In most geometry courses, we learn that there's no such thing as "SSA Congruence". That is, if we have triangles ABC and DEF such that AB = DE, BC = EF, and angle A = angle D, then we cannot deduce that ABC and DEF are congruent.
However, there are a few special cases in which SSA "works". That is, suppose we have AB = DE = x, BC = EF = y, and angle A = angle D = theta. For some values of x, y, and theta, we can deduce that triangle ABC is congruent to triangle DEF. Use the Law of Cosines or Law of Sines to explain the conditions x, y, and/or theta must satisfy in order for us to be able to deduce that triangle ABC is congruent to triangle DEF. (In other words, find conditions on x, y, and theta, so that given these values, you can uniquely reconstruct triangle ABC.)
How to do so? I'm confused. Help is appreciated, thanks.
if Theta is 90 deg, you have to conclude that all sides are congruent.
To determine the conditions for triangles ABC and DEF to be congruent using SSA, we can use the Law of Cosines or the Law of Sines. Let's explore both methods step-by-step:
Method 1: Using the Law of Cosines
1. Recall the Law of Cosines, which states that in a triangle with sides a, b, and c, and angle C opposite side c:
c^2 = a^2 + b^2 - 2ab * cos(C)
2. In triangle ABC, if AB = DE = x, BC = EF = y, and angle A = angle D = θ, we can label the remaining sides of triangle ABC as AC = p and angle B as β.
3. Applying the Law of Cosines to triangle ABC, we have:
p^2 = x^2 + y^2 - 2xy * cos(β)
4. In triangle DEF, with angle E labeled as α, we can label the remaining side as DF = q.
5. Applying the Law of Cosines to triangle DEF, we have:
q^2 = x^2 + y^2 - 2xy * cos(α)
6. In order for triangles ABC and DEF to be congruent, AC = DF = p = q. Therefore, we can equate the two expressions for p and q, giving us:
x^2 + y^2 - 2xy * cos(β) = x^2 + y^2 - 2xy * cos(α)
7. Simplifying the equation, we have:
cos(β) = cos(α)
8. The cosine function has a periodicity of 2π, meaning that the cosine values repeat every 2π radians. Therefore, β and α must satisfy:
β = α + 2πk, where k is an integer
9. Since angle β and angle α are interior angles of triangles, they must lie between 0 and π radians. Therefore, we have:
0 ≤ β < π and 0 ≤ α < π
10. Combining the inequalities from steps 8 and 9, we can conclude that the condition for triangles ABC and DEF to be congruent using SSA is:
β = α
Method 2: Using the Law of Sines
1. Recall the Law of Sines, which states that in a triangle with sides a, b, and c, and angles A, B, and C, respectively:
a/sin(A) = b/sin(B) = c/sin(C)
2. Applying the Law of Sines to triangle ABC, we have:
x/sin(θ) = y/sin(β)
3. Applying the Law of Sines to triangle DEF, we have:
x/sin(θ) = y/sin(α)
4. In order for triangles ABC and DEF to be congruent, β must equal α and the ratio of corresponding sides must be the same. Therefore, we have:
sin(β)/sin(θ) = sin(α)/sin(θ)
5. Canceling out sin(θ) from both sides of the equation, we get:
sin(β) = sin(α)
6. The sine function has a periodicity of 2π, meaning that the sine values repeat every 2π radians. Therefore, β and α must satisfy:
β = α + 2πk, where k is an integer
7. Since angle β and angle α are interior angles of triangles, they must lie between 0 and π radians. Therefore, we have:
0 ≤ β < π and 0 ≤ α < π
8. Combining the inequalities from steps 6 and 7, we can conclude that the condition for triangles ABC and DEF to be congruent using SSA is:
β = α
In conclusion, the condition for triangles ABC and DEF to be congruent using SSA is that the angles β and α (opposite the corresponding sides) must be equal. This condition ensures that the triangles have the same shape and size, allowing for a unique reconstruction of triangle ABC given the values of x, y, and θ.
To determine the conditions under which triangle ABC is congruent to triangle DEF using the SSA (side-side-angle) condition, we can utilize either the Law of Cosines or the Law of Sines. Let's examine both methods:
1. Law of Cosines:
The Law of Cosines states that in any triangle ABC with sides a, b, and c, and angle A opposite side a, the following equation holds:
c^2 = a^2 + b^2 - 2ab cos(A)
If we apply this to our problem, we have triangle ABC with sides AB = DE = x, BC = EF = y, and angle A = angle D = θ. Similarly, we have c = AC and f = DF as the unknown sides.
Applying the Law of Cosines to triangle ABC, we have:
AC^2 = x^2 + y^2 - 2xy cos(θ)
Applying the Law of Cosines to triangle DEF, we have:
DF^2 = x^2 + y^2 - 2xy cos(θ)
Since we know that AB = DE = x, BC = EF = y, and angle A = angle D = θ, we have AC = DF = c^*, which allows us to simplify the equations:
c^*^2 = x^2 + y^2 - 2xy cos(θ)
f^*^2 = x^2 + y^2 - 2xy cos(θ)
For triangle ABC and DEF to be congruent, c^* must equal f^*^, which gives us:
x^2 + y^2 - 2xy cos(θ) = x^2 + y^2 - 2xy cos(θ)
Simplifying further, we find the condition under which triangle ABC and DEF are congruent using the SSA condition:
cos(θ) = 1
Therefore, the condition that x, y, and θ must satisfy in order to deduce congruence using SSA is that the cosine of θ must be equal to 1.
2. Law of Sines:
The Law of Sines states that for any triangle ABC with sides a, b, and c, and angles A, B, and C opposite their respective sides, the following relationship holds:
a / sin(A) = b / sin(B) = c / sin(C)
If we apply the Law of Sines to our problem, we have triangle ABC with sides AB = DE = x, BC = EF = y, and angle A = angle D = θ. Let c^* represent AC and f^* represent DF.
Applying the Law of Sines to triangle ABC, we have:
x / sin(θ) = y / sin(B)
=> y * sin(θ) = x * sin(B)
=> sin(B) = (y / x) * sin(θ)
Applying the Law of Sines to triangle DEF, we have:
x / sin(θ) = y / sin(B')
=> y * sin(θ) = x * sin(B')
=> sin(B') = (y / x) * sin(θ)
Since we know that B = B' due to the given conditions, we have sin(B) = sin(B'). Thus:
(y / x) * sin(θ) = (y / x) * sin(θ)
Simplifying further, we find the condition under which triangle ABC and DEF are congruent using the SSA condition:
sin(θ) = sin(θ)
Therefore, the condition that x, y, and θ must satisfy in order to deduce congruence using SSA is that the sine of θ must be equal to itself.
In summary, the SSA condition can only determine congruence between triangles ABC and DEF when either the cosine or the sine of θ is equal to 1 (or when the sine of θ is equal to itself).