Triangle congruence is all about matching shapes. When two triangles have the same size and shape, they're congruent. This means their sides and angles line up perfectly. It's like finding a perfect twin for a triangle!

To prove triangles are congruent, we use special rules. These rules, called postulates and theorems, help us match up sides and angles. By using these tools, we can solve tricky geometry problems and find missing measurements in triangles.

Triangle Congruence

Triangle congruence conditions

• Two triangles are congruent if their corresponding sides and angles are equal in measure
• Corresponding sides must be equal in length
• Corresponding angles must be equal in measure
• Congruent triangles denoted using the symbol $\cong$
• Corresponding vertices labeled with the same letter, often uppercase for one triangle and lowercase for the other ($\triangle ABC \cong \triangle abc$)

SSS, SAS, and ASA postulates

• Side-Side-Side (SSS) postulate states if three sides of one triangle are equal in length to three sides of another triangle, then the triangles are congruent ($\overline{AB} \cong \overline{DE}$, $\overline{BC} \cong \overline{EF}$, and $\overline{AC} \cong \overline{DF}$ $\implies \triangle ABC \cong \triangle DEF$)
• Side-Angle-Side (SAS) postulate states if two sides and the included angle of one triangle are equal in measure to two sides and the included angle of another triangle, then the triangles are congruent ($\overline{AB} \cong \overline{DE}$, $\angle B \cong \angle E$, and $\overline{BC} \cong \overline{EF}$ $\implies \triangle ABC \cong \triangle DEF$)
• Angle-Side-Angle (ASA) postulate states if two angles and the included side of one triangle are equal in measure to two angles and the included side of another triangle, then the triangles are congruent ($\angle A \cong \angle D$, $\overline{AB} \cong \overline{DE}$, and $\angle B \cong \angle E$ $\implies \triangle ABC \cong \triangle DEF$)

AAS theorem for congruence

• Angle-Angle-Side (AAS) theorem states if two angles and a non-included side of one triangle are equal in measure to two angles and the corresponding non-included side of another triangle, then the triangles are congruent ($\angle A \cong \angle D$, $\angle B \cong \angle E$, and $\overline{BC} \cong \overline{EF}$ $\implies \triangle ABC \cong \triangle DEF$)
• AAS is a theorem, not a postulate, and can be proven using the ASA postulate and the fact that the sum of the measures of the angles in a triangle is 180°

Applications of congruence postulates

• Identify the given information in the problem and determine which sides or angles are equal in measure
• Choose the appropriate postulate or theorem based on the given information (SSS, SAS, ASA, or AAS)
• Apply the postulate or theorem to prove triangle congruence by writing the congruence statement using the correct notation
• Use the congruence to solve the problem by finding the measures of missing sides or angles based on the corresponding parts of the congruent triangles