ASA and AAS Examples

Is it true that ∆ABC ≅ ∆XYZ? How do you know?

We can say that two triangles are congruent if any of the SSS, SAS, ASA, or AAS postulates are satisfied. In this case, we know that two corresponding angles are congruent (∠B ≅ ∠Y and ∠C ≅ ∠Z) and corresponding segments not in between the angles are congruent (AB ≅ XY).

Since all the angles and segments match up to each corresponding location on the triangles, we can say that ∆ABC ≅ ∆XYZ according to AAS.

Is it true that ∆PRS ≅ ∆PTS?

We know that the two triangles have two congruent sides (RS ≅ TS and PS ≅ PS) and one congruent angle (∠R ≅ ∠T).

In looking through our four postulates, the only one with two sides and one angle is Side Angle Side. For SAS to work, we need the angle to be included between the two congruent sides. Unfortunately, that's not the case here.

Since we can't prove that the two triangles are congruent, we can't say whether or not it's true. All we can say is that we'd need more information before we could prove or disprove it.

Is it possible to construct only one possible triangle from two given sides and an angle that isn't included between them?

Basically, this problem calls the SSA idea into question. While it's totally possible, it forms a very special kind of triangle. Let's start with an angle (∠A) and a known side (AB).

If we include another known length from point B down toward our base, we'd be able to orient that segment in two different ways. That would give two possibilities for ∆ABC (with an isosceles triangle contained inside it). See the isosceles triangle?

The only way we wouldn't be able to draw that isosceles triangle is if the length of BC was so short that it only touches the horizontal line in one place. That would form a very specific kind of triangle: a right triangle.