Week 9: October 9th – 13th

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TeacherFelicia Taylor
Subject AreaGeometry Honors
Grade Level8
Week #9
Unit of Instruction2(A): Triangle Congruence and Proofs
Standard(s) Taught

MAFS.912.G-CO.1.2
Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

MAFS.912.G-CO.1.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

MAFS.912.G-CO.1.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

MAFS.912.G-CO.1.5
Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

MAFS.912.G-CO.2.6
Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

MAFS.912.G-CO.2.7
Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

MAFS.912.G-CO.2.8
Explain how the criteria for triangle congruence (ASA, SAS, SSS, and Hypotenuse-Leg) follow from the definition of congruence in terms of rigid motions.

MAFS.912.G-CO.3.10
Prove theorems about triangles; use theorems about triangles to solve problems. (Theorems include: measures of interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.)

MAFS.912.G-SRT.2.5
(Congruence)
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

Learning Targets and Learning Criteria

• draw transformations of reflections, rotations, translations, and combinations of these using graph paper, transparencies, and/or geometry software.
• find the coordinates for the image (output) of a figure when a transformation rule is applied to the pre-image (input).
• distinguish between transformations that are rigid and those that are not.

• describe and illustrate how a rectangle, parallelogram, and trapezoid are mapped onto themselves using transformations.
• calculate the number of lines of reflection symmetry and the degree of rotational symmetry of any regular polygon.

• define pre-image and image, and use correct notation (A vs. A’)
• construct a reflection, translation, and rotation.
• model the reflection definition by connecting any point on the pre-image to its corresponding point on the reflected image and describing the line segment’s relationship to the line of reflection.
• model the translation definition by connecting any point on the pre-image to its corresponding point on the translated image, and connecting a second point on the pre-image to its corresponding point on the translated image, and describing how the two segments are equal in length, point in the same direction, and are parallel.
• model the rotation definition by connecting the center of rotation to any point on the pre-image and to its corresponding point on the rotated image, and describing the measure of the angle formed and the equal measures of the segments that formed the angle as part of the definition.

• draw a specific transformation when given a geometric figure and a rotation, reflection, or translation.
• predict and verify the sequence of transformations (a composition) that will map a figure onto another.

• define and use rigid motions as reflections, rotations, translations and combinations of these, all of which preserve distance and angle measure.
• define congruent figures as figures that have the same shape and size and state that a composition of rigid motions will map one congruent figure onto the other.
• predict the composition of transformations that will map a figure onto a congruent figure.
• determine if two figures are congruent by determining if rigid motions will turn one figure into the other.

• identify corresponding sides and corresponding angles of congruent triangles.
• explain that in a pair of congruent triangles, corresponding sides are congruent (distance is preserved) and corresponding angles are congruent (angle measure is preserved).
• demonstrate that when distance is preserved (corresponding sides are congruent) and angle measure is preserved (corresponding angles are congruent) the triangles must also be congruent.

• list the conditions to prove triangles are congruent.
• map a triangle with one of the conditions (e.g., SSS) onto the original triangle and show that corresponding sides and corresponding angles are congruent.
• explain AAS as criterion for congruence, and demonstrate that SSA and AAA do not prove congruence.

• apply triangle congruence and triangle similarity to solve problems (e.g., indirect measure, missing sides/angle measures, side splitting).
• apply triangle congruence and triangle similarity to prove relationships in geometric figures.
• use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to find missing sides/angles AFTER triangles are proved congruent.• classify triangles by sides and angles.
• prove theorems about triangles.
• solve problems using definitions, theorems, postulates, about triangles, including:
a) Measures of interior angles of a triangle sum to 180;
b) Base angles of isosceles triangles are congruent;
c) An exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

Classroom Activities

Monday (2nd Period) / Tuesday (7th & 8th Periods)

Bellringer – Translations

Lecture on – Isosceles Triangles, Equilateral Triangles, Compositions of Transformations, Dilations

Classwork:

Practice on IXL (not graded assignments):

  1. K.11 Hypotenuse-Leg Theorem
  2. K.9 Congruence in isosceles and equilateral triangles
  3. L.10 Composition of congruence transformations: graph the image
  4. L.11 Transformations that carry a polygon onto itself
  5. L.13 Dilations: graph the image
  6. L.14 Dilations: find the coordinates
  7. L.15 Dilations: scale factors and classifications

Homework (graded assignment – formative):

MathSpace – Find sides and angles with congruence

Wednesday (Early Release Day – Short Class Period)

Review for Unit 2(A) test by reviewing theorems and practicing solving for congruence.

Thursday (2nd Period) / Friday (7th & 8th Periods)

Unit 2(A) Test (summative)

If complete test before class period ends, complete missing and late assignments as the grading period ends on Monday, October 16th.

 

Assignments Due

Summative:

Unit 2(A) Test – Thursday, October 12th (2nd Period) / Friday, October 13th (7th & 8th Periods)

Formatives:

MathSpace

  1.  Congruence in triangles – due Monday, October 9th (2nd Period) / due Tuesday, October 10th (7th & 8th Periods)
  2. Find sides and angles with congruence – due Wednesday, October 11th
Additional Resources

Please refer to attached Lecture Notes and Unit 2(A) Study Guide.