Ariella Lindenfeld & Nikki Colona Math Project Ariella Lindenfeld & Nikki Colona

5.1 - Parallelograms EFGH EF = HG;FG=EH 1. H G 3 1 2. EFGH <E = <G 4 2 3. EFGH <3 = <4 E F A parallelogram is a quadrilateral with both pairs of opposite parallel sides. Theorem 5-1: Opposite sides of the parallelogram are congruent. Theorem 5-2: Opposite angles of a parallelogram are congruent. Theorem 5-3: Diagonals of a parallelogram bisect each other.

Example: 38 X y 7 120 X equals 120 degrees because of OAC which says that Opposite Angles are congruent Y equals 22 because it supplementary to 120 So 60 minus 38 is 22

5.2 – Proving Quadrilaterals are Parallelograms 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 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 bisects each other, then the quadrilateral is a parallelogram. T < S 3 2 << M 1 << 4 < Q R

5.3 – Theorems Involving Parallel Lines Theorem 5- 8: If two lines are parallel, then all points on the line are equidistant from the other line. A B > L > M C D AB=CD

5.3 continued… Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A X B 1 2 3 Y C 4 5 Z AX II BY II CZ and AB = BC XY = YZ

Section 3 – theorems Involving Parallel Lines 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 Theorem 5-11 The segment that joins the midpoints of two sides of a triangle Is parallel to the third side Is half as long as the third side

Example: 2(4x+3) = 13x+1 8x+6=13x+1 5=5x X=1 5y+7 = 6y +3 4=y X=1, Y=4

5-4 Special parallelograms Rectangle  is a quadrilateral with four right angles. Therefore, every rectangle is a parallelogram. Rhombus  A quadrilateral with four congruent sides. Therefore, every rhombus is a parallelogram.

5-4 Special parallelograms Square  a quadrilateral with four right angles and four congruent sides. Therefore, every square is a rectangle, a rhombus, and a parallelogram *** since rectangles, rhombuses, and squares are parallelograms, they have all the properties of parallelograms.

5-4 Special parallelograms 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 are perpendicular

5-4 Special parallelograms Theorem 5-15 The midpoint of the hypotenuse of a right triangle 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

5-5 Trapezoids Trapezoid a quadrilateral with exactly one pair of parallel sides Bases  the parallel sides Legs  the other sides Isosceles Trapezoid  a trapezoid with congruent legs Base angles are congruent

5-5 Trapezoids Theorem 5-18 Theorem 5-19 Base angles of an isosceles trapezoid are congruent *** Median (of a trapezoid )the segment that joins the midpoints of the legs Theorem 5-19 The Median of a Trapezoid Is parallel to the bases Has a length equal to the average of the base lengths

Practice Questions: 1. 4x-y 7x-4y 5 9 2. z X y 75

Practice Question 42 3. 26 3x-2y 4x+y

Proof: Statement Reason Rectangle QRST RKST JQST 1.Given KS =RT JT = QS 2. OSC 3. RT =QS 3. DB 4. JT = KS 4. substitution T S J Q R K