Mathematics · Plane geometry · Grade 7

Congruent triangles

Why is a bridge truss made of triangles? How much information fixes a triangle?

The situation

A bridge truss, the guy wires of a pole and a gate railing are all assembled from triangles. Push a hinged quadrilateral and it leans over, but a triangle does not budge: there is only one triangle with three given sides. How much information fixes a triangle completely — two sides and an angle, a side and two angles, or three sides? These are the congruence criteria; with them you can even measure the width of a river without crossing it.

Topic

Triangles: types, elements, congruence criteria, properties of an isosceles triangle. A bridge truss and the guy wires of a pole: why is a triangle rigid? Congruence criteria — how much information fixes a triangle

Key ideas

Congruence criteria
Isosceles triangle
Hardness

Formulas

△ A B C = △ A_{1} B_{1} C_{1}

Congruent triangles are triangles that coincide when placed one on top of the other: their corresponding sides and corresponding angles are equal.

A B = A_{1} B_{1}, A C = A_{1} C_{1}, ∠ A = ∠ A_{1} \Rightarrow △ A B C = △ A_{1} B_{1} C_{1}

The first criterion: by two sides and the angle between them (SAS).

A B = A_{1} B_{1}, ∠ A = ∠ A_{1}, ∠ B = ∠ B_{1} \Rightarrow △ A B C = △ A_{1} B_{1} C_{1}

The second criterion: by a side and the two angles adjacent to it (ASA).

A B = A_{1} B_{1}, B C = B_{1} C_{1}, A C = A_{1} C_{1} \Rightarrow △ A B C = △ A_{1} B_{1} C_{1}

The third criterion: by three sides (SSS) — which is why a triangle is rigid.

A B = B C \Rightarrow ∠ A = ∠ C

In an isosceles triangle the base angles are equal; the bisector drawn from the apex is also a median and an altitude.

Theory

Elements and types

The triangle has vertices $A, B, C$ , sides $a = B C$ , $b = A C$ , $c = A B$ (each side is named after the letter of the opposite vertex) and angles $α, β, γ$ . Its perimeter is $P = a + b + c$ . By sides: scalene, isosceles (two equal sides, the legs; the third is the base), equilateral (all three sides equal). By angles: acute (all three angles less than $90^{\circ}$ ), right-angled (one angle is $90^{\circ}$ : the legs and the hypotenuse), obtuse (one angle greater than $90^{\circ}$ ). In the first step, move the vertices — the name of the type changes.

Congruent triangles and the first criterion

Two triangles are called congruent if they coincide when placed one on top of the other — then their corresponding sides and corresponding angles are equal. We do not have to check everything: the first criterion says that triangles with two sides and the angle between them correspondingly equal are congruent. The idea of the proof is superposition: put $A_{1}$ on $A$ and the ray $A_{1} B_{1}$ along $A B$ ; since $A_{1} B_{1} = A B$ , the point $B_{1}$ falls on $B$ ; since the angles are equal, the ray $A_{1} C_{1}$ falls along $A C$ , and from $A_{1} C_{1} = A C$ the point $C_{1}$ falls on $C$ . The third step shows exactly this movement.

Properties of an isosceles triangle

If $A B = B C$ , then $B$ is the apex and $A C$ is the base. The base angles are equal: $∠ A = ∠ C$ . Proof: draw the bisector $B D$ from $B$ ; in $△ A B D$ and $△ C B D$ we have $A B = C B$ , $B D$ is common, $∠ A B D = ∠ C B D$ — by the first criterion they are congruent, so $∠ A = ∠ C$ . The same congruence also gives $A D = D C$ and $∠ A D B = ∠ C D B = 90^{\circ}$ : the bisector is also a median and an altitude. The converse is true as well: a triangle with two equal angles is isosceles. In an equilateral triangle all three angles are $60^{\circ}$ .

The second and third criteria

The second criterion: triangles with a side and the two angles adjacent to it correspondingly equal are congruent. Why: the rays leaving the ends of the side $A B$ at the given angles meet at only one point — the third vertex is there. The third criterion: triangles with three sides correspondingly equal are congruent: arcs of radius $b$ about $A$ and radius $a$ about $B$ meet at one point on one side of $A B$ (the other is its mirror image). That is why a triangle is rigid: a quadrilateral on hinges leans over, a triangle does not. If the arcs do not meet ( $a + b \leq c$ or $| a - b | \geq c$ ), there is no triangle — this is the triangle inequality (the “Angles of a triangle” experiment).

Application: the width of a river

We need the distance from a point $A$ on one bank to a tree $B$ on the opposite bank. Along the bank we mark the points $O$ and $C$ so that $A O = O C$ , then walk from $C$ perpendicular to the bank and stop at the point $D$ where the stake $O$ and the tree $B$ are in line. In $△ A O B$ and $△ C O D$ : $A O = O C$ , $∠ A = ∠ C = 90^{\circ}$ , $∠ A O B = ∠ C O D$ (vertically opposite angles) — they are congruent by the second criterion, so $A B = C D$ . We measure $C D$ on dry land.

About the model

In the first step the vertices “snap” to grid points with whole-number coordinates and the angles are shown rounded to the degree — so an equilateral triangle cannot be built exactly on the grid (it is shown in the animation). Superposition is a visual model; in Grade 10 the criteria are proved strictly from the axioms (the logical structure of geometry). Two sides and an angle that is not between them do not fix a triangle — there is no such “criterion”. In the river problem the bank is taken to be a straight line.

Did you know?

In Euclid’s “Elements” (3rd century BC) the congruence criteria for triangles are Theorems 4, 8 and 26 of Book I; Theorem 5, on the isosceles triangle, was nicknamed the “bridge of asses” (pons asinorum) in the Middle Ages.
The most common type of bridge truss is the Warren truss, patented in 1848 by the British engineers James Warren and Willoughby Monzani; the truss consists of a row of equilateral or isosceles triangles.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.