Geometry Honors Study Guide: Transformations, Congruence, and Triangle Proofs

Fundamental Definitions and Concepts of Geometric Transformations

• Transformation: The change in position, size, or shape of a geometric figure.
• Pre-Image: The original figure before a transformation is applied.
• Image: The new figure resulting from a transformation.
• Mapping: A transformation maps the pre-image to the image.
• Notation:
  • Arrow notation ($\rightarrow$) is used to describe a transformation (e.g., $\triangle ABC \rightarrow \triangle A'B'C'$).
  • Primes ($'$) are used to label the image figure (e.g., $A'$ is the image of $A$).
• Three Basic Rigid Transformations:
  • Reflection (or Flip): A transformation across a line, called the line of reflection. Each point on the pre-image and its corresponding image point are the exact same distance from the line of reflection.
  • Rotation (or Turn): A transformation about a point $P$, called the center of rotation. Each point on the pre-image and its corresponding image point are the exact same distance from the center of rotation $P$.
  • Translation (or Slide/Shift): A transformation in which all the points of a figure move the exact same distance in the exact same direction.
• Application to Sample Figures: Transformations can be performed on target figures or letters such as $E$, $K$, or $P$.

Transformation Coordinate Rules and Examples

• Identifying and Describing Transformations:
  • Example 1A: Rotation of $90^\circ$ counterclockwise.
  • Example 1B: Reflection over the $x$-axis.
  • Example 1C: Translation right $2$ units, down $3$ units.
• Applying Coordinate Rules ($M$):
  • Rule A: $M: (x, y) \rightarrow (x - 4, y + 1)$
  • Description: Translation left $4$ units, up $1$ unit.
  • Vertices Application: Pre-image polygon with vertices $P(1, 3)$, $Q(1, 1)$, and $R(4, 1)$ maps to image vertices $P'(-3, 4)$, $Q'(-3, 2)$, and $R'(0, 2)$.
  • Rule B: $M: (x, y) \rightarrow (x, -y)$
  • Description: Reflection over the $x$-axis.
  • Vertices Application: Pre-image polygon with vertices $A(1, 2)$, $B(4, 2)$, and $C(3, 1)$ maps to image vertices $A'(1, -2)$, $B'(4, -2)$, and $C'(3, -1)$.
  • Rule C: $M: (x, y) \rightarrow (y, x)$
  • Description: Rotation $90^\circ$ clockwise about the origin.
  • Vertices Application: Pre-image polygon with vertices $R(-3, 0)$, $E(-3, 3)$, $C(-1, 3)$, and $T(-1, 0)$ maps to image vertices $R'(0, 3)$, $E'(3, 3)$, $C'(3, 1)$, and $T'(0, 1)$.
• Board Practice Transformations:
  • 1. Translation right $6$ units, down $2$ units.
  • 2. Reflection over the $y$-axis.
  • 3. Rotation $90^\circ$ clockwise.
  • 4. Rotation $180^\circ$ clockwise or counterclockwise about the origin.
  • 5. Reflection.
  • 6. Reflection over the $y$-axis applied to coordinates $B(-5, -1)$, $A(-0.6, -2)$, and $C(3, -4)$ resulting in $B'(5, -1)$, $A'(0.6, -2)$, and $C'(-3, -4)$.

Congruence and Rigid Motion Transformations

• Connection Between Transformations and Congruence: Reflections, rotations, and translations result in an image that is congruent to the pre-image.
• Definition of Congruent Figures: Figures are congruent when all pairs of corresponding sides and all pairs of corresponding angles are congruent.
• Listing Corresponding Congruent Parts:
  • Given Statement: $\triangle DOG \cong \triangle CAT$
  • Corresponding Angles:
  • $\angle D \cong \angle C$
  • $\angle O \cong \angle A$
  • $\angle G \cong \angle T$
  • Corresponding Sides:
  • $DO \cong CA$
  • $OG \cong AT$
  • $DG \cong CT$

Geometric Postulates, Properties, and Proof Rules

• Reflexive Property Postulate: Any segment or angle is congruent to itself.
  • Examples: $KT \cong KT$, $\angle Q \cong \angle Q$, $RC \cong RC$, $MT \cong MT$
• Specific Triangle Congruence & Property Examples:
  • $\triangle ERC \cong \triangle TCR$ because $RC \cong RC$ by the Reflexive Property.
  • Given $E$ is the midpoint of $TP$, $\triangle SPE \cong \triangle RTE$.
  • $\triangle BOW \cong \triangle YTI$ where $\angle 1 \cong \angle 2$ because vertical angles are congruent.
• Shortcuts and Direct Reasoning Rules in Proofs:
  • Straight Angles: Diagrams do not require assuming straight angles; direct statements can be made such as "straight angles are congruent" or "If the sum of $2$ angles is a straight angle, then they are supplementary."
  • Complementary Angles:
  • Angles complementary to the same angle are congruent.
  • Angles complementary to congruent angles are congruent.
  • Supplementary Angles:
  • Angles supplementary to the same angle are congruent.
  • Angles supplementary to congruent angles are congruent.
  • Perpendicular Lines and Right Angles:
  • Perpendicular segments form right angles.
  • All right angles are congruent.
  • All straight angles are congruent.
  • Midpoints and Bisectors:
  • A midpoint divides a segment into $2$ congruent segments.
  • A segment bisector divides a segment into $2$ congruent segments.
  • Angle Bisectors and Trisection:
  • An angle bisector divides an angle into $2$ congruent angles.
  • An angle trisector divides an angle into $3$ congruent angles.
  • Segment Trisection:
  • A segment trisector divides a segment into $3$ congruent segments.
• Triangle Correspondence Notation: Line up corresponding vertices when writing triangle congruence statements (e.g., $\triangle HMT \cong \triangle AMT$).

Triangle Congruence Shortcuts

• Proving Triangles Congruent: It is not necessary to show that all three pairs of sides and all three pairs of angles are congruent to prove two triangles are congruent. Four valid shortcuts exist:
  • $SSS$ (Side-Side-Side): The three sides of one triangle must be congruent to the three sides of the other triangle.
  • $SAS$ (Side-Angle-Side): Two sides and the included angle of one triangle must be congruent to two sides and the included angle of the other triangle.
  • $ASA$ (Angle-Side-Angle): Two angles and the included side of one triangle must be congruent to two angles and the included side of the other triangle.
  • $AAS$ (Angle-Angle-Side): Two angles and the nonincluded side of one triangle must be congruent to two angles and the nonincluded side of the other triangle.
• Invalid Methods:
  • $AAA$ (Angle-Angle-Angle) cannot be used in proofs.
  • $SSA$ (Side-Side-Angle) cannot be used in proofs.

Step-by-Step Formal Geometric Proofs

• Proof 1:

  • Given: $ID$ bisects $\angle WIL$, $ID \perp WL$
  • Prove: $\triangle WID \cong \triangle LID$
  • Proof Steps:
  1. $ID$ bisects $\angle WIL$, $ID \perp WL$ | Given
  2. $\angle WID \cong \angle LID$ | Bisector divides an angle into $2$ congruent angles
  3. $\angle WDI$ and $\angle LDI$ are right angles | Perpendicular segments form right angles
  4. $\angle WDI \cong \angle LDI$ | Right angles are congruent
  5. $ID \cong ID$ | Reflexive Property
  6. $\triangle WID \cong \triangle LID$ | $ASA$

• Proof 2:

  • Given: $AD = AE$, $AF = AG$, $CD = BE$, $FG$ bisects $AB$ and $AC$
  • Prove: $\triangle ACE \cong \triangle ABD$
  • Proof Steps:
  1. $AD = AE$, $AF = AG$, $CD = BE$, $FG$ bisects $AB$ and $AC$ | Given
  2. $CE = DB$ | Addition
  3. $AC = AB$ | Addition / Multiplication
  4. $\triangle ACE \cong \triangle ABD$ | $SSS$

• Proof 3:

  • Given: $GM = OH$, $\angle G$ is complementary to $\angle HEM$, $\angle O$ is complementary to $\angle HEM$, $EG = EO$
  • Prove: $\triangle EGH \cong \triangle EOM$
  • Proof Steps:
  1. $GM = OH$, $\angle G$ comp. to $\angle HEM$, $\angle O$ comp. to $\angle HEM$, $EG = EO$ | Given
  2. $\angle G \cong \angle O$ | Angles complementary to the same angle are congruent
  3. $GH = OM$ | Subtraction
  4. $\triangle EGH \cong \triangle EOM$ | $SAS$

• Proof 4:

  • Given: $B$ is the midpoint of $AE$, $AE$ bisects $CD$
  • Prove: $\triangle ACB \cong \triangle EDB$
  • Proof Steps:
  1. $B$ is the midpoint of $AE$, $AE$ bisects $CD$ | Given
  2. $AB \cong BE$ | Midpoint divides a segment into $2$ congruent segments
  3. $CB \cong BD$ | Bisector divides a segment into $2$ congruent segments
  4. $\angle CBA \cong \angle EBD$ | Vertical angles are congruent
  5. $\triangle ACB \cong \triangle EDB$ | $SAS$