Isosceles Triangle

Pronunciation: /aɪˈsɒ səˌliz ˈtraɪˌæŋ gəl/ ?

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)
Manipulative 1: Isosceles triangle. Created with GeoGebra.

An isosceles triangle is a triangle where two of the sides are congruent (are equal). Note that the definition of an Isosceles triangle does not rule out three equal sides.[1] This means that an Equilateral triangle is also an Isosceles triangle. Manipulative 1 is an example of an Isosceles triangle.

Click on the blue points in the manipulatives and drag them to change the figure.

Euclid, Elements Book 1 Proposition 5: The base angles of an Isosceles triangle are congruent.

StepExampleDescriptionJustification
1 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Let ΔABC be a triangle where side CA is the same length as side CB. We will show that ∠CAB ≅ ∠CBA and ∠FAB ≅ ∠GBA.

Starting conditions.

2 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Extend sides CA and BC.

Euclid. Elements Book 1, Postulate 2: A line segment of a specific length can be drawn in a straight line.

3 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Place an arbitrary point F on the extended line segment CA on the opposite side of point A from point C.

Although Euclid does not justify picking an arbitrary point on a line in Elements, modern geometry considers a line to be made up of infinite points, so any point may be picked.

4 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Place a point G on the extended segment CB such that CG is the same length as CF.

Euclid. Elements Book 1, Proposition 3: A line segment the same length as a given line can be drawn on a larger line.

5 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Draw line segments FB and GA.

Euclid. Elements Book 1, Postulate 1: A straight line can be drawn between any two points.

6 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Since CF = CG and CA = CB, and ∠ACB is in common, ΔCFB ≅ ΔCGA.

Euclid. Elements Book 1, Proposition 4: Two triangles with corresponding side-angle-side equal are equal to each other. See also SAS Congruence All Math Words Encyclopedia.

7 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now) Since CF = CG and CA = CB, then the remainders AF = BG.

Euclid. Elements Book 1, Common Notation 3: If equals are subtracted from equals, then the remainders are equal.

8 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

In step 6, it was shown that ΔCFB ≅ ΔCGA. All of the corresponding parts of the two triangles are also equal. So FB ≅ GA and angles CFB ≅ CGA.

Euclid. Elements Book 1, Proposition 4: Two triangles with corresponding side-angle-side equal are equal to each other, and the corresponding parts are equal. See also SAS Congruence All Math Words Encyclopedia.

9 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Since AF ≅ BG (step 7), FB ≅ GA and angles ∠CFB ≅ ∠CGA (step 8), triangles ΔAFBΔBGA by SAS congruence.

The area shared by the two triangles is in purple.

Euclid. Elements Book 1, Proposition 4: Two triangles with corresponding side-angle-side equal are equal to each other, and the corresponding parts are equal. See also SAS Congruence All Math Words Encyclopedia.

10 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Since triangles ΔAFB ≅ ΔBGA, we can conclude that angles ∠FAB ≅ ∠GBA and angles ∠FBA ≅ ∠GAB.

Euclid. Elements Book 1, Proposition 4: Two triangles with corresponding side-angle-side equal are equal to each other, and the corresponding parts are equal. See also SAS Congruence All Math Words Encyclopedia.

11 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Since ∠CAF is a straight angle and ∠CBG is a straight angle, they must be equal.

Euclid. Elements Book 1, Common Notion 4: Things which coincide with one another equal one another.

12 Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

But, since ∠FAB ≅ ∠GBA, the remaining angles ∠CAB ≅ ∠CBA. QED.

Euclid. Elements Book 1, Common Notion 4: Things which coincide with one another equal one another.

References

  1. isosceles triangle. merriam-webster.com. Encyclopedia Britannica. (Accessed: 2010-03-03). http://www.merriam-webster.com/dictionary/isosceles triangle.
  2. Casey, John, LL.D., F.R.S.. The First Six Books of the Elements of Euclid, pp 8,19-23. Translated by Casey, John, LL.D. F.R.S.. Hodges, Figgis & Co., 1890. (Accessed: 2010-01-02). http://www.archive.org/stream/firstsixbooksofe00caseuoft#page/8/mode/1up/search/isosceles.
  3. MacDonald, J. W.. Principles of Plane Geometry, pp 14-15. Allyn and Bacon, 1894. (Accessed: 2010-03-03). http://www.archive.org/stream/principlesofplan00macdrich#page/14/mode/1up/search/isosceles.
  4. Boyd, Burrill, and Cummins. Glencoe Geometry, pp 181-185, 222-224. Glencoe/McGraw-Hill, 2001. (Accessed: 2010-03-03).

More Information

  • McAdams, David. Triangle. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-03-03. http://www.allmathwords.org/article.aspx?lang=en&id=Triangle.
  • McAdams, David. Isosceles Right Triangle. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 2010-03-03. http://www.allmathwords.org/article.aspx?lang=en&id=Isosceles Right Triangle.
  • Euclid. Elements. 2009-03-12. D. Joyce. http://babbage.clarku.edu/~djoyce/java/elements/bookI/propI5.html.

Printed Resources

Cite this article as:


Isosceles Triangle. 2010-03-03. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/isoscelestriangle.html.

Translations

Image Credits

Revision History


2010-03-03: Added "References" (McAdams, David.)
2009-12-21: Added reference to Euclid's Elements; Expanded table of angle classes. (McAdams, David.)
2008-11-19: Changed manipulatives to GeoGebra (McAdams, David.)
2008-03-26: Changed More Information to match current standard (McAdams, David.)
2007-08-27: Add Elements Book 1 Proposition 5 (McAdams, David.)
2007-07-12: Initial version (McAdams, David.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2005-2011 Life is a Story Problem LLC. All rights reserved.
Creative Commons License This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License