Decide whether the conclusion is valid. Given, if two angles in a triangle add up to 70 degrees, then the triangle is obtuse, and given, the triangle 𝐴𝐵𝐶 is an obtuse triangle, the conclusion being two of the angles in the triangle 𝐴𝐵𝐶 add up to 70 degrees, is the conclusion valid or not?
First, we need to consider our two given statements. When we’re using if-then statements, we sometimes use the letters p and q to represent the arguments. Here p would be if two angles in a triangle add up to 70 degrees, and q would be the triangle being obtuse.
In conditional statements like these, p is a subset of q. We have a large group q, and within that group q falls the subset p. Our q is obtuse triangles, and our subset, our conditional p, is triangles whose two angles add up to 70 degrees. What we’re not saying is that q, obtuse triangles, and p, triangles whose two angles add up to 70 degrees, fall in the same set. Our first statement models this, p being a subset of q.
Now our second statement says 𝐴𝐵𝐶 is an obtuse triangle. We have no more information than that. In our Venn diagram, we put triangle 𝐴𝐵𝐶 in the larger subset of obtuse angles. Just because triangle 𝐴𝐵𝐶 is an obtuse triangle, that does not automatically make its two angles add up to 70 degrees. We would need more information to determine that. Concluding that two of the angles in triangle 𝐴𝐵𝐶 add up to 70 degrees based on only the information we’re provided is not a valid conclusion.