Geometric Transformations Unit 1 Geometric Transformations

Standard: m8g.5 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.

Element: m8g.5 Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.

Element: m8g.5 Apply properties of angle pairs formed by parallel lines cut by a transversal.

Unit 1 Vocabulary (Ch. 6 pg 256) Acute Angle – measures less than 90°. Right Angle – measures equal to 90°. Obtuse Angle – have measures between 90° and 180°. Straight Angle – measures equal to 180°.

Unit 1 vocabulary cont. Vertical Angles – opposite angles formed by intersecting lines. Vertical angles are congruent. Adjacent Angles – have the same vertex (angle), share a common side, and do NOT overlap.

Unit 1 vocabulary cont. Complimentary Angles – angles whose sum of their measures is equal to 90°. If you put the angles together, the angles should form a right angle. Supplementary Angles – angles whose sum of their measures is equal to 180°. If you put the angles together, the angles should form a straight angle.

Perpendicular Lines - lines that intersect at right angles. Unit 1 vocabulary cont. Perpendicular Lines - lines that intersect at right angles. Parallel Lines – two lines in a plane that never intersect or cross. Draw the diagram on page 257.

Transversal – a line that intersects two or more lines. Unit 1 vocabulary cont. Transversal – a line that intersects two or more lines. Draw diagram on page 258.

Complimentary angles

Parallel lines cut by a transversal

Angle Relationships

Transfer Notes to your math journal Transfer the notes from the Buckle Down Workbook on pages 120- 121 to your math journal.

Angle Relationships

Geometric city

Acute and obtuse angles Interior and exterior angles Vertical angles Review your notes Acute and obtuse angles Interior and exterior angles Vertical angles Corresponding angles Same side angles Complementary and Supplementary angles

Combining like terms Term – the part of an expression the is separated by a + or – sign. Variable – a symbol used to represent a quantity that can change. Coefficient – the number that is multiplied by a variable. Constant – a term without a variable; the value never changes. Like terms – terms that have the same variable(s) raised to the same powers(s).

Combing like terms

Distributive property

Distributive property

Find the missing angles

Transfer to Math Journal MGSE8.G.1 Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines. 

Transfer to math journal MGSE8.G.2 Understand that a two‐dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

Transfer to math journal MGSE8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two‐ dimensional figures using coordinates.

Coordinate System vocabulary Coordinate plane – formed by two number lines that form right angles and intersect at their zero points.  Y-axis – the vertical number line. X-axis – the horizontal number line. 

Coordinate system vocabulary cont. Origin – The point of intersection of the two number lines. The ordered pair for the origin is (0,0). Quadrants – The number lines separate the coordinate plane into four quadrants (sections).  Ordered Pair – Any point on the coordinate plane can be graphed by using a pair of numbers. The first number in the ordered pair is the x-coordinate. The second number in the ordered pair is the y- coordinate. (x,y).

Coordinate system vocabulary cont. X-coordinate – The first number in an ordered pair. Also called the abscissa. Y-Coordinate – The second number in an ordered pair. Also called the ordinate. 

Review of coordinate system

Transfer to math journal MGSE8.G.4 Understand that a two‐dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two‐dimensional figures, describe a sequence that exhibits the similarity between them.

Transformation – mapping of a geometric figure. Vocabulary Transformation – mapping of a geometric figure. Reflection – mirror image produced by flipping a figure over a line. Line of Reflection – the line an image is flipped over.

BrainPop quiz for transformations https://www.brainpop.com/m ath/geometryandmeasuremen t/transformation/ 

Cut and paste into your math journal. Please use a glue stick!

Translations

Cut and paste into your math journal. Please use a glue stick!

Transfer to Math Journal MGSE8.G.1 Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines. 

Translation test PRacice http://www.regentsprep.org/regents/math/geometry/GT2/PracT.htm

Translations

Translations

Rotations

Complete packet (packet will be your test grade) complete transformation Artwork (follow instructions on page) complete transformation airplane (follow instructions on page)

Properties preserved (invariant) under a rotation: 1 Properties preserved (invariant) under a rotation: 1. distance is preserved (lengths of segments are the same) 2. angle measures (remain the same) 3. parallelism (parallel lines remain parallel) 4. colinearity (points stay on the same lines) 5. midpoint (midpoints remain the same in each figure) 6. orientation (lettering order remains the same)

Reflections

Write question and answer in math journal How would you reflect an image over the x- axis? How would you translate a figure T(3,-5)? How would you rotate a point R(90)? How would you rotate an image R(270)? How would you rotate an image R(-90)?

Transfer to Math journal Dilation - A dilation is a transformation (notation ) that produces an image that is the same shape as the original, but is a different size. A dilation stretches or shrinks the original figure. The description of a dilation includes the scale factor (or ratio) and the center of the dilation.

Scale factor definition In two similar geometric figures, the ratio of their corresponding sides is called the scale factor. To find the scale factor, locate two corresponding sides, one on each figure. Write the ratio of one length to the other to find the scale factor from one figure to the other.

MGSE8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two‐ dimensional figures using coordinates.

Math mashup

Discovery Education https://app.discoveryeducation.com/learn/videos/5792d4bc-4fc1-4267-bdee-f151af95dc1d

Transfer to math journal A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries. A periodic tiling has a repeating pattern.

Angles in a triangle