• In which country did philosophers know about the fact about 1000 years earlier?
• How do we lable the vertices of a triangle?
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THE TRIANGLE
Polygons. A portion of a plane surface may be enclosed by straight lines. The least number by which it can be so enclosed is three; it may be bounded by four, five, or any higher number of straight lines or by a curved line. Such an enclosed plane is called a plane figure or a polygon.
Triangles. A triangle is a three-sided plane figure bounded by three straight lines and containing three angles. The three sides of a triangle meet at points called vertices. The vertex at the top of a triangle may be called the apex, and the line at the bottom may be called the base. The perpendicular distance from the opposite vertex to the base is called the altitude. A triangle is identified by naming its vertices in any order. The sum of every triangle is 180°.
Kinds of triangles. There are various types of triangles. A scalene triangle is a triangle with no two sides equal. An isosceles triangle is a triangle which has two equal sides.
The equal sides are called the legs, the third side is called the base. The angles at the base are called the base angles. The angle formed by the two equal sides, is called the vertex angle.
An equilateral triangle is a triangle with all three sides equal.
A triangle having one right angle is called a right triangle; having one obtuse angle, an obtuse triangle; having three acute angles, an acute triangle.
In a right- angled triangle the side opposite the right angle is called the hypotenuse.
The theorem of Pythagoras states: "In a right-angled triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides."
In triangle ABC, line BC is produced to point X (Fig. 15.1). ABC is an interior angle, and ACX is an exterior angle.
Fig. 15.1
Congruence, similarity and symmetry. If the following parts of two triangles are equal:
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b) a right angle, the hypotenuse and a side; or, c) two angles and a corresponding side; or, d) all three sides;
then the two triangles are congruent.
If two triangles have their corresponding angles equal and corresponding sides proportional, they are similar. Similar figures have the same shape but not necessarily the same size (Fig. 15.2).
Fig. 15.2 Fig. 15.3
A line has been drawn in Fig. 15.3. The figure is now divided into two parts that have the same size and shape. Two triangles are on either side of an axis of symmetry (or centre line). They are symmetrical triangles. If we fold the figure along the axis of symmetry, the two parts will coincide. The axis of symmetry AX is perpendicular to, and bisects BC.
We speak about line symmetry (Fig. 15.3), point symmetry and plane symmetry. The circle possesses point symmetry as well as line symmetry. If a solid can be divided into two equal solids by a plane, and if every part on one side of the plane has a corresponding part on the other, the original solid has plane symmetry.
Vocabulary and Language Focus
1. Translate into English . Use some of the expressions in sentences of your own.
l. telgsümmeetria, 2.tsentraalsümmeetria, 3. sümmeetria tasandi suhtes, 4. kummalgi pool sümmeetriatelge, 5. kolmnurga alust poolitama, 6. tasapinnaline kujund, 7. tipp, tipud, 8.
alusnurgad, 9. tipunurka moodustama, l0.ükskõik millises järjekorras, 11. kolmnurga kõrgust
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kolmnurk, 16. võrdkülgne kolmnurk, 17. kaks külge ja nendevaheline nurk, 18. sama kuju ja suurust omama, 19. kongruentne olema, 20. kongruentsus ja sarnasus, 21. sarnased kolmnurgad, 22. kolme sirgega piiratud olema, 23. tasapind, 24. kokku langema, ühtima , 25. mitmed liigid, 26. hulknurk.
2. Fill in the blanks.
1. Each triangle has three points, or __________ . 2. A line which meets another __________ at 90° is called a __________ line. 3. If two angles of a triangle are equal to 45°, the triangle is called a __________ __________ __________ triangle. 4. If we __________ a right angle, we have two __________ angles of 45°. 5. If each of the angles in a triangle is equal to 60°, the triangle is called __________ . 6. If a figure has __________ symmetry, then the line which connects every two corresponding points must pass through the centre of symmetry and be bisected by it. 7. Any diameter of the circle is an __________ of symmetry.
8. If we fold a figure along the axis of symmetry and the two parts __________ , we may speak about __________ symmetry.
3. Complete the following theorems.
1. The __________ angle of a triangle is equal to the __________ of the two __________ opposite angles.
2. In an __________ triangle __________ sides are equal and the angles at the base of these sides are also __________ .
3. In an __________ triangle __________ sides are equal and __________ angles are equal to 60°.
4. If two angles and a side of one triangle are equal to two angles and the __________
side of another triangle, the triangles are __________.
4. State the meaning of the prefixes in the following words.
triangle
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statements.
1. The exterior angle of a triangle is always obtuse.
2. Only two angles of a triangle may be acute.
3. The smallest angle of a triangle is opposite the shortest side.
4. The point where the sides of an angle meet is called the vertex.
5. A triangle with two obtuse angles is called an obtuse triangle.
Pair Work
1. Describe each triangle. Discover any relationships between the triangles (i.e.
symmetry, similarity or congruence).
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1. How many dimensions does a polygon have? 2. What is a polygon? 3. What is a plane figure? 4. What is a triangle? 5. What do we call the line at the bottom of a triangle? 6.
What are some special forms of triangles? Describe them. 7. What does the theorem of Pythagoras state? 8. What figures are similar figures? 9. When are two polygons similar? 10.
When are two triangles congruent? 11. What figure has line symmetry? 12. How many axes of symmetry can figures have? 13. What points are called corresponding ones? 14. What is meant by point symmetry? 15. What is plane symmetry?
Group Work
Solve the puzzle.
How many triangles are there in this figure?
Answer the question.
In 1816, while studying the Brocard points of a triangle, a mathematician exclaimed, "It is indeed wonderful that so simple a figure as the triangle is so inexhaustible in properties. How many as yet unknown properties of other figures may there not be?"
Who was the mathematician?
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