#### Math - 2017-18

# 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.6The student, given information in the form of a figure or statement, willprovetwo triangles are congruent.

**Bloom's Level: ** Evaluate

*Adopted: 2009*

### 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.6_{1}
**Use** definitions, postulates, and
theorems to **prove** triangles
congruent.

· G.6_{3} **Use**
direct proofs to **prove** two triangles
congruent.

G.6_{3}
**Use** algebraic methods to **prove** 2 triangles are congruent.

· G.6_{1} **Prove**
two triangles congruent given relationships among angles and sides of triangles
expressed numerically or algebraically.

G.6_{2}
**Use** coordinate methods, such
as distance formula and the slope formula to **prove** 2 triangles are congruent.

· G.6_{2} **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*