Geometry

Unit Review Sheet

Facts and Definitions

Lesson 1: Congruence and Similarity

• Angles are the spaces formed when two sides meet at a corner, called a vertex.
• Corresponding parts are matching parts of two shapes that are in the same relative position, even if the shapes are flipped or rotated.
• Congruent shapes are shapes that match exactly, meaning they have the same size and the same shape (symbol: ≅).
• Similar shapes are shapes that have the same shape but different sizes, and their corresponding sides are proportional (symbol: ∼).
• Scale factor is the number you multiply a shape's side lengths by to make a similar shape larger or smaller.
• Corresponding parts are the matching sides or angles in two shapes that are in the same position, even if the shapes are rotated, flipped, or resized.

Lesson 2: Translations

• A translation is a transformation that slides a figure from one place to another without changing its size or rotating it.
• The notation for translation is: T_a,b → P′(x + a, y + b)

Lesson 3: Reflections

• A reflection is a type of transformation that flips a figure over a line, called the line of reflection, to create a mirror image.
• Perpendicular Distance Method: A way to reflect a point, line, or shape by drawing a straight line at a right angle (perpendicular) from the original point to the line of reflection, measuring that distance, and placing the reflected point the same distance on the opposite side.

Lesson 4: Rotations

• A rotation is a transformation that turns a figure around a fixed point, called the center of rotation.
• Rotation Coordinate Rules
  *90° Counterclockwise: (x, y)→(−y, x)
  *90° Clockwise: (x, y)→(y, −x)
  *180° (either direction): (x, y)→(−x, −y)
  *270° Counterclockwise: (x, y)→(y, −x)
  *270° Clockwise: (x, y)→(−y, x)

Lesson 5: Sequences of Rigid Transformations

• A rigid transformation moves a figure without changing its size or shape.

Lesson 6: Dilations

• A dilation is a transformation that changes the size of a figure but keeps its shape the same.
• A scale factor is the number that tells how much a shape is enlarged or reduced during a dilation.
• An enlargement is a dilation with a scale factor greater than 1 and makes the shape bigger.
• A reduction is a dilation with a scale factor less than 1 and makes the shape smaller.
• New side length = Original side length × scale factor

Lesson 7: Sequences of Transformations

• A sequence of transformations is a set of two or more transformations performed one after the other on the same figure. Each transformation changes the figure's position, orientation, or size, and the result of one transformation becomes the starting point for the next.

Lesson 8: Triangles and Transversals

• Vertical angles are the pairs of angles directly across from each other when two lines intersect. They are always equal.
• Adjacent angles share a side and a vertex, and they do not overlap.
• Supplementary angles are two angles that add up to 180°.
• Complementary angles are two angles that add up to 90°.
• Congruent angles are any two angles that have exactly the same measure.
• Triangle Sum Rule: The sum of the interior angles of any triangle is always 180°.
• Exterior Angle Rule: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• Angle-Angle (AA) Similarity Rule: If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar.
• Transversal: A line that crosses two or more other lines. When it crosses parallel lines, it creates multiple pairs of related angles.
• Alternate Interior Angles: Angles on opposite sides of the transversal, between the two parallel lines. They are equal when the lines are parallel.
• Alternate Exterior Angles: Angles on opposite sides of the transversal, outside the two parallel lines. They are equal when the lines are parallel.
• Same-Side Interior: Angles on the same side of the transversal, between the two parallel lines. They are supplementary (add up to 180°) when the lines are parallel.
• Corresponding Angles: Angles that are in the same position at each intersection where the transversal crosses the parallel lines (same side of transversal, same side of parallel lines). They are equal when the lines are parallel.

Lesson 9: Using the Pythagorean Theorem

• A right triangle is a triangle with one 90° angle.
• The legs are the two shorter sides of a right triangle that form the right angle and are usually labeled a and b.
• The hypotenuse is the longest side of a right triangle and is located directly across from the right angle, usually labeled c.
• The Pythagorean Theorem is a formula that shows the relationship between the sides of a right triangle:
  $a^2+b^2=c^2$
  .
• The slant height is the distance measured along the middle of a triangular face, from the midpoint of a base edge up to the top vertex (apex) of a pyramid.

Lesson 10: Volume

• Volume is the amount of space a three-dimensional shape takes up.
• The formula to find the area of a circle is: $A$
• Radius of a circle = Half the diameter
• $\pi$
• The formula to find the volume of a cylinder is: $V$
• The formula to find the volume of a cone is: $V$
• The formula to find the volume of a sphere is: $V$

Lesson 11: Unit 6 Test

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Final Project: Abstract Art Gallery

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