Triangles and Congruence

Triangles

  • Definition of a Triangle: A triangle is a closed figure in a plane consisting of three segments called sides. Any two sides intersect at exactly one point called a vertex.

  • Naming Triangles: A triangle can be named using the capital letters assigned to its vertices in a clockwise or counterclockwise direction. For example, triangle ABC is named counterclockwise beginning with vertex A.

Alternative Names for Triangle ABC

  • Other Names: Triangle ABC can also be referred to as:

    • ∆ACB

    • ∆BAC

    • ∆BCA

    • ∆CAB

    • ∆CBA

Classification of Triangles

  • Triangles can be classified according to their

    • Sides

    • Angles

Classification by Sides

  1. Equilateral Triangle: A triangle with three congruent sides.

    • Properties: All angles are equal, and each angle measures 60 degrees.

  2. Isosceles Triangle: A triangle with at least two sides congruent.

    • Properties: The angles opposite the congruent sides are equal.

  3. Scalene Triangle: A triangle with no two sides congruent.

    • Properties: All sides and angles are different.

Answers to Classification Questions

  • b. Are all equilateral triangles isosceles? Yes, because every equilateral triangle has at least two sides that are congruent.

  • c. Are some isosceles triangles equilateral? Yes, if all three sides of an isosceles triangle are congruent, it is also equilateral.

Classification by Angles

  1. Acute Triangle: A triangle with three acute angles (all angles less than 90 degrees).

  2. Obtuse Triangle: A triangle with one obtuse angle (greater than 90 degrees).

  3. Right Triangle: A triangle with one right angle (exactly 90 degrees).

    • Hypotenuse: In right triangles, the side opposite the right angle is called the hypotenuse, and the other two sides are called legs.

Exercises

  1. True or False Questions (with justifications):

    1. An equilateral triangle is always acute. True; all angles are 60 degrees.

    2. An obtuse triangle can also be isosceles. True; it can have two sides congruent and one angle greater than 90 degrees.

    3. The acute angles of a right triangle are complementary. True; they sum to 90 degrees.

  2. Coordinate Geometry Exercise:

    • Given points for triangle ABC: A(1,5), B(5,5), C(5,1)

      • a. Graph ∆ABC on the coordinate plane.

      • b. Classify this triangle by its sides and angles.

Combined Classification

  • Examples of Combined Classifications:

    • Right Isosceles Triangle

    • Right Scalene Triangle

    • Obtuse Isosceles Triangle

Exploration of Triangle Formation

  • Experiment using segments of linguine:

    • Snap segments to create lengths of 3, 5, 6, and 9 inches.

    1. Determine which sets of three lengths can make a triangle.

    2. Identify the sets that did not form a triangle.

    • Example: 3, 6, and 9 did not form a triangle since 3 + 6 = 9.

    1. Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Theorem Example

  • Finding Possible Lengths for the Third Side: Given two sides of lengths 4 cm and 7 cm:

    • 4 + 7 > x

    • 4 + x > 7

    • 7 + x > 4

    • From above, 11 > x > 3; therefore, possible lengths for the third side are given by the intersection of these inequalities.

Segments of Triangles

  • Definitions:

    1. Altitude: A segment drawn from a vertex perpendicular to the opposite side or its extension.

    2. Median: A segment having one endpoint at a vertex and the other at the midpoint of the opposite side.

    3. Angle Bisector: A segment that bisects an angle of a triangle.

  • Properties:

    • Every triangle has three altitudes that meet at a point called the orthocenter.

    • Every triangle has three medians that intersect at the centroid, or center of mass.

    • Angle bisectors meet at the incenter, the center of the inscribed circle.

Examples of Triangle Segments

  1. Exploring the intersection of altitudes using paper folding.

  2. Verifying the meeting point of medians.

  3. Investigating angle bisectors using paper folding.

Point of Concurrency

  • Altitudes, medians, and angle bisectors have common points of intersection known as points of concurrency.

Congruent Triangles

  • Definition: Two triangles are congruent if and only if their corresponding sides and their corresponding angles are congruent.

How to Note Congruence

  • Notation: ∆ABC ≅ ∆DEF

  • Corresponding parts of congruent triangles are referred to as corresponding parts of congruent triangles (CPCTC).

Transitive and Reflexive Properties

  • Reflexive Property: A triangle is congruent to itself (∆RST ≅ ∆RST).

  • Transitive Property: If ∆RST ≅ ∆UVW and ∆UVW ≅ ∆XYZ, then ∆RST ≅ ∆XYZ.

Congruence Postulates

  1. SSS (Side-Side-Side): Two triangles are congruent if three sides of one triangle are congruent to three sides of another triangle.

  2. SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle.

  3. ASA (Angle-Side-Angle): Two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle.

Examples for Each Postulate

  1. SSS: Identify and demonstrate congruency through corresponding sides.

  2. SAS: Show congruency using given sides and the included angle.

  3. ASA: Prove triangle congruency via corresponding angles and the included side.

Exercises and Conjecture Testing

  • Verifying if any two angles and a non-included side (AAS) make triangles congruent.

  • Considerations of scenarios and assessing the possibility of an SSA arrangement.

Additional Proofs and Congruence Applications

  1. Complete various proofs using postulates to establish triangle congruency.

  2. Apply learned properties and theorems to justify congruency in geometrical contexts.