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.)
Thanks to bobpursley I found the answer to the LAw of cosines, that if theta = 90 then both triangles are equal. But what about the law of sines, and any other ones? Thanks, Help is appreciated
Determine if there are zero, one or two possible triangles. Drae the triangles, if possible, including the unknown measurements.
The question is, In triangle JKL, angle J is 55, side j is 10.4 and side k is 11.6.
To determine the conditions under which triangle ABC is congruent to triangle DEF using the Law of Sines, we will consider the sine rule for triangles. Let's denote the angles opposite sides AB and DE as angle C and angle F, respectively.
According to the Law of Sines, we have:
AB/sin(C) = BC/sin(A) (1)
DE/sin(F) = EF/sin(D) (2)
Since AB = DE = x and BC = EF = y, we can substitute these values into equations (1) and (2):
x/sin(C) = y/sin(A) (3)
x/sin(F) = y/sin(D) (4)
We also know that angle A = angle D = θ, so sin(A) = sin(D) = sin(θ). Substituting these values:
x/sin(C) = y/sin(θ) (5)
x/sin(F) = y/sin(θ) (6)
From equations (5) and (6), we can conclude that in order for triangle ABC to be congruent to triangle DEF, the angles C and F must be equal. Mathematically:
C = F
If C ≠ F, the two triangles will not be congruent.
Therefore, the condition for triangle ABC to be congruent to triangle DEF using the Law of Sines is that the angles C and F must be equal.
In summary, the conditions for triangle ABC to be congruent to triangle DEF when using the Law of Sines are:
- AB = DE = x
- BC = EF = y
- Angle A = angle D = θ
- Angle C = angle F
It's important to note that these conditions apply only when the given angles and sides allow for a unique construction of the triangles. In other cases, when the conditions are not satisfied, triangle ABC and DEF may have multiple possible configurations or may not be congruent.