#### Math - 2018-19

G.9   The student will verify and use properties of quadrilaterals to solve problems, including practical problems.

### 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

2016 VDOE Curriculum Framework - G.9 Understanding

·  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.91  Solve problems, including practical problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.

·  G.92  Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula.

·  G.93  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