Chapter 9 — Geometric Twins

It is about congruence — figures that have the same shape and size. The chapter explains what makes two figures congruent, develops five conditions to test congruence of triangles (SSS, SAS, ASA, AAS, RHS), and uses congruence to prove properties of isosceles and equilateral triangles.

Figures that have the same shape and size are said to be congruent. Congruent figures can be superimposed exactly, one over the other. While checking for congruence, a figure may be rotated or flipped before superimposing.

The five conditions that guarantee congruence are: (a) SSS — same three sidelengths, (b) SAS — two sides and the included angle, (c) ASA — two angles and the included side, (d) AAS — two angles and a non-included side, and (e) RHS — in right-angled triangles, equality of the hypotenuse and one other side.

SSS stands for Side Side Side. If two triangles have the same three sidelengths, then they are congruent. This is because all triangles constructed with the same three sidelengths turn out to be congruent to each other.

SAS stands for Side Angle Side. When two sides and the included angle (the angle between those two sides) of one triangle are equal to the corresponding two sides and included angle of another triangle, the triangles are congruent.

SSA stands for Side Side Angle — two sides and a non-included angle. It does not guarantee congruence because two non-congruent triangles can be constructed with the same two sides and a non-included angle. For example, with sides 6 cm and 4 cm and an angle of 30° (not included), the arc from one vertex cuts the opposite ray at two different points, giving two distinct triangles, ΔPQR and ΔPQS, which are not congruent.

ASA (Angle Side Angle) requires that two angles and the side included between them are equal in both triangles. AAS (Angle Angle Side) requires two equal angles and a side that is NOT included between them. The chapter shows that both conditions guarantee congruence, because once two angles are known, the third angle is determined (angles sum to 180°), which converts an AAS situation into an ASA one.

RHS stands for Right Hypotenuse Side. In a right-angled triangle, the side opposite the right angle is called the hypotenuse. The RHS condition says that if two right-angled triangles have equal hypotenuses and one other equal side, then they are congruent.

The notation ΔABC ≅ ΔXYZ means vertex A corresponds to X, vertex B to Y, and vertex C to Z. The order of letters matters: writing ΔACB ≅ ΔXYZ would be incorrect for this correspondence, but ΔACB ≅ ΔXZY would be another correct way to express the same congruence.

No. Two triangles that have the same set of angles need not be congruent. They will have the same shape but can differ in size, so knowing only the three angles is not sufficient to guarantee congruence.

In an isosceles triangle with AB = AC, drawing the altitude from A to BC creates two smaller triangles (ΔADB and ΔADC). These share the side AD, both have a right angle at D, and AB = AC. This satisfies the RHS condition, so ΔADB ≅ ΔADC. Since the triangles are congruent, ∠B = ∠C — the angles opposite the equal sides are equal.

In an equilateral triangle all three sides are equal. Using the property that angles opposite equal sides are equal, AB = AC gives ∠B = ∠C, and AB = BC gives ∠A = ∠C, so all three angles are equal. Since they must sum to 180°, each angle is 180° ÷ 3 = 60°.

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