The presentation "Signs of parallelism of two lines" is an excellent educational resource that will be useful for both teachers, tutors, and students and their parents.
In many European countries, projectors and screens have long replaced dusty boards. Fortunately, due to the active development of technologies in various fields, including education, they have become available in our schools. Now lessons can be accompanied by video tutorials and presentations. This will save the teacher's time, and he will have more opportunity to conduct a full-fledged lesson according to the curriculum.
slides 1-2 (Presentation topic "Signs of parallelism of two lines", definition of a secant)
This presentation is devoted to one of the interesting and necessary topics from the cycle of school geometry. The first slide defines the secant, which students will meet more than once in various textbooks and problem books. The drawing shows an example of a secant c in relation to straight lines a and b. Also indicated in a convenient way are all the angles that are formed as a result of the intersection of lines. Their names will be in front of the students' eyes during the lesson if this presentation is shown while the teacher comments on the drawing. By reviewing this page on their own, students can understand the conditions of the problems associated, for example, with crossed corners.
slides 3-4 (theorems)
On the next slide, you can see the theorem on the equality of cross-lying angles. At first, many students have difficulty understanding this theorem. However, if it is clearly explained and shown with examples of drawings, then over time, it will not seem so complicated.
In addition to the formulation of the theorem, there is also a geometric proof. If the student is explained the proof, he will remember, understand and remember the essence of the theorem much more successfully.
The next slide is also devoted to the theorem. This theorem states that lines are parallel if their intersection with a secant makes the corresponding angles equal. The proof is shown below in the form of a drawing. Verbal evidence can be omitted, because the picture speaks for itself. During the lesson, the teacher will no doubt comment and explain the proof. After that, students can once again review, repeat and try to apply in practice in tasks.
slide 5 (theorem)
And finally, the last slide. It demonstrates the theorem on the parallelism of two lines, for which, at the intersection of the secant, the sum of one-sided angles is equal to 180 degrees. On the given drawing it is possible to formulate the proof of the theorem.
Thanks to such resources, students will not fall behind their classmates, as before, in case of omissions or inability to perceive in the environment that prevails in the classroom. After all, you can view in a relaxed home environment on your own or with your parents. Thus, students will feel more confident in the lesson and there will be an interest in science.
Purpose: To update students' knowledge of parallel lines obtained in grade 5; Enter the definition of parallel lines, parallel segments; To acquaint with cross lying, one-sided, corresponding angles, with signs of parallelism of two straight lines.
Definitions of a b c A line c is called a secant with respect to lines a and b if it intersects them at two points.
1. Mark the middle of the segment AB. AO=OB Theorem If at the intersection of two lines of a transversal, the lying angles are equal, then the lines are parallel a b А В 1 2 Given: AB intersects lines a and b. Prove: Proof Let's make the constructions: О Н 3. On the straight line b from point В let's set aside and draw the segment Н 1 ?
Theorem If at the intersection of two lines crosswise the lying angles are equal, then the lines are parallel and b А В 1 2 Given: AB intersects lines a and b. Prove: Proof of OH H Point H lies on the continuation of the ray OH, I.e. points H, O and H lie on the same straight line 1 1
Geometry grade 7, p.24-25,
pp.54-57
- Review the concept of parallel lines
- Introduce the concept of cross lying, one-sided and corresponding angles
- Consider signs of parallelism of two lines
- Learn to solve problems on the use of signs of parallelism of two lines
Everyone knows these lines.
storing direction,
Together they run away from me into infinity.
We often see them
it is impossible to name everything:
A pair of rails by the tram
there are five in the staff...
Even if there are many lines
Do not mix one with the other:
They are very strict
Distance between each other.
Parallel lines - glorious,
polite people:
2. Complete the statements by selecting the desired item:
Intersecting lines have...
A) in the drawing one common point;
B) one common point.
4. Indicate the wrong ending of the definition:
Two lines in a plane are called parallel...
A) if they are permanent
distance from each other;
B) if they do not intersect on a plane;
C) if they are both perpendicular to the third line;
D) if they do not intersect in the drawing.
6. Indicate the correct ending of the definition:
Two line segments are said to be parallel if they are...
A) both are perpendicular to the third line;
B) lie on parallel lines;
C) have the same distance between the ends;
D) do not intersect in a plane.
a, b- straight, With towards them secant
Crossed angles - 3 and 5, 4 and 6;
Unilateral corners - 4 and 5, 3 and 6;
Corresponding angles - 1 and 5, 4 and 8,
2 and 6, 3 and 7.
Choose the correct statements
- 5 and 8 - vertical
- 6 and 2 - one-sided
- 7 and 3 - corresponding
- 3 and 4 - adjacent
- 1 and 8 - lying crosswise
- 4 and 6 - one-sided
- 5 and 4 - lying crosswise
- 6 and 2 - corresponding
- 2 and 7 - vertical
Signs of parallel lines
- If at the intersection of two lines by a transversal, the lying angles are equal, then the lines are parallel.
2. If at the intersection of two lines of a secant, the corresponding angles are equal, then the lines are parallel.
3. If, at the intersection of two straight lines, the secant sum of one-sided angles is 180 degrees.
Sit right!
Take care of your eyesight!
Given: = 32˚
Prove:
Tasks for fixing the signs of parallelism of straight lines on the finished drawings:
º,
Prove:
Tasks for fixing the signs of parallelism of straight lines on the finished drawings:
º,
Prove:
n. 24, 25 of the theorem to teach, questions 1 - 5 on page 68, Workbook No. 91,96
№ 186, №187,
№ 188,№189,№190
Are the lines parallel? a And b , If 1 = 3?
Task 1
Yes , because 1 and 3 - NL with straight lines a and b and secant d.
Task 2:
Are the lines parallel?
a And b , If 1= 4?
Yes , because 1 and 4 - corresponding with straight lines a and b and secant d .
Task 3 :
Are the lines parallel?
a And b ,
If 1 + 2 = 180?
Yes , because 1 and 2 - one-sided with straight lines a and b and secant with.
Task 4 :
Are the lines parallel?
a And b ,
If 5 = 6 = 90°?
Yes , since two straight lines, perpendicular to the third, are parallel.
Task 6
k
Are the lines parallel? d And e ?
d
ANSWER:
d e, because 3 = 2 = 141°, as vertical, 3 + 1= 39° + 141° =180°, they are one-sided.
e
Task 7
Given: EO=LO; FO = KO.
Prove: EF KL.
ANSWER:
EOF And LOK 1) OE= OL 2)OK=OF EOF= LOK 3) 1 = 2
E= L- NL with direct EF And KL and secant EL EF KL
Task 8
Given: 1 = 2; 2+3==180°
Prove: a c
Solution:
Because 1 = 2, corresponding, A b.
2 = 4 as vertical ,
2 + 3 = 180° 4 + 3 = 180° , and they are one-sided, c b. A b, c b A c.
Sides VA and EO combined. Ray BO is the bisector of the angle AVM. MN > CD. Scale millimeter ruler, caliper, tailor centimeter. 1dm. Comparing shapes using overlay. Vertices B and E are aligned. Comparison of segments and angles. Other units of measure. Meter standard. The sides of the VM and the EU have come together. What is the largest number of parts a plane can be divided into by 4 distinct lines? F3 = f4. Angle comparison.
"Geometric tasks for construction" - Working with the parameter line. Building a rectangle manually. Through the vertex of the angle A and the point of intersection of the circles E draw a straight line. Mark points A and B on the drawing. Construct a circle centered at point O. KOMPAS-ZD computer-aided design window. We construct two circles of radius BC with centers at points B and C. Coordinate fields. Let's construct a circle of arbitrary radius with the center at the vertex.
"Proof of 3 signs of equality of triangles" - Case. Twisted triangle. Triangle view. From the collection of impossible objects. The student showed a triangle. Student. Clue. Sign of equality of triangles on three sides. Properties of angles in an isosceles triangle. Compass. The third sign of the equality of triangles. Sign. Examination. Prove. Think. Wonderful triangles. Strange rooms. Triangles are equal on three sides. Triangles are equal on two sides.
"Isosceles Triangle" - Guess the rebus. Equality of triangles. Theorem. A triangle is called isosceles if its two sides are equal. Define the view. In an isosceles triangle AMK AM = AK. Lateral sides. AFD is isosceles. Triangle. Classification of triangles by sides. ABC is isosceles. Isosceles triangle. List the congruent elements of the triangles. Problem solving. Find the angle KBA. Classification of triangles according to the size of the angles.
"Cartesian coordinate system on the plane" - The plane on which the Cartesian coordinate system is specified. Cartesian coordinate system on the plane. Introduction. Coordinates in people's lives. Ancient Greek astronomer Claudius. Hipparchus. Algebra project. The value of the Cartesian coordinate system. Rectangular coordinate system. Scientists who are the authors of the coordinates. System geographical coordinates. Introduction of simpler notation to algebra. Place in the cinema.
"Angle" grade 7 - Straight lines that intersect at an angle of 90. Perpendicular lines. Tasks for oral counting. Half corner. The sum of adjacent angles is 180. The vertical angles are equal. A right angle is an angle that is 90. Geometric figure. A bisector is a ray that emerges from the vertex of an angle. Lines that do not intersect. An angle that is 180. An angle that is 90. An angle is a geometric figure.
Goals:
To consolidate students' knowledge of the types of angles formed as a result of the intersection of two straight lines;
to study the signs of parallel lines;
formation of skills to analyze the studied material and skills to apply it to solve problems; show the importance of the concepts being studied;
to consolidate the skills of solving problems on the use of signs of parallel lines;
development cognitive activity and self-sufficiency in obtaining knowledge;
education of interest in the subject, independence.
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Slides captions:
Signs of parallel lines 12/16/2014 Loginova N.V. mathematics teacher, MBOU "Secondary School No. 16", Izhevsk
How can two lines be located in a plane? 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 2 a c a b O Two straight lines either have one common point, i.e. intersect; or do not have any common point, i.e. do not intersect.
Select patterns with intersecting lines. a b A a b B a b C 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 3
Indicate the numbers of the figures that show parallel lines. a b A a b B a b c C 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 4
Define parallel lines. a b 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 5 Two straight lines in a plane are called parallel if they do not intersect. Parallelism of lines a and b is denoted by a b
Lines a and b are perpendicular to line c. How are they located among themselves? Make a conclusion. 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 6 c a b
Point out the figures that show parallel segments. a b A B C D b A A B C D B A C D C b B A C D D a 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 7
What is mutual arrangement segments AB, SR, MK? A B C R M K 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 8 Define parallel segments and parallel rays What is a secant?
The angles formed at the intersection of two secant lines are called: Corresponding: 2 and 6, 3 and 7, 1 and 5, 4 and 8. Crosswise lying: 3 and 5, 4 and 6. Internal unilateral: 4 and 5, 3 and 6. 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 9
What corners are highlighted? A). b). V). a c b 1 2 1 and 2 - lying crosswise a b 1 2 a b 1 2 c c 1 and 2 - unilateral 1 and 2 - corresponding 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 10
The lines m and n are intersected by the secant p. Name all the pairs of angles out of the eight formed angles: a) lying crosswise; b) internal unilateral; c) relevant. 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 11
Which line in the figure is a secant with respect to the other two lines? 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 12
Illusion of Goering (illusion of a fan) 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 13
The illusion of the cafe "Wall" 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 14
Signs of parallelism of two lines. 1) If 1 = 2, then a ║ b. Sign 1. If at the intersection of two lines of a secant, the lying angles are equal, then the lines are parallel. 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 15 2) If 3 = 2, then a ║ b. 3) If 2 + 4 \u003d 180 , then a ║ b.
a b c 2 1 4 3 5 7 6 8 Choose the correct statements: Lines a and b are parallel if … MBOU "Secondary School No. 16" 16
Tasks for fixing signs of parallelism of straight lines on finished drawings: a b c 1 2 = 32˚ 1 2 Given: = 32˚ Prove: a b 1 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 17
2 a b 1 2 1 = 48 º, 2 = 132º Prove: a b MBOU "Secondary School No. 16" 18
Tasks for fixing signs of parallelism of straight lines on finished drawings: a b c 1 3 4 2 5 1 = 47 º, 2 = 133º a b Prove: 3 MBOU "Secondary School No. 16" 19
Given: 1 =47, 2 = 133. Prove: d ║ s. Tasks for fixing the signs of parallelism of straight lines on the finished drawings: 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 20
Given: 1 =125, 2 =55. Prove: k ║ f. Tasks for fixing the signs of parallelism of straight lines on the finished drawings: 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 21
Prove: d || a. d a 1 2 3 4 5 6 7 8 p Tasks for fixing the signs of parallelism of lines on the finished drawings: 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 22
Given: AD=BC, AB=CD. Prove: AD ⃦ BC. A B C D MBOU "Secondary School No. 16" 23
A B C D F Prove: AB || DF Tasks for fixing the signs of parallelism of lines on the finished drawings: 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 24 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 24
A B O C D Prove: AB || CD. Tasks for fixing the signs of parallelism of straight lines on the finished drawings: 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 25
According to the figure, find the angle 1 a b 1 c d 2 3 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 26
Find the degree measure of each angle shown in the drawing a b 4 c 5 2 1 3 6 a b 4 c 5 1 3 2 6 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 27
Independent work option 1 option 2 d b c 3) 1 3 2 d b c 3) 1 3 2 16.12.2014 Loginova N.V. MBOU "Secondary School No. 16" 28
A B C D Through points A and C draw lines a and c parallel to BD. Is it true that a ⃦ c? a c Tasks for fixing signs of parallel lines 12/16/2014 Loginova N.V. MBOU "Secondary School No. 16" 29
