Geometry Proof Triangles

Question

Show that if two of the corresponding angles of two triangles are equal then so is the third.

Is there a formal way to prove this? I wanted to just say in one sentence that if two angles are the same, then the angles of a triangle must add up to 180, so we are done.

But my professor said that that was insufficient. Please help!

Answer 1

You're pretty much right. To make things slightly more rigorous, suppose that:

$(1)$ The interior angles of one triangle are $a,b,c$.

$(2)$ The interior angles of the other triangle are $x,y,z$.

$(3)$ We have $a = x$ and $b = y$.

We want to prove that $c = z$. Indeed, observe that from $(1)$, we know that: $$a+b+c=180^\circ \tag{4}$$ Similarly, we know from $(2)$ that: $$x + y + z = 180^\circ \tag{5}$$ Hence, it follows that: \begin{align*} c &= (a + b + c) - (a + b) \\ &= 180^\circ - (a+b) & \text{by (4)}\\ &= 180^\circ - (x + y) & \text{by (3)}\\ &= (x + y + z) - (x + y) & \text{by (5)}\\ &= z \end{align*} as desired.