Chapter 5 Pre-AP Geometry Quadrilaterals Chapter 5 Pre-AP Geometry

Objectives Apply the definition of a parallelogram and the theorems about properties of a parallelogram. Prove that certain quadrilaterals are parallelograms. Apply theorems about parallel lines. Apply the midpoint theorems for triangles. Apply the definitions and identify the special properties of a rectangle, a rhombus, and a square. Determine when a parallelogram is a rectangle, rhombus, or square. Apply the definition and identify the properties of a trapezoid and an isosceles trapezoid.

Lesson 5.1 Pre-AP Geometry Parallelograms Lesson 5.1 Pre-AP Geometry

Objective Apply the definition of a parallelogram and the theorems about properties of a parallelogram.

Definition Parallelogram A quadrilateral with two sets of parallel sides.

Properties of Parallelograms The diagonals of a parallelogram bisect each other. Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. Each diagonal bisects the parallelogram into two congruent triangles. Parallelogram

Theorems 5-1 Opposite sides of a parallelogram are congruent. 5-2 Opposite angles of a parallelogram are congruent. 5-3 Diagonals of a parallelogram bisect each other.

Practice Name all pairs of parallel lines. Name all pairs of congruent angles. Name all pairs of congruent segments. What is the sum of the measures of the interior angles of a parallelogram? What is the sum of the measures of the exterior angles of a parallelogram?

Review – True or False Every parallelogram is a quadrilateral. Every quadrilateral is a parallelogram. All angles of a parallelogram are congruent. All sides of a parallelogram are congruent.

Problem Set 5.1, p. 169: # 2 – 32 (even) skip # 16 Written Exercises Problem Set 5.1, p. 169: # 2 – 32 (even) skip # 16

Proving Quadrilaterals are Parallelograms Lesson 5.2 Pre-AP Geometry

Objective Prove that certain quadrilaterals are parallelograms.

Quadrilaterals and Parallelograms A quadrilateral is a polygon with 4 sides. A parallelogram is a quadrilateral whose opposite sides are parallel (the top and bottom are parallel and the left and right are parallel).

Theorem 5-4 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 5-5 If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.

Theorem 5-6 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem 5-7 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Five ways to Prove that a Quadrilateral is a Parallelogram Show that both pairs of opposite sides are parallel. Show that both pairs of opposite sides are congruent. Show that one pair of opposite sides are both congruent and parallel. Show that both pairs of opposite angles are congruent. Show that the diagonals bisect each other.

Review Answer with always, sometimes, or never. The diagonals of a quadrilateral bisect each other. If the measures of two angles of a quadrilateral are equal, then he quadrilateral is a parallelogram. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is a parallelogram. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. To prove a quadrilateral is a parallelogram, it is enough to show that one pair of opposite sides is parallel.

Practice BE = EX; CE = EA BAD  DCB; ADC  CBA BC || AD; AB || DC State the definition of theorem that enables you to deduce, from the information provided, that quadrilateral ABCD is a parallelogram. BE = EX; CE = EA BAD  DCB; ADC  CBA BC || AD; AB || DC BC  AD; AB  DC A B C D E

Written Exercises Problem Set 5.2 p. 174 # 2, 4, 8, 10, 14, 20, 22

Theorems Involving Parallel Lines Lesson 5.3 Pre-AP Geometry

Objective Apply theorems about parallel lines and the segment that joins the midpoints of two sides of a triangle.

Theorem 5-8 If two lines are parallel, then all points on one line are equidistant from the other line. l m

Theorem 5-9 If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. l m n

Theorem 5-10 A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. D B N M A C

Theorem 5-11 (1) is parallel to the third side; The segment that joins the midpoints of two sides of a triangle: (1) is parallel to the third side; (2) is half as long as the third side. B N M A C BM = AM, CN = AN BC = 2(MN) MN = ½(BC)

Problem Set 5.3 p. 180 # 2-20 even, p. 182 # 1-6 Written Exercises Problem Set 5.3 p. 180 # 2-20 even, p. 182 # 1-6

Special Parallelograms Lesson 5.4 Pre-AP Geometry

Objectives Apply the definitions and identify the special properties of a rectangle, a rhombus, and a square. Determine when a parallelogram is a rectangle, rhombus, or square.

Every rectangle is a parallelogram. A parallelogram with four right angles. Both pairs of opposite angles are congruent. Every rectangle is a parallelogram.

Every rhombus is a parallelogram. A quadrilateral with four congruent sides. Both pairs of opposite sides are congruent. Every rhombus is a parallelogram.

A square is also a rectangle, a rhombus, and a parallelogram. A quadrilateral with four right angles and four congruent sides. Both pairs of opposite angles and opposite sides are congruent. A square is also a rectangle, a rhombus, and a parallelogram.

Theorem 5-12 The diagonals of a rectangle are congruent.

Theorem 5-13 The diagonals of a rhombus are perpendicular.

Theorem 5-14 Each diagonal of a rhombus bisects two angles of the rhombus.

Theorem 5-15 The midpoint of the hypotenuse of a right angle is equidistant from the three vertices.

Theorem 5-16 If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

Theorem 5-17 If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus.

Practice Reply with always, sometimes, or never. A square it a rhombus. The diagonals of a parallelogram bisect the angles of the parallelogram. A quadrilateral with one pair of sides congruent is a parallelogram. The diagonals of a rhombus are congruent. A rectangle has consecutive sides congruent. A rectangle has perpendicular diagonals. The diagonals of a rhombus bisect each other. The diagonals of a parallelogram are perpendicular bisectors of each other.

Written Exercises Problem Set 5.4 p. 187 # 1-10, 12-30 evens

Lesson 5.5 Pre-AP Geometry Trapezoids Lesson 5.5 Pre-AP Geometry

Objective Apply the definitions and identify the properties of a trapezoid and an isosceles trapezoid.

Definition Trapezoid A quadrilateral with exactly one pair of parallel sides. leg base

Definition Isosceles Trapezoid In an isosceles trapezoid, the base angles are equal, and so are the other pair of opposite sides AD and BC. A B C D

Theorem 5-18 Base angles of an isosceles trapezoid are congruent.

Median of a Trapezoid The segment that joins the midpoints of the legs of a trapezoid. Median of a Trapezoid

Theorem 5-19 The median of a trapezoid: (1) is parallel to the base; (2) has a length equal to the average of the base lengths.

Practice In trapezoid ABCD, EF is a median. If AB = 25 and DC = 13, then EF = _____. If AE = 11 and FB = 8, then AD = _____ and BC = _____. If AB = 29 and EF = 24, then DC = _____. If AB = 7y + 6 and EF = 5y – 3 and DC = y – 5, then y = _____.

Written Exercises Problem Set 5.5 p. 192 # 2-26 evens