G.6 and *G.6 - Congruent Triangles

G.6    The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs.

G.6  The student, given information in the form of a figure or statement, will prove two triangles are congruent.

Bloom's Level:  Evaluate

BIG IDEAS

• I can construct a bridge, design structural beams for strength, and design sails for a boat that maximize wind for speed.
• I will know that congruence describes a special similarity relationship between objects and is a form of equivalence, and will apply to new situations some techniques for finding solutions.

UNDERSTANDING THE STANDARD

• Congruence has real-world applications in a variety of areas, including art, architecture, and the sciences.
• Congruence does not depend on the position of the triangle.
• Concepts of logic can demonstrate congruence or similarity.
• Congruent figures are also similar, but similar figures are not necessarily congruent.

·  Deductive or inductive reasoning is used in mathematical proofs. In this course, deductive reasoning and logic are used in direct proofs. Direct proofs are presented in different formats (typically two-column or paragraph) and employ definitions, postulates, theorems, and algebraic justifications including coordinate methods.

·  Congruence has practical applications in a variety of areas, including art, architecture, and the sciences.

·  Congruence does not depend on the position of the triangles.

·  Congruent triangles are a result of rigid isomorphic transformations.

·  Concepts of logic can demonstrate congruence or similarity.

·  Congruent figures are also similar, but similar figures are not necessarily congruent.

·  Corresponding parts of congruent triangles are congruent.

·  Two triangles can be proven congruent using the following criterion:

­  Side-Angle-Side (SAS);

­  Side-Side-Side (SSS);

­  Angle-Angle-Side (AAS); and

­  Angle-Side-Angle (ASA).

·  Two right triangles can be proven congruent using the criteria
Hypotenuse-Leg (HL).

·  Triangle congruency can be explored using geometric constructions such as an angle congruent to a given angle or a line segment congruent to a given line segment.

·  The construction for the bisector of a given angle can be justified using congruent triangles.

·  The construction for an angle congruent to a given angle can be justified using congruent triangles.

·  The construction of the perpendicular to a given line from a point on the line can be justified using congruent triangles.

·  The construction of the perpendicular to a given line from a point not on the line can be justified using congruent triangles.

ESSENTIALS

G.61  Use definitions, postulates, and theorems to prove triangles congruent.

·  G.63  Use direct proofs to prove two triangles congruent.

G.63  Use algebraic methods to prove 2 triangles are congruent.

·  G.61  Prove two triangles congruent given relationships among angles and sides of triangles expressed numerically or algebraically.

G.62  Use coordinate methods, such as distance formula and the slope formula to prove 2 triangles are congruent.

·  G.62  Prove two triangles congruent given representations in the coordinate plane and using coordinate methods (distance formula and slope formula).

KEY VOCABULARY

figure, statement, triangle, congruent, algebraic method, coordinate method, deductive proof, postulate, theorem, distance formula, slope formula, Side-Side-Side Triangle Congruence, Side-Angle-Side  Triangle Congruence, Angle-Angle-Side Triangle Congruence, Angle-Side-Angle Triangle Congruence, position, similarity/similar

Updated: Oct 27, 2017