2 5.1 Properties of Parallelograms A parallelogram ( ) is a quadrilateral with both pairs of opposite sides parallel.Theorem 5-1( OSC): Opposite sides of a parallelogram are congruent.Theorem 5-2( OAC): Opposite angles of a parallelogram are congruentTheorem 5-3( DB): Diagonals of a parallelogram bisect each other.
3 5-1 ExamplerWe know that WE CR, so angle COW is congruent to angle EOW. Since the diagonals of parallelograms bisect each other CO is congruent to OE. So Triangle COW is congruent to Triangle EOW.Corresponding sides of triangles are congruent so CW is congruent to WEAccording to 5-1 WE is congruent to CR, and CW is congruent to REBy transitive property you can prove that: RE RC, CW CEocew
4 5.2 Ways to Prove that Quadrilaterals are Parallelograms Theorem 5-4 (OSC ): If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 5-5(One Pair CP ): If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram.Theorem 5-6(OAC ): If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.Theorem 5-7(DB ): If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.Note: A parallelogram ( ) is a quadrilateral with both pairs of opposite sides parallel.
5 s a o k c 5-2 Example Its given that SA KC, SK AC SA and KC, SK and AC are two pairs of opposite sides.According to the definition of parallelogram, a parallelogram is a quadrilateral with both pairs of opposite sides parallel.So, the definition of a parallelogram allows us to reduce that quadrilateral SACK is a parallelogram.okc
6 5.3 Theorems Involving Parallel Lines Theorem 5-8(//PE): If two lines are parallel, then all points on one line are equidistant from the other line.Theorem 5-9(3//CST): If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal.
7 5.3 cont.Theorem 5-10(C. Mid-segment): 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 sideTheorem 5-11(Mid-segment): The segment that joins the midpoints of two sides of a triangle1: is parallel to the third side2: is half as long as the third side.
8 5.3 practice Given: L//M; A and B are any point on L; Prove AC=BD A B
9 AnswerSince AB and CD are contained in parallel lines, AB//CD. Since AC and BD are coplanar and are both perpendicular to M, they are parallel. Thus ABDC is a parallelogram, by definition of a parallelogram. Since opposite sides AC and BD are congruent, AC=BD.
10 5-4 Special Parallelograms A rectangle is a quadrilateral with four right angles therefore, every rectangle is a parallelogram.A rhombus is a quadrilateral with four sides therefore, every rhombus is a parallelogram.A square is a quadrilateral with four right angles and four congruent sides. It’s a rhombus, square, and parallelogram together.
11 5-4 Theorems 5-12 (DRC) 5-13 (DRH ) 5-14 The diagonals of a rectangle are congruent5-13 (DRH )The diagonals of a rhombus are perpendicular5-14Each diagonal of a rhombus bisects two angles of the rhombus
12 5-4 cont.5-15The midpoint of the hypotenuse of a right triangle is equidistant form the three vertices.5-16If an angle of a parallelogram is a right angle, then the parallelogram is a rectangle5-17If two consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus
13 Practice for 5-4 Name each figure shown A-parallelogram B- rectangle C- rhombus D- square E-None of the above184.108.40.206.5.6.
14 More practice 5.4 MP is 6.5 then AM is ? <KAP is a right angle, and AM is a median.Not drawn to scale.PKM
15 5-5 TrapezoidsTrapezoid: a quadrilateral with exactly one pair of parallel sideIsosceles trapezoid: a trapezoid with congruent sides.Theorem 15-8Base angles of an isosceles trapezoid are congruent.15-9The median of a trapezoid is parallel to the bases and has a length equal to the average of the base lengths
16 YEEAAAHHH! Looking good Ms. Sulkes Thanks for learning with us today. From AniThanks for learning with us today.Goodbye.