[b][size=200]#7) When Does "SSA" Determine a Triangle?[/size][/b]
Despite the fact that "SAS" and "ASA" determine triangles, [b]"SSA" does [u]not[/u] always define a triangle. [/b][br][br]This is the case when we have two sides and a NON-included angle, so we can think of these three parts as:[br][list][*]The [b]opposite side[/b] (the side "unattached" to the given angle)[/*][*]The [b]adjacent side[/b] (the side "attached" to the given angle),[/*][*]and the [b]given angle[/b].[/*][/list][br]In some cases, "SSA" produces ZERO possible triangles. [br][br]In other cases, it produces two different triangles! [br][br]Still, there are some specific cases in which "SSA" will determine a triangle. [br][br]This final section of the Chapter 5 Construction Packet is all about exploring these different possible cases.[br]
[size=150][i]We'll use capital "S" for the longer side and lowercase "s" for the shorter side.[br][/i][b][br][color=#0000ff]Case I[/color][/b][b][color=#0000ff]: "SsA" + obtuse angle[/color][/b][/size][size=150][b][br][/b][list][*][b]The opposite side is [u]longer[/u] than the adjacent side [/b](ex. BC > BA).[/*][*][b]The given angle is [u]obtuse[/u][/b] (ex. angle A is obtuse).[/*][/list][/size][br]In the window below:[br][list][*]On the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[/*][*]On the right: try to find a way to create a different triangle that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[/*][/list][br]Test it out with a different triangle on the left, until you are confident in your answer.[br][br][b]In this case, does "SsA" determine a triangle?[/b]
[size=150][i]We'll use capital "S" for the longer side and lowercase "s" for the shorter side.[br][/i][b][br][color=#0000ff]Case II[/color][/b][b][color=#0000ff]: "sSA" + obtuse angle[/color][/b][/size][size=150][list][*][b]The opposite side is [u]shorter[/u] than the adjacent side [/b](ex. BC < BA).[/*][*][b]The given angle is [u]obtuse[/u][/b] (ex. angle A is obtuse).[/*][/list][/size][br]In the window below:[br][list][*]On the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[/*][*]On the right: try to find a way to create a different triangle that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[/*][/list][br]Test it out with a different triangle on the left, until you are confident in your answer.[br][br][b]In this case, does "sSA" determine a triangle?[/b]
[size=150][i]We'll use capital "S" for the longer side and lowercase "s" for the shorter side.[br][/i][b][br][color=#0000ff]Case III[/color][/b][b][color=#0000ff]: "SsA" + right angle[/color][/b][/size][size=150][list][*][b]The opposite side is [u]longer[/u] than the adjacent side [/b](ex. BC > BA).[/*][*][b]The given angle is [u]right[/u] [/b](ex. angle A is right).[/*][/list][/size][br]In the window below:[br][list][*]On the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[/*][*]On the right: try to find a way to create a different triangle that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[/*][/list][br]Test it out with a different triangle on the left, until you are confident in your answer.[br][br][b]In this case, does "SsA" determine a triangle?[/b]
[size=150][i]We'll use capital "S" for the longer side and lowercase "s" for the shorter side.[br][/i][b][br][color=#0000ff]Case IV[/color][/b][b][color=#0000ff]: "sSA" + right angle[/color][/b][/size][size=150][list][*][b]The opposite side is [u]shorter[/u] than the adjacent side [/b](ex. BC < BA).[/*][*][b]The given angle is [u]right[/u] [/b](ex. angle A is right).[/*][/list][/size][br]In the window below:[br][list][*]On the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[/*][*]On the right: try to find a way to create a different triangle that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[/*][/list][br]Test it out with a different triangle on the left, until you are confident in your answer.[br][br][b]In this case, does "sSA" determine a triangle?[/b]
[size=150][b][color=#0000ff]Consider the isosceles case when the given angle is right or obtuse.[br][/color][/b][/size][size=150][list][*][b]The opposite side is [u]equal[/u] in length to the adjacent side [/b](ex. BC = BA).[/*][*][b]The given angle is [u]right[/u] or [u]obtuse[/u] [/b](ex. angle A is right or obtuse).[/*][/list][/size][br]In the window below:[br][list][*]On the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[/*][*]On the right: try to find a way to create a different triangle that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[/*][/list][br]Test it out with a different triangle on the left, until you are confident in your answer.[br][br][b]In this case, does "ssA" determine a triangle?[/b]
[size=150][i]We'll use capital "S" for the longer side and lowercase "s" for the shorter side.[br][/i][b][br][color=#0000ff]Case V[/color][/b][b][color=#0000ff]: "Ssa" + acute angle[/color][/b][/size][size=150][list][*][b]The opposite side is [u]longer[/u] than the adjacent side [/b](ex. BC > BA).[/*][*][b]The given angle is [u]acute[/u][/b] (ex. angle A is acute).[/*][/list][/size][br]In the window below:[br][list][*]On the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[/*][*]On the right: try to find a way to create a different triangle that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[/*][/list][br]Test it out with a different triangle on the left, until you are confident in your answer.[br][br][b]In this case, does "Ssa" determine a triangle?[/b]
[size=150][i]We'll use capital "S" for the longer side and lowercase "s" for the shorter side.[br][/i][b][br][color=#0000ff]Case VI[/color][/b][b][color=#0000ff]: "ssa" + acute angle[/color][/b][/size][size=150][list][*][b]The opposite side is [u]equal[/u] in length to the adjacent side [/b](ex. BC = BA).[/*][*][b]The given angle is [u]acute[/u][/b] (ex. angle A is acute).[/*][/list][/size][br]In the window below:[br][list][*]On the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[/*][*]On the right: try to find a way to create a different triangle that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[/*][/list][br]Test it out with a different triangle on the left, until you are confident in your answer.[br][br][b]In this case, does "ssa" determine a triangle?[/b]
[size=150][i]We'll use capital "S" for the longer side and lowercase "s" for the shorter side.[br][/i][b][br][color=#0000ff]Case VII[/color][/b][b][color=#0000ff]: "sSa" + acute angle[/color][/b][/size][size=150][list][*][b]The opposite side is [u]shorter[/u] than the adjacent side [/b](ex. BC < BA).[/*][*][b]The given angle is [u]acute[/u][/b] (ex. angle A is acute).[/*][/list][/size][br]In the window below, on the left: move around points A, B, and C, to make a triangle satisfying the above conditions.[br][br]You should find that this case can produce [b][u]two[/u] different triangles[/b]! [br][br][u]Show this[/u] by putting together [b]one triangle on the left [/b]that satisfies the conditions in this case,[br]and then showing how you can make [b]another, noncongruent, triangle on the right [/b]that still has BC = B'C', BA = B'A', and angle A congruent to angle A'.[br][br][b]Therefore, in this case, "sSa" does NOT determine a triangle, as there are two possible triangles that can be formed.[/b]
[size=150][b][color=#0000ff]...However, there is a special case of "sSa" + acute angle.[br][br][/color][/b][/size]In the window below: move around points A, B, and C to make a triangle where the conditions from above still hold (BC < BA and angle A is acute), but now BC is juuuust long enough to [b]touch [/b]the opposite side [b]without crossing[/b] it.[br][br][b]What kind of angle is angle C?[/b]
[b]In this case, does "sSa" determine a triangle?[/b]
[size=200][b]Summary of "SSA" Cases[/b][/size]
a) If the opposite side is [b]longer[/b] than the adjacent side, for which kinds of given angles will "SSA" determine exactly one triangle? (Select all that apply)[br][br][i](obtuse [/i]and/or[i] right [/i]and/or[i] acute)[/i]
b) If the opposite side is [b]equal[/b] in length to the adjacent side, for which kind of given angle does "SSA" determine exactly one triangle?[br][br][i](obtuse, right, or acute)?[/i]
c) If the opposite side is [b]shorter [/b]than the adjacent side, for which kind of given angle [i][b]might [/b][/i]"SSA" determine exactly one triangle? [br][br][i](obtuse, right, or acute)?[/i]
d) [b]If [/b]SSA does determine a triangle in condition (c), what type of triangle must it be?