Congruent Triangle Proof
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Theory
A congruence proof is a short list: one fact per line with its reason, then the congruence statement with the test, then what follows from it, such as \(PA = PB\) (matching sides of congruent triangles). This Year 8 maths page shows how.
Three facts, then the test, then the conclusion. Give exactly three matching facts, each with its reason, name the test (SSS, SAS, AAS or RHS), then draw the conclusion asked for.
Reasons you may use without being told: common side (a side in both triangles), common angle, vertically opposite angles, alternate angles on parallel lines (“alternate angles, \(AB \parallel DC\)”), radii of a circle are equal, a midpoint cuts a segment in half, a bisector cuts an angle in half, and any property of a named shape.
Given facts come from the question or from marks on the figure: ticks for equal sides, arcs for equal angles, a square for a right angle.
The conclusion. Once the triangles are congruent, every pair of matching parts is equal: “matching (corresponding) sides of congruent triangles” or “matching angles of congruent triangles”.
Write the vertices in matching order. The conclusion is read straight off that order, so swapping letters changes what the statement says.
A proof set out in full (the figure on the Diagrams tab):
| Statement | Reason |
|---|---|
| \(AM = BM\) | \(M\) is the midpoint of \(AB\) |
| \(\angle PMA = \angle PMB\) | both \(90^\circ\), \(PM \perp AB\) |
| \(PM = PM\) | common side |
| \(\triangle PMA \equiv \triangle PMB\) | SAS |
| \(PA = PB\) | matching sides of congruent triangles |
Writing a congruence proof
- Name the two triangles: “In \(\triangle \ldots\) and \(\triangle \ldots\):”.
- List three matching facts, one per line, each with a reason.
- Write the congruence statement with matching vertices in order, and the test in brackets.
- Conclude the required fact: matching sides or angles of congruent triangles.
| Statement | Reason |
|---|---|
| \(PR = XZ\) | given (hypotenuses) |
| \(\angle PQR = \angle XYZ\) | both \(90^\circ\) (given) |
| \(QR = YZ\) | given |
| \(\triangle PQR \equiv \triangle XYZ\) | RHS |
\(\triangle PQR \equiv \triangle XYZ\) (RHS).
| Statement | Reason |
|---|---|
| \(PR = TR\) | given |
| \(QR = SR\) | given |
| \(\angle PRQ = \angle TRS\) | vertically opposite angles |
| \(\triangle PRQ \equiv \triangle TRS\) | SAS |
The angle at \(R\) is between the two given sides in each triangle, so the test is SAS.
In \(\triangle TUV\) and \(\triangle TWV\):
| Statement | Reason |
|---|---|
| \(TU = TW\) | given |
| \(\angle UTV = \angle WTV\) | \(TV\) bisects \(\angle UTW\) |
| \(TV = TV\) | common side |
| \(\triangle TUV \equiv \triangle TWV\) | SAS |
| \(UV = WV\) | matching sides of congruent triangles |
\(WY\) is opposite both right angles, so it is the hypotenuse of each triangle. In \(\triangle WXY\) and \(\triangle WZY\):
| Statement | Reason |
|---|---|
| \(\angle WXY = \angle WZY\) | both \(90^\circ\) |
| \(WY = WY\) | common side (hypotenuse) |
| \(XY = ZY\) | given |
| \(\triangle WXY \equiv \triangle WZY\) | RHS |
| \(WX = WZ\) | matching sides of congruent triangles |
The matching vertices are \(A \leftrightarrow D\), \(B \leftrightarrow F\), \(C \leftrightarrow E\).
The statement should be \(\triangle ABC \equiv \triangle DFE\) (SAS). Jai’s order would claim \(AC = DF\), which he was never given; the correct order gives \(AC = DE\).
\(TB = TC\), so \(\angle TBC = \angle TCB\) (base angles of an isosceles triangle). Taking each from \(180^\circ\) gives \(\angle TBA = \angle TCD\) (angles on a straight line). In \(\triangle TBA\) and \(\triangle TCD\):
| Statement | Reason |
|---|---|
| \(TB = TC\) | given |
| \(\angle TBA = \angle TCD\) | shown above |
| \(BA = CD\) | given |
| \(\triangle TBA \equiv \triangle TCD\) | SAS |
| \(TA = TD\) | matching sides of congruent triangles |
Common pitfalls
Frequently asked questions
How do you write a congruence proof?
Name the two triangles, list three matching facts with reasons, write the congruence statement and the test, then state what follows.
What is a common side?
A side shared by both triangles; its reason is “common side”.
What reason do I give for angles where two lines cross?
Vertically opposite angles are equal.
How do I prove two sides are equal using congruence?
Prove the triangles that contain them are congruent, then use “matching sides of congruent triangles”.
Why does the order of the vertices matter in a proof?
The conclusion is read from the statement, so the matching vertices must be in the same positions.