# Proving Congruent Triangles

### {{ 'ml-heading-theory' | message }}

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

$\overline{AC} \cong \overline{DF}, \quad \overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}$## Showing Congruence in Triangles using Rigid Motion

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

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

## 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.## 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.## 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.## Exercises

*settings_overscan*

## {{ 'ml-heading-exercise' | message }} {{ focusmode.exercise.exerciseName }}

*keyboard_backspace*

{{ 'ml-tooltip-premium-exercise' | message }}

{{ 'ml-tooltip-recommended-exercise' | message }}

Programmeringsuppgift | {{ 'course' | message }} {{ exercise.course }}

*keyboard_backspace*{{ 'ml-btn-previous' | message }}

*keyboard_backspace*