“Triangle Congruence Theorems” Geometry “Triangle Congruence Theorems”

Congruent Triangles This topic focuses on proving triangles congruent (isometric). We have first learned the process of justifying steps. We must follow a general process, with justifications how to prove triangles are congruent. The proof is out there...

The 4 Triangle Congruence Theorems By comparing sides and/or angles, we can prove triangles to be congruent. Don’t be an ASS! Unless you’re right... RHS

Theorem If two angles in one triangle are congruent to two angles in another triangle, the third angles must also be congruent. Think about it… they have to add up to 180°

Let’s see what this means... If two triangles have two pairs of angles congruent, then their third pair of angles is congruent. 85° 30° 85° 30° But do the two triangles have to be congruent?

Example Draw two non-congruent triangles with angles of 30 and 90. 30° 30° Don’t these triangles have to be congruent? This leads us to our first theorem of congruent triangles…

ASA (Angle, Side, Angle) If two angles and the included side of one triangle are congruent to two angles and the included side of the other, then the triangles are congruent. A C B X X F E D X AD AB  DE BE Then the 2 triangles are congruent.

AAS (Angle, Angle, Side) Special case of ASA B If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent. X X F E D X The third pair of angles must be congruent, so it’s considered ASA C  F, A  D, AB  DE, then the 2 triangles are congruent.

SAS (Side, Angle, Side) CA  FD A  D AB  DE If in two triangles, two sides and the contained angle of one are congruent to two sides and the contained angle of the other, then the triangles are congruent. F E D CA  FD A  D AB  DE … then the 2 triangles are congruent.

SSS (Side, Side, Side) CA  FD AB  DE CB  FE In two triangles, if 3 sides of one are congruent to three sides of the other then the triangles are congruent. F E D CA  FD AB  DE CB  FE Then the triangles are congruent.

RHS (Right Angle, Hypotenuse, Side) C B If both triangles have a right angle, both hypotenuses are congruent, and another pair of sides are congruent, then the triangles are congruent. F E D A  D = 90° CB  FE AB  DE … then the triangles are congruent.

Example A C B Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D E Yes, by SAS F

Example No, No, ASS does not guarantee congruence No, Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? A C B No, No, ASS does not guarantee congruence No, G I H F E D The angle must be between the congruent sides

Example D A C B Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? Yes, by SSS Yes, by SSS ABC  ? DBC

Summary: The four congruence theorems: ASA - Pairs of congruent sides contained between two congruent angles SAS - Pairs of congruent angles contained between two congruent sides SSS - Three pairs of congruent sides RHS - ASS condition where matching angles are 90°

That’s all folks...