Congruent Triangles, Chapter 4

Types of Triangles

• Acute Triangle: Contains three angles, each less than 90 degrees.
• Obtuse Triangle: Has one angle greater than 90 degrees.
• Right Triangle: Features one angle exactly at 90 degrees.
• Equilateral Triangle: All three sides are equal in length; also has three congruent angles (60 degrees each).
• Isosceles Triangle: At least two sides are equal, leading to two congruent angles.
• Scalene Triangle: No sides are congruent, and all angles are different.

Triangle Congruence

• Congruent Polygons: Two shapes are congruent if their corresponding sides and angles match in measure.
• Side-Side-Side (SSS) Congruence: Triangles are congruent if all three sides of one are equal to the three sides of another.
• Side-Angle-Side (SAS) Congruence: Congruency is established when two sides and the included angle of one triangle match those of another.
• Angle-Side-Angle (ASA) Congruence: Two triangles are congruent if two angles and the included side are congruent.
• Angle-Angle-Side (AAS) Congruence: If two angles and the non-included side of one triangle match those of another, the triangles are congruent.
• Leg-Leg (LL) Congruence: Specifically for right triangles, both legs must be congruent.
• Hypotenuse-Angle (HA) Congruence: The hypotenuse and one acute angle of a right triangle must be congruent to those of another.
• Leg-Angle (LA) Congruence: One leg and an acute angle of a right triangle must match those of another.
• Hypotenuse-Leg (HL) Congruence: In right triangles, the hypotenuse and one leg must be congruent between the two triangles.

Angle Relationships

• Triangle Angle-Sum Theorem: The total measure of angles in a triangle is always 180 degrees.
• Exterior Angle: Formed by extending one side of the triangle; it is supplementary to the adjacent interior angle.
• Remote Interior Angles: Two interior angles that are not adjacent to a given exterior angle of a triangle.

Isosceles Triangle Theorem

• Isosceles Triangle Theorem: States that if two sides of a triangle are congruent, then the angles opposite those sides are also congruent.
• Converse of the Isosceles Triangle Theorem: If two angles in a triangle are congruent, their opposite sides are also congruent.

Proof Techniques

• Flow Proof: A structured proof using boxes and arrows to illustrate logical connections.
• Two-Column Proof: Presents statements and the corresponding reasons in a two-column format for clarity.
• Coordinate Proof: Involves plotting figures on a coordinate plane for proof.

Transformations

• Transformation: An operation that maps a figure onto another.
• Rigid Transformation: Any transformation that preserves the shape and size of the figure.
• Isometry: Another term for rigid transformations.
• Reflections: A flip of the figure over a line.
• Translations: A slide in a specific direction.
• Rotations: A turn around a fixed point.

Angle Relationships

• Complementary Angles: Two angles sum up to 90 degrees.
• Supplementary Angles: Two angles sum up to 180 degrees.

Distance and Midpoint Formulas

• Distance Formula: Calculated as (d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}).
• Midpoint Formula: Determines the midpoint between two coordinates as (\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)).