This construction depicts the SSA (Side-Side-Angle) condition for triangles, in which two sides and one of their opposite angles are given. It is intended to demonstrate how the [u]number[/u] of triangles changes (0, 1, or 2) as the sides and angle values are changed. This construction does NOT numerically solve all sides/angles of the SSA triangle. [url=https://www.geogebra.org/m/z6neks9e]Try out my triangle solver[/url] for numerical solutions of all congruence conditions (SSS, SAS, ASA, SAA, SSA).[br][br]If you have access to a keyboard:[br][list][*]For finer control of the sliders, click the slider and use ←→ keys to adjust.[/*][*]Tab forward through input boxes and sliders, or shift-tab backward.[/*][/list][br]The [color=#0000ff][b]labels[/b][/color] slider covers all permutations of labeling this triangle with angles/vertices A, B, & C and opposite sides a, b, & c. All questions/info below assume the default setting of [b][color=#0000ff]labels=1[/color][/b].[br][br]I suggest playing around with the construction and then answering the exercises below it. Or perhaps even better, just play around with it and draw your own observations and conclusions without any guidance.
[size=150]The number of triangles in this construction are as follows:[/size]
[u]0 triangles[/u][br]In it's default state, the swinging side (labeled [b][color=#0000ff]a[/color][/b] by default) is not long enough to form a real triangle. [color=#333333]For [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, w[/color]rite an inequality in terms of given SSA parts ([b][color=#0000ff]a[/color][/b], [b][color=#0000ff]b[/color][/b], & [b][color=#0000ff]A[/color][/b]) that establishes when 0 triangles are formed.[br]How about when [b][color=#0000ff]A [/color][/b][color=#333333]≥ 90°?[/color]
[color=#333333]When [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, no triangles are formed when [/color][color=#0000ff][b]a[/b][/color] < [b][color=#0000ff]b[/color][/b]·sin([b][color=#0000ff]A[/color][/b]).[br][br]When [b][color=#0000ff]A [/color][/b]≥ 90°, there are no triangles formed when [b][color=#0000ff]a[/color][/b] ≤ [b][color=#0000ff]b[/color][/b].
[u]1 triangle[/u][br][color=#333333]For [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, d[/color]rag the slider for side [b][color=#0000ff]a[/color][/b] until it is just long enough to form 1 triangle. What type of triangle is this? Write an equation in terms of given SSA parts ([b][color=#0000ff]a[/color][/b], [b][color=#0000ff]b[/color][/b], & [b][color=#0000ff]A[/color][/b]) that establishes when this 1 triangle is formed.[br][br]As an alternative to using the slider in the GeoGebra construction, you may type such a trig expression into the input box for [color=#0000ff][b]a[/b][/color]. However, you'll need to enter numerical values in the input box instead of variables (e.g. 16 instead of [b][color=#0000ff]b[/color][/b], 30° instead of [b][color=#0000ff]A[/color][/b]).
[color=#333333]Right triangle;[br][color=#333333]When [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, o[/color]ne right triangle is formed when [/color][b][color=#0000ff]a[/color][/b] = [b][color=#0000ff]b[/color][/b]·sin([b][color=#0000ff]A[/color][/b]).[br][br](When [b][color=#0000ff]A [/color][/b]≥ 90°, no such right triangle may be formed.)
[u]2 triangles[/u][br][color=#333333]For [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, k[/color]eep on lengthening side [b][color=#0000ff]a[/color][/b] until there are 2 possible triangles formed by the given info. This is known as the "[b]ambiguous[/b]" SSA case because the given information doesn't specify which of the 2 triangles is desired for a particular situation. Keep on lengthening [b][color=#0000ff]a[/color][/b] until there are no longer two triangles anymore.[br]Write a compound inequality in terms of given SSA parts ([b][color=#0000ff]a[/color][/b], [b][color=#0000ff]b[/color][/b], & [b][color=#0000ff]A[/color][/b]) that establishes when there are 2 triangles formed.
[color=#333333]When [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, t[/color]wo triangles are formed when [b][color=#0000ff]b[/color][/b][b]·sin([/b][b][color=#0000ff]A[/color][/b][b])[/b] < [b][color=#0000ff]a[/color][/b] < [b][color=#0000ff]b[/color][/b][color=#333333].[br][br](When [b][color=#0000ff]A [/color][/b]≥ 90°, no "ambiguous" case of two triangles is possible.)[/color]
[u]1 triangle[/u][br][color=#333333]For [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, u[/color]pon continuing to lengthen side [b][color=#0000ff]a[/color][/b], at some point 1 of the 2 triangles becomes invalid and there is only 1 valid triangle left. Why did the 1 triangle become invalid? Write an inequality in terms of given SSA parts ([b][color=#0000ff]a[/color][/b], [b][color=#0000ff]b[/color][/b], & [b][color=#0000ff]A[/color][/b]) that establishes when only 1 valid triangle remains.[br]How about when [b][color=#0000ff]A [/color][/b][color=#333333]≥ 90°?[/color]
One of the triangles becomes invalid because the measure of an interior angle is the supplement of [b][color=#0000ff]A[/color][/b] (measures 180°–A) rather than [b][color=#0000ff]A[/color][/b] itself.[br][color=#333333]When [/color][b][color=#0000ff]A [/color][/b][color=#333333]< 90°, o[/color]ne valid (non-right) triangle is formed when [b][color=#0000ff]b[/color][/b] ≤ [b][color=#0000ff]a[/color][/b].[br][br]When [b][color=#0000ff]A [/color][/b]≥ 90°, it is found that [color=#333333]o[/color]ne valid (non-right) triangle is formed when [b][color=#0000ff]b[/color][/b] < [b][color=#0000ff]a[/color][/b].