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Copy of Chapter 5: Triangle Constructions (part 3: the many cases of "SSA")

Auteur :mbader97
#7) When Does "SSA" Determine a Triangle?
Despite the fact that "SAS" and "ASA" determine triangles, "SSA" does not always define a triangle. This is the case when we have two sides and a NON-included angle, so we can think of these three parts as:
  • The opposite side (the side "unattached" to the given angle)
  • The adjacent side (the side "attached" to the given angle),
  • and the given angle.
In some cases, "SSA" produces ZERO possible triangles. In other cases, it produces two different triangles! Still, there are some specific cases in which "SSA" will determine a triangle. This final section of the Chapter 5 Construction Packet is all about exploring these different possible cases.

We'll use capital "S" for the longer side and lowercase "s" for the shorter side. Case I: "SsA" + obtuse angle

  • The opposite side is longer than the adjacent side (ex. BC > BA).
  • The given angle is obtuse (ex. angle A is obtuse).
In the window below:
  • 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'.
Test it out with a different triangle on the left, until you are confident in your answer. In this case, does "SsA" determine a triangle?

Testing "SSA": Case 1

We'll use capital "S" for the longer side and lowercase "s" for the shorter side. Case II: "sSA" + obtuse angle

  • The opposite side is shorter than the adjacent side (ex. BC < BA).
  • The given angle is obtuse (ex. angle A is obtuse).
In the window below:
  • 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'.
Test it out with a different triangle on the left, until you are confident in your answer. In this case, does "sSA" determine a triangle?

Testing "SSA": Case II

We'll use capital "S" for the longer side and lowercase "s" for the shorter side. Case III: "SsA" + right angle

  • The opposite side is longer than the adjacent side (ex. BC > BA).
  • The given angle is right (ex. angle A is right).
In the window below:
  • 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'.
Test it out with a different triangle on the left, until you are confident in your answer. In this case, does "SsA" determine a triangle?

Testing "SSA": Case III

We'll use capital "S" for the longer side and lowercase "s" for the shorter side. Case IV: "sSA" + right angle

  • The opposite side is shorter than the adjacent side (ex. BC < BA).
  • The given angle is right (ex. angle A is right).
In the window below:
  • 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'.
Test it out with a different triangle on the left, until you are confident in your answer. In this case, does "sSA" determine a triangle?

Testing "SSA": Case IV

Consider the isosceles case when the given angle is right or obtuse.

  • The opposite side is equal in length to the adjacent side (ex. BC = BA).
  • The given angle is right or obtuse (ex. angle A is right or obtuse).
In the window below:
  • 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'.
Test it out with a different triangle on the left, until you are confident in your answer. In this case, does "ssA" determine a triangle?

Testing "ssA"

We'll use capital "S" for the longer side and lowercase "s" for the shorter side. Case V: "Ssa" + acute angle

  • The opposite side is longer than the adjacent side (ex. BC > BA).
  • The given angle is acute (ex. angle A is acute).
In the window below:
  • 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'.
Test it out with a different triangle on the left, until you are confident in your answer. In this case, does "Ssa" determine a triangle?

Testing "SSA": Case V

We'll use capital "S" for the longer side and lowercase "s" for the shorter side. Case VI: "ssa" + acute angle

  • The opposite side is equal in length to the adjacent side (ex. BC = BA).
  • The given angle is acute (ex. angle A is acute).
In the window below:
  • 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'.
Test it out with a different triangle on the left, until you are confident in your answer. In this case, does "ssa" determine a triangle?

Testing "SSA": Case VI

We'll use capital "S" for the longer side and lowercase "s" for the shorter side. Case VII: "sSa" + acute angle
  • The opposite side is shorter than the adjacent side (ex. BC < BA).
  • The given angle is acute (ex. angle A is acute).
 In the window below, on the left: move around points A, B, and C, to make a triangle satisfying the above conditions. You should find that this case can produce two different triangles! Show this by putting together one triangle on the left that satisfies the conditions in this case, and then showing how you can make another, noncongruent, triangle on the right that still has BC = B'C', BA = B'A', and angle A congruent to angle A'. Therefore, in this case, "sSa" does NOT determine a triangle, as there are two possible triangles that can be formed.

Testing "SSA": Case VII

...However, there is a special case of "sSa" + acute angle. 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 touch the opposite side without crossing it. What kind of angle is angle C?

In this case, does "sSa" determine a triangle?

Testing "SSA": Case VII, special edition

Summary of "SSA" Cases

a) If the opposite side is longer than the adjacent side, for which kinds of given angles will "SSA" determine exactly one triangle? (Select all that apply) (obtuse and/or right and/or acute)

b) If the opposite side is equal in length to the adjacent side, for which kind of given angle does "SSA" determine exactly one triangle? (obtuse, right, or acute)?

c) If the opposite side is shorter than the adjacent side, for which kind of given angle might "SSA" determine exactly one triangle? (obtuse, right, or acute)?

d) If SSA does determine a triangle in condition (c), what type of triangle must it be?

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