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# Triangle Congruence Theorems

## 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.

Based on the diagram above, the theorem can be written as follows.

### Proof

Side-Angle-Side Congruence Theorem

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

The primary purpose of the 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 the ways will be shown here.
1
Translate So That Two Corresponding Vertices Match
expand_more
Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more
Rotate counterclockwise about so that a pair of corresponding sides match. If the image of this transformation is the proof is complete. Note that this rotation maps onto Therefore, the rotation maps onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That Two More Corresponding Sides Match
expand_more
Reflect across Because reflections preserve angles, is mapped onto Additionally, it is given that Therefore, is mapped onto which gives that is mapped onto
This time the image matches
Consequently, after a sequence of rigid motions, can be mapped onto This means that and are congruent triangles. The proof is complete.

## 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 Congruent Theorem

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

The goal of the proof is to find a rigid motion or sequence of rigid motions that maps one triangle onto the other. This can be done in several ways. One of the ways will be shown here.
1
Translate So That Two Corresponding Vertices Match
expand_more
Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more
Rotate counterclockwise about so that a pair of corresponding sides match. If the image of this transformation is the proof is complete. Note that this rotation maps onto Therefore, the rotation maps onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That All Corresponding Sides Match
expand_more
Reflect across Because reflections preserve angles, and are mapped onto and respectively. Then, the point of intersection of the original rays is mapped onto the point of intersection of the image rays
This time the image matches
Consequently, after a sequence of rigid motions can be mapped onto This means that and are congruent triangles.

## 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.

Based on the diagram above, the theorem can be written as follows.

### Proof

Side-Side-Side Congruence Theorem

This proof will be developed based on the given diagram, but it is valid for any pair of triangles.

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 So That Two Corresponding Vertices Match
expand_more
Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more
Rotate counterclockwise about so that a pair of corresponding sides matches. If the image of this transformation is the proof is complete. Note that this rotation maps onto Consequently, is mapped onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That Two More Corresponding Sides Match
expand_more

The points and are on opposite sides of Now, consider Let 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 by a reflection across by reflecting across Because reflections preserve angles, and are mapped onto and respectively.
This time the image matches
Consequently, the application of a sequence of rigid motions allows to be mapped onto This means that and are congruent triangles. The proof is complete.

## Angle-Angle-Side Congruence Theorem

If two angles and a non-included side of a triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.

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.

The primary purpose of the 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 the ways will be shown here.
1
Translate So That Two Corresponding Vertices Match
expand_more
Translate so that is mapped onto If this translation maps onto the proof is complete.
Since the image of the translation does not match at least one more transformation is needed.
2
Rotate So That Two Corresponding Sides Match
expand_more
Rotate clockwise about so that a pair of corresponding sides match. If the image of this transformation is the proof is complete. Note that this rotation maps onto Therefore, the rotation maps onto
As before, the image does not match Therefore, a third rigid motion is required.
3
Reflect So That Two More Corresponding Sides Match
expand_more

It is given that two angles of are congruent to two angles of Hence, by the Third Angle Theorem, is congruent to

Reflect across Because reflections preserve angles, and are mapped onto and respectively. Then, the point of intersection of the original segments is mapped onto the point of intersection of the image segments
This time the image matches
Consequently, after a sequence of rigid motions, can be mapped onto This means that and are congruent triangles. The proof is complete.