Congruent triangles

Some tools

That ↑ is your task!
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In order to investigate the different cases for triangles, you will need tools for copying lengths and angles. One key feature is to use circles for copying lengths. Make sure you know how to do these tasks before you continue to do the triangle cases

Make a triangle. If you have no other objects, the vertices of the triangle will be the points A, B and C. The sides of the triangle will be denoted AB=c, BC=a and AC=b

Copy length

Start with: A triangle ΔABC

Construct: A circle with radius c (c=AB). You should be able to move the circle around.

Construct: A segment with length c. You should be able to move the segment around.

The letters a, b and c are the names of the sides of the triangle. They are also so called variables, i.e. they have numerical values which you can use in mathematical expressions. The numerical value of the variable is, in this case, the length of the side.

Use the tool Icon Segment with Given Length from Point. A pop up will show up where you can enter a number.

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Try making a few segments of various lengths. Enter c in the pop up window.

Right click on an object and choose Properties... Change the settings of the labels from Name to Value.

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In a similar way you can make a circle with radius c. Use the tool Icon Circle with Centre and Radius.

Copy angle

Start with: A triangle ΔABC

Construct: A segment and a ray having a common endpoint. The angle between the segment and the ray should have the same value as one of the angles of the triangle. Make sure that you know how to place the angle at any endpoint, clockwise or counter clockwise.

Use the tool Icon Angle to measure the angle BAC. The angle will appear in the Algebra View as α.

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Enter a segment Icon DE. Use the tool Icon Angle with Given Size on the segment DE; enter α in the pop up window.

You can use the tool Icon Angle with Given Size in two different ways.

If you start by clicking on a segment and then enter the angle, the angle will automatically appear at the first point of the segment.

If you first click on D, then on E, and then enter the angle; the angle DED' will be created; the angle will hence be placed at the point E.

Copy length to a segment on a given line

Start with: A triangle ΔABC and a line l through the points D and E

Construct: A segment having one endpoint at D. The segment should have the same length as the side AB and it should lie on the line l.

 

Like this ↑

Triangle cases

Case 1: side - angle - side

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Start with: A triangle ΔABC with labels as in the picture above. Enter the angle BAC and call it α.

Construct: A new triangle having one side of length c and another side of length b. The angle between these two sides should be α.

When you are done, you should be able to drag all the vertices of the original triangle. You should be able to move and rotate the new triangle.

side - angle - side: Conjecture -- Change the original triangle. Make a conjecture about the copy!

Case 2: side - side -side

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Start with: A triangle ΔABC with labels as in the picture above.

Construct: A new triangle with sides having the same lengths as the sides of the original triangle.

 

side - side -side: Conjecture -- Change the original triangle. Make a conjecture about the copy!

Case 3: angle - side - angle

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Start with: A triangle ΔABC with labels as in the picture above.

Construct: A new triangle having the two angles α and β . The side between these two angles should have the length c.

 

angle - side - angle: Conjecture -- Change the original triangle. Make a conjecture about the copy!

Case 4: side - side - angle

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Start with: A triangle ΔABC with labels as in the picture above.

Construct: A new triangle having one side of the length b, another side of the length c, and such that one of the angles not lying between these two sides is equal to corresponding angle in the original triangle.

 

side - side - angle: Conjecture -- Change the original triangle. Make a conjecture about the copy!

Quadrilateral

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Start with: A quadrilateral with the sides a, b, c and d.

Construct: A new quadrilateral having the same side lengths.

 

quadrilateral: Conjecture -- Can you make the same conjecture as you could when doing triangles?

Summary

Euclid's Elements

The first book of the Elements consists of definitions and postulates. Another word for postulate is axiom. An axiom is a proposition that can not be proved but is considered to be self-evident.

The first postulate is: To draw a straight line from any point to any point

This should be read as: It is possible to draw a straight line from any point to any point

By using the wordings of the postulates it is possible to make so called ruler and straightedge constructions, see Geometry - Ruler and Compass. Once a ruler and straightedge construction has been made, it is then assumed that it's possible to divide a segment in two equal part, it's possible to draw a perpendicular height in a triangle, it's possible to bisect an angle, and so on.

After the definitions, the postulates, and the ruler and straightedge constructions, a number of theorems are stated and proved. Once a theorem has been proved, you may assume that the theorem is true.

What follows is a simplified version:

Definitions and postulates

Definition 1 Two lines in a plane that do not intersect are parallel.

Definition 2 Two triangles are congruent if the corresponding sides and the corresponding angles are pair-wise the same. The fact that the triangles ΔABC and ΔDEF are congruent, is written like this ΔABC≅ΔDEF

Postulate 1 You can draw one and only one line through two points.

Postulate 2 The Parallel Postulate Given a straight line and a point not on the line, you can draw one and only one line through the point that is parallel to the first line.

Postulate 3 The Method of Superposition You can move, rotate and flip any geometrical figure without changing the shape or the size. (This is not a postulate in the Elements but it is used in the books anyway. For a detailed discussion check this out.)

Theorems

The three first cases you did as GeoGebra-exercises are called the congruence theorems.

Theorem 1 The Side-Angle-Side Congruence Theorem, SAS

Theorem 2 Isosceles triangles If two sides of a triangle are equal, then the opposite angles are equal.

Theorem 3 The Side-Side-Side Congruence Theorem, SSS

Theorem 4 The Angle-Side-Angle Congruence Theorem, ASA

Comment

Some text books call the SAS, SSS, and ASA congruence cases for postulates (or axioms) instead of theorems. In the Elements they are proved using superposition of triangles, which is a method that later has been criticised. When Hilbert in his book The foundations of geometry (pdf) formalized Euclid's Elements, he made part of the SAS case an axiom.

Exercise

Use Theorem 1 to prove Theorem 2!

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