Geometric Proofs with Congruent Triangles

A proof is a chain of statements, each one justified, leading from what you are given to what you must show. What counts as a justification is limited: facts you were given, marks on the diagram, named theorems like the angle sum of a triangle, and things you proved earlier in the same proof. What does not count is 'they look equal', 'it is obvious', or measuring the diagram with a ruler. Diagrams are not drawn to scale and are not evidence, so two sides that look equal are only equal if you are told or you prove it. In practice, marks lost in this topic are almost always missing reasons rather than wrong mathematics.

Here is the layout to copy every time. Start by naming both triangles: in triangle ABD and triangle CBD. Then give three statements, each with a reason. AB equals CB, given. AD equals CD, given. And BD is common, meaning the two triangles share that side, so it is equal to itself. Then conclude: therefore triangle ABD is congruent to triangle CBD, by SSS — and naming the test is compulsory. That third line about the common side is worth noticing, because students frequently miss it and end up one fact short. Any time two triangles share a side or an angle, that is a free equality you are entitled to use.

Most students get two facts easily and then stall on the third. It is nearly always one of five things. A common side or angle, where the triangles share a part. Vertically opposite angles, wherever two lines cross. Alternate or corresponding angles, wherever the diagram has parallel arrows. A midpoint, which cuts a segment into two equal halves. Or a given right angle, marked by the small square. So when you are stuck with only two facts, run down that list: look for a shared part, a crossing, parallel arrows, a midpoint, or a right angle. One of them is almost certainly sitting in the diagram waiting to be named.

Usually congruence is the middle of a proof rather than the end. Take the classic result that the base angles of an isosceles triangle are equal. Draw AM from the apex to the midpoint of BC. In triangles ABM and ACM: AB equals AC because the triangle is isosceles; BM equals CM because M is the midpoint; and AM is common. So the triangles are congruent by SSS. Then comes the line that answers the actual question: therefore angle ABM equals angle ACM, because matching angles of congruent triangles are equal. That final line needs its own reason, and stopping at the congruence statement leaves the question unanswered.

Here is the whole method in six steps. One: mark everything you are given onto the diagram, ticks for equal sides and arcs for equal angles. Two: identify the two triangles and write both in matching vertex order. Three: list the equal parts you can justify — given, common, midpoint, parallel lines, vertically opposite. Four: stop at three and check they fit a valid test. Five: conclude with the congruence symbol and name the test. Six: add the line that answers the question, with its own reason. Step one does most of the work, because a fully marked diagram nearly writes the proof for you, and step six is the one most often forgotten.