Isosceles Triangle

Pronunciation: /aɪˈsɒ səˌliz ˈtraɪˌæŋ gəl/ Explain

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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

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

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

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

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

Draw line segments FB and GA.

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

6

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
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

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

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

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

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

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. Merriam-Webster. Last Accessed 8/6/2018. 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. www.archive.org. D. F.R.S.. Hodges, Figgis & Co.. 1890. Last Accessed 8/6/2018. http://www.archive.org/stream/firstsixbooksofe00caseuoft#page/8/mode/1up/search/isosceles. Buy the book
  3. MacDonald, J. W.. Principles of Plane Geometry. pp 14-15. www.archive.org. Allyn and Bacon. 1894. Last Accessed 8/6/2018. http://www.archive.org/stream/principlesofplan00macdrich#page/14/mode/1up/search/isosceles. Buy the book
  4. Boyd, Burrill, and Cummins. Glencoe Geometry. pp 181-185, 222-224. Glencoe/McGraw-Hill. 2001. Last Accessed 8/6/2018. Buy the book

More Information

  • McAdams, David E.. Triangle. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/3/2010. http://www.allmathwords.org/en/t/triangle.html.

Cite this article as:

McAdams, David E. Isosceles Triangle. 8/7/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/i/isoscelestriangle.html.

Image Credits

Revision History

8/6/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
3/3/2010: Added "References". (McAdams, David E.)
12/21/2009: Added reference to Euclid's Elements; Expanded table of angle classes. (McAdams, David E.)
11/19/2008: Changed manipulatives to GeoGebra. (McAdams, David E.)
3/26/2008: Changed More Information to match current standard. (McAdams, David E.)
8/27/2007: Add Elements Book 1 Proposition 5. (McAdams, David E.)
7/12/2007: Initial version. (McAdams, David E.)

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