline{AB} \cong \overline{DE},\, \angle A \cong \angle D, \text{ and }\ \overline{AC} \cong \overline{DF}. If either of these can be mapped onto the other using rigid motion, then they are congruent. As AB\overline{AB} is congruent with DE,\overline{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 ABF\triangle ABF' can now be reflected in the line AB.\overleftrightarrow{AB}. If the image of FF' then falls onto C,C, the triangles will completely overlap. As the angles CAB\angle CAB and FAB\angle F'AB are congruent, the ray AF\overrightarrow{AF'} will be mapped onto the ray AC.\overrightarrow{AC}. This, combined with ACAF, \overline{AC} \cong \overline{AF'}, means that FF' will indeed be mapped onto CC when ABF\triangle ABF' gets reflected in AB.\overleftrightarrow{AB}.

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

Side-Side-Side Congruence Theorem

If each side of two triangles are congruent to their respective counterpart, then the triangles are congruent.

This can be proven using rigid motions.

Proof

Side-Side-Side Congruence Theorem

Consider the triangles ABC\triangle ABC and DEF,\triangle DEF, where ABDE,ACDF, and  BCEF. \overline{AB} \cong \overline{DE},\, \overline{AC} \cong \overline{DF}, \text{ and }\ \overline{BC} \cong \overline{EF}. If there exists a rigid motion that maps one of these onto the other, then they are congruent. As the sides AB\overline{AB} and DE\overline{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 CF\overleftrightarrow{CF'} can now be drawn, dividing the angle C\angle C into 1\angle 1 and 3,\angle 3, and F\angle F' into 2\angle 2 and 4.\angle 4.

Notice that the BCF\triangle BCF' is an isosceles triangle, leading to 1\angle 1 and 2\angle 2 being congruent. Similarly, 3\angle 3 and 4\angle 4 are congruent as ACF\triangle ACF' is also an isosceles triangle. This leads to m1+m3=m2+m4, m\angle 1 + m\angle 3 = m\angle 2 + m\angle 4, which by construction means that mC=mF. m\angle C = m\angle F'. Thus, C\angle C and F\angle F' 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 in a triangle are congruent with corresponding parts in another triangle, then the triangles are congruent.

This can be proven using rigid motions.

Proof

Angle-Side-Angle Congruence Theorem

Consider the triangles ABC\triangle ABC and DEF,\triangle DEF, where AD,ABDE, and  BE. \angle A \cong \angle D,\, \overline{AB} \cong \overline{DE}, \text{ and }\ \angle B \cong \angle E. If either of these can be mapped onto the other using rigid motion, then they are congruent. As AB\overline{AB} is congruent with DE,\overline{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 ABF\triangle ABF' can now be reflected in the line AB.\overleftrightarrow{AB}. If the image of FF' falls onto C,C, the triangles will completely overlap. As the angles CAB\angle CAB and FAB\angle F'AB are congruent, the ray AF\overrightarrow{AF'} will be mapped onto the ray AC.\overrightarrow{AC}. Similarly, BF\overrightarrow{BF'} will be mapped onto BC.\overrightarrow{BC}.


Thus, the intersection of AF\overrightarrow{AF'} and BF,\overrightarrow{BF'}, which is F,F', will be mapped onto the intersection of AC\overrightarrow{AC} and BC,\overrightarrow{BC}, which is C.C.

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

Show that the triangles below are congruent.

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\angle A \cong \angle D and ABDE.AB \cong DE. 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\angle B is congruent to E.\angle E.


Since the measure of B\angle B is given, it is necessary to show that E=56.\angle E=56^\circ.The angles D\angle D and F\angle F are known, 3434 ^\circ and 90.90 ^\circ. Since the sum of the angles in a triangle is 180,180^\circ, we can write an equation that can be solved for mE.m\angle E.

mD+mE+mF=180m\angle D + m\angle E + m\angle F = 180^\circ
Solve for mEm\angle E
90+mE+34=180{\color{#0000FF}{90^\circ}}+m\angle E +{\color{#009600}{34^\circ}} = 180 ^\circ
mE+124=180m\angle E + 124^\circ=180^\circ
mE=56m\angle E=56^\circ
Thus, the measure of E=56.\angle E = 56^\circ. 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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Exercises

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