U18 Congruence and Similarity: Essential Formulas

This section covers the fundamental conditions and properties for congruent and similar triangles, which are crucial for solving geometric proofs and problems in the DSE exam.

1 Conditions for Congruent Triangles

Four Standard Tests (A.S.A., S.A.S., S.S.S., R.H.S.)

Two triangles are congruent if they satisfy any one of the following four conditions. The corresponding sides and angles are equal.

$$\triangle ABC \cong \triangle PQR$$
A B C P Q R ≅

2 Conditions for Similar Triangles

Three Standard Tests (A.A.A., Three sides proportional, Two sides proportional and included angle equal)

Two triangles are similar if they satisfy any one of the following three conditions. The corresponding angles are equal and the corresponding sides are proportional.

$$\triangle ABC \sim \triangle XYZ$$

If $\triangle ABC \sim \triangle XYZ$, then: $$\frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX}$$ and $\angle A = \angle X$, $\angle B = \angle Y$, $\angle C = \angle Z$.

A B C X Y Z ∼

3 Ratio of Areas and Volumes

Area Ratio for Similar Figures

For any two similar plane figures, the ratio of their areas is equal to the square of the ratio of their corresponding lengths.

$$\frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle XYZ} = \left( \frac{AB}{XY} \right)^2 = k^2$$

Volume Ratio for Similar Solids

For any two similar solids, the ratio of their volumes is equal to the cube of the ratio of their corresponding lengths.

$$\frac{\text{Volume of solid A}}{\text{Volume of solid B}} = \left( \frac{l_A}{l_B} \right)^3 = k^3$$

Struggling with complex problems?

Learner App features AI step-by-step analysis technology. Snap a photo and it will guide you through the solution!

Download Learner Now