# Isosceles Triangle

Pronunciation: /aɪˈsɒ.səˌliz ˈtraɪˌæŋ.gəl/ Explain

 Click on the blue points and drag them to change the figure. When you change the figure, what remains the same? What changes? 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.[3] 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.

 Step Example Description Justification 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 ΔAFB ≅ ΔBGA 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. McAdams, David E.. All Math Words Dictionary, isosceles triangle. 2nd Classroom edition 20150108-4799968. pg 103. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. isosceles triangle. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 8/7/2018. http://www.merriam-webster.com/dictionary/isosceles triangle. Buy the book
3. 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 8/7/2018. http://www.archive.org/stream/firstsixbooksofe00caseuoft#page/8/mode/1up/search/isosceles. Buy the book
4. MacDonald, J. W.. Principles of Plane Geometry. pp 14-15. www.archive.org. Allyn and Bacon. 1894. Last Accessed 8/7/2018. http://www.archive.org/stream/principlesofplan00macdrich#page/14/mode/1up/search/isosceles. Buy the book
5. Boyd, Burrill, and Cummins. Glencoe Geometry. pp 181-185, 222-224. Glencoe/McGraw-Hill. 2001. Last Accessed 8/7/2018. Buy the book

• McAdams, David E.. Triangle. allmathwords.org. All Math Words Encyclopedia. Life is a Story Problem LLC. 3/3/2010. https://www.allmathwords.org/en/t/triangle.html.
• Euclid of Alexandria. Elements. Clark University. 9/6/2018. https://mathcs.clarku.edu/~djoyce/elements/elements.html.

McAdams, David E. Isosceles Triangle. 12/21/2018. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/i/isoscelestriangle.html.

### Revision History

12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)