Triangle Theorems
UNIT 5: Triangle Theorems
Triangle Congruence, Similarity, and Special Segments
Page 2: Posters and Anchor Charts
Polygons: Interior angle sums and exterior angle sums.
Triangle Sum Theorem: The sum of interior angles in a triangle is 180°.
Triangle Inequality & Pythagorean Inequality: Rules governing triangle side lengths and right triangles respectively.
Triangle Congruence and Similarity Theorems: The criteria for determining triangle congruence and similarity; includes solving proportions.
Page 4: DAY 1 - Prerequisite Knowledge
Triangle Symbol (△): Represents triangle, e.g., △ABC.
Conditional Statements: Review of IF-THEN statements.
2-Column Proofs: A method of proof using statistical reasoning.
Triangle Classification:
Acute: All angles < 90°
Right: One angle = 90°
Obtuse: One angle > 90°
Equilateral: All sides and angles equal
Isosceles: Two sides and angles equal
Scalene: No sides or angles equal
Naming triangles: Use vertices in a clockwise (CW) or counterclockwise (CCW) manner, e.g., △ABC.
Page 5: DAY 1 - Triangle Classification by Sides and Angles
Triangle Inequality Theorem: A new theorem explored this day.
Classification:
By sides: scalene, isosceles, equilateral.
By angles: acute, right, obtuse.
Activity: Triangle classification exercise.
Page 6: DAY 2 - Range of Possible Side Lengths
Triangle Inequality Theorem: Conditions for side lengths to form a triangle:
The length of any side must be less than the sum of the other two sides.
The length of any side must be greater than the difference of the other two sides.
Charts and Examples: Review inequalities for triangle formations.
Page 8: DAY 1 - Triangle Inequality Theorem
Definition: The sum of any two sides must always be greater than the length of the third side, and the difference must be less than the third side.
Proof: Illustrated through positioning and manipulation of triangle angles and sides.
Page 9: DAY 2 - Pythagorean Inequality in Classification
Establishing if a triangle is acute, right, or obtuse by using the squares of side lengths.
Acute Triangle: c² < a² + b²
Right Triangle: c² = a² + b²
Obtuse Triangle: c² > a² + b²
Page 11: DAY 3 - Interior vs Exterior Angles
Interior Angles: Angles within the triangle.
Exterior Angles: Formed by extending the triangle's side.
Remote Interior Angles: Non-adjacent interior angles relative to an external angle.
Page 12: DAY 3 - Triangle Sum Theorem
Theorem: Sum of interior angles in a triangle = 180°.
Proof Process: Steps to demonstrate validity using parallel lines and angles.
Page 13: DAY 3 - Exterior Angle Theorem
Theorem: The exterior angle equals the sum of the two remote interior angles.
Proof Process: Similar structured proof as The Triangle Sum Theorem.
Page 15: DAY 4-5 - Review & Practice
Isosceles and Equilateral Triangles: Emphasis on their properties for the test preparation.
Isosceles: Two congruent sides and angles.
Equilateral: All sides and angles equal (60 degrees each).
Page 19: Properties of Congruent Triangles
Key Concept: If two triangles are congruent, all corresponding parts are congruent (CPCTC).
Other properties (area, perimeter) are also the same.
Page 20: Triangle Congruence Theorems
Triangles are congruent based on:
SSS (Side-Side-Side)
SAS (Side-Angle-Side)
ASA (Angle-Side-Angle)
AAS (Angle-Angle-Side)
HL (Hypotenuse-Leg for right triangles).
Page 21: Groups that Do Not Work
Triangles are NOT congruent if:
AAA (All angles equal) - Similar but not congruent.
SSA (Two sides, non-included angle).
Page 28: Corresponding Parts of Congruent Triangles
CPCTC reiterated as an important principle in proofs.
Page 56-63: Special Segment Constructions
Centroid, Circumcenter, Incenter: Definitions and methods for constructing special segments and bisectors within triangles.