#### Math - 2018-19

# G.9 - Quadrilaterals

G.9The student willverifyanduseproperties of quadrilaterals tosolveproblems, including practical problems.

*Adopted: 2016*

### BIG IDEAS

- I can lay
out a football or baseball field, create floor plans, and plan building construction
with engineering and architecture.

- I will
develop analytical skills through comparing, contrasting, and sorting while improving my ability to look at things through multiple
perspectives making me a better problem solver.

### UNDERSTANDING THE STANDARD

· 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.

· Quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals.

· Properties of quadrilaterals can be used to identify the quadrilateral and to determine the measures of sides and angles.

· Given coordinate representations of quadrilaterals, the distance, slope, and midpoint formulas may be used to verify that quadrilaterals have specific properties.

· The angle relationships formed when parallel lines are intersected by a transversal can be used to prove properties of quadrilaterals.

· Congruent triangles can be used to prove properties of quadrilaterals.

· A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Properties of a parallelogram include the following:

opposite sides are congruent;

opposite angles are congruent;

consecutive angles are supplementary; and

diagonals bisect each other.

· A rectangle is a quadrilateral with four right angles. Properties of rectangle include the following:

opposite sides are parallel and congruent; and

diagonals are congruent and bisect each other.

· A rhombus is a quadrilateral with four congruent sides. Properties of a rhombus include the following:

all sides are congruent;

opposite sides are parallel;

opposite angles are congruent;

diagonals are perpendicular bisectors of each other;

diagonals bisect opposite angles; and

diagonals divide the rhombus into four congruent right triangles.

· A square is a quadrilateral that is a regular polygon with four congruent sides and four right angles. Properties of a square include the following:

opposite sides are parallel;

diagonals are congruent;

diagonals are perpendicular bisectors of each other; and

diagonals
divide the square into four congruent

45°-45°-90° triangles.

· A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called bases. The nonparallel sides of a trapezoid are called legs.

· An isosceles trapezoid has the following properties:

nonparallel sides are congruent;

diagonals are congruent; and

base angles are congruent.

· The construction of the perpendicular bisector of a line segment can be justified using the properties of quadrilaterals.

· The construction of the perpendicular to a given line from a point on, or not on, the line can be justified using the properties of quadrilaterals.

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

· The construction of a bisector of a given angle can be justified using the properties of quadrilaterals.

· The construction of a square inscribed in a circle can be justified using the properties of squares.### ESSENTIALS

**The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to**

·
G.9_{1} **Solve**
problems, including practical problems, using the properties specific to
parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and
trapezoids.

·
G.9_{2} **Prove**
that quadrilaterals have specific properties, using coordinate and algebraic
methods, such as the distance formula, slope, and midpoint formula.

·
G.9_{3} **Prove**
the properties of quadrilaterals, using direct proofs.

### KEY VOCABULARY

verify,
characteristic, quadrilateral, properties of quadrilaterals, solve,
parallelogram, rectangle, rhombus, square, isosceles trapezoid, trapezoid,
coordinate method, algebraic method, distance formula, slope formula, midpoint
formula, deductive reasoning, sides, angles, inscribed, circle, diagonal,
mid-segment, base angles, opposite angles, right angle, base, segment,
characteristics, identify, measure

*Updated: Aug 23, 2018*