# Congruent triangles

## Some tools

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

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`

.

In a similar way you can make a circle with radius `c`. Use the tool
`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
`Angle`

to measure the angle `BAC`. The angle will appear
in the Algebra View as `α`.

Enter a segment
`DE`. Use the tool
`Angle with Given Size `

on the segment `DE`; enter `α`
in the pop up window.

You can use the tool
`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`.

## Triangle cases

### Case 1: side - angle - side

** 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

** 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

** ****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

** 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

** 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!

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License