# 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