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# Subject Mathematics Geometry Total 68 hours

Theme of the lecture Phales Theorem 2 hours/ week

Lecturer Grachyova Svetlana Semenovna. Profile Economics

Lesson 13,14

Statement of theorem

If parallel lines cross one side of angle and cut equal segments, then they cut equal segments on the other side too. Proof

Lets A1, A2, A3 – points of crossing parallel lines with one side of angle. Lets denote B1, B2, B3 corresponding points of crossing this lines with another side of angle. Prove that if A1A2=A2A3 than respectively B1B2= B2B3. Lets draw through point B2 straight line of FE parallel to A1A2. According to the property of the parallelogram A1A2=FB2, A2A3=B2E. So if A1A2=A2A3 than FB2=B2E. Triangles B1B2F and B2B3E are equal according to the second test of equality. Really B2F=B2E, angles FB2B1=B3B2E as vertical angles, angles B2FB1 and B2EB3 are equal as interior alternate angles at parallel lines A1B1 and A3B3 and crossing line FE. From the equality of triangles B2B1F and B2B3F we can say that B1B2=B2B3. The theorem proved.

Divide segment into definite number of parts.

Glossary

 English Russian Kazakh Quadrilateral Четырехугольник Төртбұрыш Vertex Вершина Төбесі Convex Выпуклый Дөңес Angle Угол Бұрыш Square Квадрат Квадрат Rhomb Ромб Ромб

Individual work of student: Collection for preparing to examination Group 15 (15B21 –15B22).

Office–hours Geometry 8grade Шыныбеков А.Н., 2012, ex 35 ( p 33-37)

Literature: Geometry 8grade Шыныбеков А.Н. 2012, [33-37], Collection for preparing to examination Group15

Notes

Kazakh-American University

Subject Mathematics Geometry Total 68 hours

Theme of the lecture Trapeze 2 hours/ week

Lecturer Grachyova Svetlana Semenovna. Profile Economics

Lesson 15-17

Definition

Trapeze – quadrilateral in which two opposite sides are parallel and called bases., not parallel sides are called lateral sides AB, CD- lateral sides. BC,AD- bases.

Definition

Rectangular trapeze – trapeze in which one angle is right angle.  BAD= 900

Definition

Isosceles trapeze – is trapeze in which lateral sides are equal. AB=CD

Definition

Middle line in trapeze – is a line which joins middle of lateral sides. EF- middle line.

Theorem

Middle line in trapeze is parallel to the base and equal to the half of sum of the bases. Prove

ABCD – trapeze. EF – midline. Lets draw through the vertex B and point F - midpoint of lateral side CD straight line CD.

Lets this straight line crosses straight line AD in the point P. So we get two equal triangles. Triangle BFC and triangle PFD are equal according to the second test of equality of triangles, because CF = FD, CFB = DFP, as vertical, BCF = PDF as alternate interior angle with parallel lines BC, DP and secant line CD.

From the equality of this triangles we write the equality of sides BF = FP. So EF is midline of triangle ABP. According to the property of midline of triangle, EF is parallel to the AD so it is parallel and BC too.

EF = AP because BC = DP and AP = AD + DP, then EF = ( AD + BC).

The theorem proved.

Middle line in trapeze is 8 cm, one base is 6 cm. Find another base.

Glossary

 English Russian Kazakh Trapeze Трапеция Трапеция Middle line Средняя линия Орташа сызық Angle Угол Бұрыш Square Квадрат Квадрат Rhomb Ромб Ромб

Lesson 1 ex 150 ,153 Lesson 2 ex 154, 155 Lesson 3 ex 163, 164

Individual work of student: Collec. for preparing to examination Group 15 (15B25 –15B26).

Office–hours Geometry 8grade Шыныбеков А.Н. 2011, ex 102,101 ( p 37- 42)

Literature: Geometry 8grade Шыныбеков А.Н. 2011, [37- 42], Collection for preparing to examination Group15

Notes

Kazakh-American University

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