# 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. 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.
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. http://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. http://www.allmathwords.org/en/i/isoscelestriangle.html.

### Revision History

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