Congruent Triangle Proofs
A congruence proof sets out three matching facts, each with a reason, and names the test. Once two triangles are congruent, their other matching sides and angles are equal too. This NSW Year 9 Advanced maths subtopic writes these formal proofs (NESA outcome MA5‑GEO‑P‑02).
You will decide whether pairs of triangles are congruent, fill in missing reasons and write full proofs. You will prove properties of shapes, including that the diagonals of a parallelogram bisect each other, the diagonals of a rectangle are equal and the diagonals of a rhombus are perpendicular, and that the perpendicular from a circle's centre bisects a chord. It is part of Year 9 Advanced Congruence and Similar Figures.
Every question with a fully worked solution.
- Congruent Triangle Proofs - Video - Congruent Triangles Geometry Proof Watch
Theory
A congruence proof shows two triangles are congruent by listing three pairs of equal parts, each with a reason, and naming the test. The congruent triangles then prove something else: equal sides, equal angles, parallel lines or perpendicular lines. This Year 9 Advanced maths page is part of the path towards Extension maths.
A proof is a chain of statements. Each statement is either given or follows from a known fact, and each one has a reason in brackets.
A congruence proof has three parts: name the two triangles; list three pairs of equal sides or angles with reasons; write the congruence statement with the test, e.g. \(\triangle ABC \equiv \triangle DEF\) (SAS).
After that, "hence" means use the result just proved. Matching parts of congruent triangles are equal. The reason is written "matching sides of congruent triangles" or "matching angles of congruent triangles".
Useful reasons: common side, vertically opposite angles, alternate angles (parallel lines), radii of a circle, base angles of an isosceles triangle, and the properties of squares, rectangles, rhombuses, parallelograms and kites.
Setting out a congruence proof:
In \(\triangle ABC\) and \(\triangle DEF\):
\(AB = DE\) (reason)
\(\angle ABC = \angle DEF\) (reason)
\(BC = EF\) (reason)
How to write a congruence proof
- Find the triangles: the question usually names them. If not, pick two triangles that contain the parts you want to prove equal.
- Mark the diagram: given facts, then hidden ones: common sides, vertically opposite angles, alternate angles, radii, sides and angles of special shapes.
- Write three pairs of equal parts, each with a reason. Keep the vertices in matching order.
- Conclude with \(\therefore\) the congruence statement and the test.
Proving something else ("hence")
- Equal sides or angles: they are matching parts of the congruent triangles.
- Midpoint or bisector: show the two halves are matching parts.
- Perpendicular: show two matching angles lie on a straight line, so each is \(90^\circ\).
- Parallel: show a pair of alternate angles are matching angles, so they are equal.
- Fill-the-gap questions: each missing reason is one of the standard reasons listed in Definitions.
Prove two triangles congruent, then use matching sides. In \(\triangle PMA\) and \(\triangle PMB\):
\(AM = BM\) (\(M\) is the midpoint of \(AB\))
\(\angle PMA = \angle PMB = 90^\circ\) (given)
\(PM\) is common
\(\therefore\ \triangle PMA \equiv \triangle PMB\) (SAS)
\(\therefore\ \boldsymbol{PA = PB}\) (matching sides of congruent triangles)
The two triangles share no side, so use the base angles of the isosceles triangle. In \(\triangle ABD\) and \(\triangle ACE\):
\(AB = AC\) (given)
\(\angle ABD = \angle ACE\) (base angles of isosceles \(\triangle ABC\))
\(BD = CE\) (given)
\(\therefore\ \triangle ABD \equiv \triangle ACE\) (SAS)
\(\therefore\ \boldsymbol{AD = AE}\) (matching sides of congruent triangles), so \(\triangle ADE\) is isosceles too.
Radii of the same circle are equal. In \(\triangle PXQ\) and \(\triangle PYQ\):
\(PX = PY\) (radii of the circle centre \(P\))
\(QX = QY\) (radii of the circle centre \(Q\))
\(PQ\) is common
\(\therefore\ \triangle PXQ \equiv \triangle PYQ\) (SSS)
\(\therefore\ \angle XPQ = \angle YPQ\) (matching angles of congruent triangles), so \(PQ\) bisects \(\angle XPY\).
(i) Prove that \(\triangle AOB \equiv \triangle COD\).
(ii) Hence prove that \(AB \parallel DC\).
(i) In \(\triangle AOB\) and \(\triangle COD\):
\(AO = CO\) (\(O\) is the midpoint of \(AC\))
\(\angle AOB = \angle COD\) (vertically opposite angles)
\(BO = DO\) (\(O\) is the midpoint of \(BD\))
\(\therefore\ \triangle AOB \equiv \triangle COD\) (SAS)
(ii) \(\angle OAB = \angle OCD\) (matching angles of congruent triangles). These are alternate angles on the transversal \(AC\), and they are equal, so \(\boldsymbol{AB \parallel DC}\).
The same argument with \(\triangle AOD\) and \(\triangle COB\) gives \(AD \parallel BC\), so \(ABCD\) is a parallelogram.
Common pitfalls
Frequently asked questions
How do you write a congruent triangle proof?
Name the two triangles. List three pairs of equal sides or angles, each with a reason. Finish with \(\therefore\), the congruence statement in matching order and the test in brackets.
What reasons can you use in a geometry proof?
Common reasons are given, common side, vertically opposite angles, alternate angles with parallel lines, radii of a circle, base angles of an isosceles triangle, sides or angles of a special quadrilateral, and matching sides or angles of congruent triangles.
What does hence mean in a proof question?
Hence means use the result you have just proved. After proving two triangles congruent, use their matching sides or angles to prove the next part.
How do you prove two lines are perpendicular using congruent triangles?
Show that two angles at the meeting point are matching angles of congruent triangles, so they are equal. They lie on a straight line and add to \(180^\circ\), so each is \(90^\circ\).
How do you prove two lines are parallel?
Show that a pair of alternate angles, or corresponding angles, are equal. Congruent triangles often give these equal angles as matching angles.
What is a common side?
A common side is a side that belongs to both triangles. It is equal to itself, so it gives one pair of equal sides in a congruence proof.