Unit 2 similarity, congruence, and proofs
Unit 2: Building on standards from Unit 1 and from middle school, students will use transformations and proportional reasoning to develop a formal understanding of similarity and congruence. Students will identify criteria for similarity and congruence of triangles, develop facility with geometric proofs (variety of formats), and use the concepts of similarity and congruence to prove theorems involving lines, angles, triangles, and other polygons.
STANDARDS
Prove geometric theorems
Understand similarity in terms of similarity transformations
MGSE9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.
MGSE9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
MGSE9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
MGSE9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, (and its converse); the Pythagorean Theorem using triangle similarity.
MGSE9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
MGSE9-12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor.
MGSE9-12.G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
MGSE9-12.G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove theorems involving similarity
MGSE9-12.G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, (and its converse); the Pythagorean Theorem using triangle similarity.
MGSE9-12.G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Understanding Dilations
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Transformation Dilation a scale factor of 2
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Dilations not about the origin
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Mathbits practice with dilations
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How to find the hypotenuse or legs of a right triangle using Pythagorean's Theorem
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Try these 20 Pythagorean Theorem problems
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Try CK-12 dilation practice
Dilations & similarity
AA condition for similar triangles
How to write congruent parts of a triangle
Triangle proportionally theorem
Triangle proportionally practice with CK12
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What are similar triangles?
When are two triangles similar?
CK12 AA similiarity practice (signin with google if prompted)
CK12 Corresponding Part (CPCTC)
Properties of parallel lines and similar trianlges
Converse of the triangle proportionally theorem
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Try 4 problems from kahn.
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Triangle Side Splitter Theorem
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Triangle Sum Theorem
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Triangle Side Splitter Pracitice
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Triangle Sum Practice
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What are the 3 triangle similarity theorems?
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IXL triangle similarity
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Properties of quadrilaterals comparison chart
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Properties of quadrilaterals quiz
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Understand congruence in terms of rigid motions
MGSE9-12.G.CO.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.
MGSE9-12.G.CO.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.
MGSE9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS.)
MGSE9-12.G.CO.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.
MGSE9-12.G.CO.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.
MGSE9-12.G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. (Extend to include HL and AAS.)
What are the 5 triangle congruence theorems?
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IXL triangle congruence
Which triangle congruence theorem can be used?
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Try this triangle side splitter theorem & get feedback!
Try these triangle sum problems with Kahn!
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Parallel lines cut by a transversal, angle relationships
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MGSE9-12.G.CO.9 Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
The vertical angle relationship
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Perpendicular bisector theorem
Vertical, complementary, supplementary angles
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Interior, exterior, parallel lines & transversal
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MGSE9-12.G.CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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.
Parallel lines, transversal & angle relationships
Parallel lines & transversal project
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Isosceles triangle theorem
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MGSE9-12.G.CO.11 Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Properties of square, rectangles, & parallelograms
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See how many quadrilateral properties you know!
Can you recognize quadrilateral?
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Try 10 angle bisector problems on Mathbits
Use for a two column proofs
Try some proofs & check your work!
Prove HK is = to GF
Proof practice
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12 slides to explain proofs
Proof: parallelogram 3
How to use CPCTC in a geometry proof
Khan academy
IXL proof
2 Column proof by
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Make geometric constructions
MGSE9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.
MGSE9-12.G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
MGSE9-12.G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon, each inscribed in a circle.
Please read…you can stop when you get to the examples (if you want)…
Let’s look at how constructions are created!
Geometric constructions
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Geometry tools joke reading
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