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Reference

Triangle Congruence Theorems

Rule

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.

Side-Angle-Side Congruence Theorem

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.

Two triangles with two congruent sides and one congruent angle
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.
SAS translation
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
SAS rotation
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
SAS reflection
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.

Rule

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.

Triangle ABC is congruent to triangle DEF

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.

Triangle ABC and triangle DEF with two pairs of congruent angles and one pair of the congruent included sides
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.
Translation that maps vertex D of triangle DEF onto vertex A of triangle ABC
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
Rotation that maps vertex E' of AE'F' onto B of ABC
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
Reflection that maps ABF'' onto ABC
This time the image matches
Consequently, after a sequence of rigid motions can be mapped onto This means that and are congruent triangles.

Rule

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.

Two congruent triangles ABC and DEF

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.

Two triangles ABC and DEF with congruent corresponding sides
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.
Translation of ABC
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
Rotation of AE'F' about A
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.
Reflecting ABF'' across line AB
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.

Rule

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.

Triangles ABC and DEF
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.
DEF is translated
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
Translation of CD'E'
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

Triangles ABC and CBD'' with a common side CB
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
Reflection of CBD'' across BC
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.