* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download ch 5 - ariella and nikki - 2012
Survey
Document related concepts
Group action wikipedia , lookup
Riemannian connection on a surface wikipedia , lookup
Perspective (graphical) wikipedia , lookup
Line (geometry) wikipedia , lookup
Steinitz's theorem wikipedia , lookup
Euler angles wikipedia , lookup
Integer triangle wikipedia , lookup
Rational trigonometry wikipedia , lookup
Trigonometric functions wikipedia , lookup
History of geometry wikipedia , lookup
Atiyah–Singer index theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Noether's theorem wikipedia , lookup
Euclidean geometry wikipedia , lookup
Riemann–Roch theorem wikipedia , lookup
Four color theorem wikipedia , lookup
Transcript
MATH PROJECT Ariella Lindenfeld & Nikki Colona H G 3 1 4 2 E EFGH EF = HG;FG=EH EFGH <E = <G EFGH <3 = <4 A parallelogram is a quadrilateral with both pairs of opposite parallel sides. Theorem 2. 3. F 1. 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: y 38 X 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 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 1 Q S 3 2 M 4 < R Theorem L M A C B D AB=CD > > 5- 8: If two lines are parallel, then all points on the line are equidistant from the other line. Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A B 1 C 4 X 2 3 Y 5 AX II BY II CZ and AB = BC XY = YZ Z SECTION 3 – THEOREMS INVOLVING PARALLEL LINES 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 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 5y+7 = 6y +3 4=y 13x+1 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 The diagonals of a rectangle are congruent Theorem 5-12 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 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 9 2. X 75 z y PRACTICE QUESTION 42 3. 26 3x-2y 4x+y PROOF: Statement Reason 1. Rectangle QRST RKST JQST 1.Given KS =RT JT = QS 2. OSC 3. RT =QS 3. DB 4. JT = KS 4. substitution T J Q S R K