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

$\overline{AC} \cong \overline{DF}, \quad \overline{AB} \cong \overline{DE}, \quad \overline{BC} \cong \overline{EF}$
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Exercise

Determine $m\angle E.$

Show Solution
Solution

To find the measure of angle $E,$ we could use that the sum of a triangle's interior angles is $180^\circ$. However, this requires that we know $m\angle D.$ Notice that angle $A$ and angle $D$ are corresponding and congruent. $\angle A \cong \angle D$ Thus, $m\angle A$ and $m\angle D$ are equal.

We can now write and solve the following equation, using that the sum of $m\angle D,$ $m\angle E,$ and $m\angle F$ is $180^\circ.$
$m\angle D + m\angle E + m\angle F = 180^\circ$
${\color{#0000FF}{29^\circ}} + m\angle E + {\color{#009600}{90^\circ}} = 180^\circ$
$m\angle E + 119^\circ = 180^\circ$
$m\angle E = 61^\circ$
The desired angle measure is $61^\circ.$
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 $\triangle ABC$ 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 $\overline{AC} \cong \overline{DF}.$
Transform

The final step is now to map the point $E''$ to $B.$ This will map all angles and sides of $\triangle AE''C$ to the corresponding ones in $\triangle ABC.$ This is done with a reflection in the line $\overleftrightarrow{AC}.$ Thus, there is a rigid motion that maps $\triangle DEF$ to $\triangle ABC$ — 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 $\triangle ABC$ and $\triangle DEF,$ where $\overline{AB} \cong \overline{DE},\, \angle A \cong \angle D, \text{ and }\ \overline{AC} \cong \overline{DF}.$ If either of these can be mapped onto the other using rigid motion, then they are congruent. As $\overline{AB}$ is congruent with $\overline{DE},$ 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 $\triangle ABF'$ can now be reflected in the line $\overleftrightarrow{AB}.$ If the image of $F'$ then falls onto $C,$ the triangles will completely overlap. As the angles $\angle CAB$ and $\angle F'AB$ are congruent, the ray $\overrightarrow{AF'}$ will be mapped onto the ray $\overrightarrow{AC}.$ This, combined with $\overline{AC} \cong \overline{AF'},$ means that $F'$ will indeed be mapped onto $C$ when $\triangle ABF'$ gets reflected in $\overleftrightarrow{AB}.$

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

Notice that the $\triangle BCF'$ is an isosceles triangle, leading to $\angle 1$ and $\angle 2$ being congruent. Similarly, $\angle 3$ and $\angle 4$ are congruent as $\triangle ACF'$ is also an isosceles triangle. This leads to $m\angle 1 + m\angle 3 = m\angle 2 + m\angle 4,$ which by construction means that $m\angle C = m\angle F'.$ Thus, $\angle C$ and $\angle F'$ 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 $\triangle ABC$ and $\triangle DEF,$ where $\angle A \cong \angle D,\, \overline{AB} \cong \overline{DE}, \text{ and }\ \angle B \cong \angle E.$ If either of these can be mapped onto the other using rigid motion, then they are congruent. As $\overline{AB}$ is congruent with $\overline{DE},$ 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 $\triangle ABF'$ can now be reflected in the line $\overleftrightarrow{AB}.$ If the image of $F'$ falls onto $C,$ the triangles will completely overlap. As the angles $\angle CAB$ and $\angle F'AB$ are congruent, the ray $\overrightarrow{AF'}$ will be mapped onto the ray $\overrightarrow{AC}.$ Similarly, $\overrightarrow{BF'}$ will be mapped onto $\overrightarrow{BC}.$

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

There is a rigid motion that maps $\triangle DEF$ onto $\triangle ABC.$ Consequently, $\triangle ABC$ and $\triangle DEF$ are indeed congruent.
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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 $\angle A \cong \angle D$ and $AB \cong DE.$ 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 $\angle B$ is congruent to $\angle E.$

Since the measure of $\angle B$ is given, it is necessary to show that $\angle E=56^\circ.$The angles $\angle D$ and $\angle F$ are known, $34 ^\circ$ and $90 ^\circ.$ Since the sum of the angles in a triangle is $180^\circ,$ we can write an equation that can be solved for $m\angle E.$

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