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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 For the sides, hatch marks are used to show that they are congruent. The following pairs of sides are congruent.

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Exercise

Determine

Show Solution
Solution

To find the measure of angle we could use that the sum of a triangle's interior angles is . However, this requires that we know Notice that angle and angle are corresponding and congruent. Thus, and are equal.

We can now write and solve the following equation, using that the sum of and 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 triangle has a corresponding congruent part in Therefore, by applying rigid motions to one triangle, it's possible to map it onto the other, showing congruence. The triangle can be translated so that the maps to The image, can then be rotated so that  maps to This is possible because
Transform

The final step is now to map the point  to This will map all angles and sides of to the corresponding ones in This is done with a reflection in the line Thus, there is a rigid motion that maps  to — 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 and where If either of these can be mapped onto the other using rigid motion, then they are congruent. As is congruent with 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 the triangles will completely overlap. As the angles and are congruent, the ray will be mapped onto the ray This, combined with means that will indeed be mapped onto when gets reflected in

Thus, there is a rigid motion that maps onto Consequently, and 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 and where If there exists a rigid motion that maps one of these onto the other, then they are congruent. As the sides and 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 into and and into and

Notice that the is an isosceles triangle, leading to and being congruent. Similarly, and are congruent as is also an isosceles triangle. This leads to which by construction means that Thus, 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 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 and where If either of these can be mapped onto the other using rigid motion, then they are congruent. As is congruent with 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 the triangles will completely overlap. As the angles 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

There is a rigid motion that maps onto Consequently, and 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 and 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 is congruent to


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

Solve for
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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