
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. 
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 sideangleside 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 sideangleside 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 sideangleside 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 sideangleside 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. 
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