FLVS Understanding Symmetry and Rigid Transformations Note Guide

Understanding Symmetry

Definition of Symmetry

• A line of symmetry divides a figure into two equal halves that are identical in size and shape.

• Symmetrical figures can be folded along the line of symmetry without overlapping edges.

• Common examples include shapes like squares and circles, which have multiple lines of symmetry.

Types of Symmetry

• Reflectional Symmetry: A figure can be divided into two identical halves by a line (the line of symmetry).

• Rotational Symmetry: A figure can be rotated around a center point and still look the same after a certain degree of rotation.

Finding Lines of Symmetry

• To find lines of symmetry, visualize how the figure can be folded in half.

• For example, a rectangle has two lines of symmetry: one vertical and one horizontal.

Examples of Symmetrical Figures

• Equilateral Triangle: Has three lines of symmetry and rotational symmetry of order 3.

• Square: Has four lines of symmetry and rotational symmetry of order 4.

Rotational Symmetry

Understanding Rotational Symmetry

• A figure has rotational symmetry if it can be rotated about a center point and still appear the same.

• The order of rotation is the number of times a figure maps onto itself during a full rotation (360°).

Calculating Angles of Rotation

• The angle of rotation can be calculated by dividing 360° by the order of rotation.

• For example, a triangle (order 3) has an angle of rotation of 120° (360°/3).

Examples of Rotational Symmetry

• Kaleidoscope: May have an order of 2, meaning it looks the same after a 180° rotation.

• Regular Decagon: Has 10 vertices, order of rotation is 10, angle of rotation is 36° (360°/10).

Special Cases in Rotational Symmetry

• A regular octagon typically has an order of rotation of 8, but specific designs may alter this to 4 due to patterns.

Rigid Transformations and Congruence

Rigid Transformations Defined

• Rigid transformations include translations, reflections, and rotations that preserve the shape and size of figures.

• These transformations justify that two figures are congruent.

Identifying Transformations

• To determine the transformation that maps one triangle to another, analyze the movement of vertices.

• Example: If triangle ABC is moved to triangle DEF, identify if it was translated, reflected, or rotated.

Practice with Rigid Transformations

• Practice identifying transformations using coordinate planes and visual aids.

• Example: Determine the transformation that justifies ΔABC ≅ ΔDEF.

Proof by Contradiction

Understanding Proof by Contradiction

• A proof by contradiction starts by assuming the opposite of the statement to be proven is true.

• This method is used to show that the assumption leads to a contradiction, thus proving the original statement.

Steps in Constructing a Proof by Contradiction

1. Identify the conclusion of the original statement.

2. Assume the opposite of the original statement is true.

3. Show that this assumption leads to a contradiction.

4. Conclude that the original statement must be true.

Example of Proof by Contradiction

• To prove that two supplementary angles cannot both be obtuse:

1. Assume both angles are obtuse (greater than 90°).

2. Show that their sum exceeds 180°, contradicting the definition of supplementary angles.