Triangles - Similarity Concepts

Flashcards covering key vocabulary and theorems related to the similarity of triangles.

Similar Figures

Figures are similar if they have the same shape but different sizes. Their corresponding angles are equal, and corresponding sides are in proportion, expressed as a constant ratio $k$, where $k$ is the scale factor.

Congruent Figures

Figures are congruent if they have the same shape and size. All corresponding sides and angles are equal. This means the ratio of corresponding sides is 1, i.e., scale factor $k = 1$. If triangle ABC is congruent to triangle XYZ, it can be written as $\triangle ABC \cong \triangle XYZ$, implying $AB = XY$, $BC = YZ$, $CA = ZX$ and $\angle A = \angle X$, $\angle B = \angle Y$, $\angle C = \angle Z$.

Similar Figures

Figures are similar if they have the same shape but different sizes. Their corresponding angles are equal, and corresponding sides are in proportion, expressed as a constant ratio $k$, where $k$ is the scale factor.

Congruent Figures

Figures are congruent if they have the same shape and size. All corresponding sides and angles are equal. This means the ratio of corresponding sides is 1, i.e., scale factor $k = 1$. If triangle ABC is congruent to triangle XYZ, it can be written as $\triangle ABC \cong \triangle XYZ$, implying $AB = XY$, $BC = YZ$, $CA = ZX$ and $\angle A = \angle X$, $\angle B = \angle Y$, $\angle C = \angle Z$.

Similarity of Polygons

Two polygons with the same number of sides are similar if: (i) their corresponding angles are equal (e.g., $\angle A = \angle X$, $\angle B = \angle Y$), and (ii) their corresponding sides are in the same ratio (e.g., $\frac{AB}{XY} = \frac{BC}{YZ} = k$), where $k$ is the scale factor.

Scale Factor

The scale factor $k$ is the ratio of corresponding sides of similar polygons. If polygon A is similar to polygon B, and the length of a side in A is $LA$ and the length of the corresponding side in B is $LB$, then $k = \frac{LA}{LB}$. If $k > 1$, polygon A is an enlargement of polygon B; if $0 < k < 1$, polygon A is a reduction of polygon B; and if $k = 1$, polygon A is congruent to polygon B.

Equiangular Triangles

Two triangles are equiangular if their corresponding angles are equal. For example, if $\triangle ABC$ and $\triangle XYZ$ are equiangular, then $\angle A = \angle X$, $\angle B = \angle Y$, and $\angle C = \angle Z$.

Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle intersecting the other two sides at distinct points, then it divides the two sides in the same ratio. Given $\triangle ABC$ with line $DE \parallel BC$, where D is on AB and E is on AC, then $\frac{AD}{DB} = \frac{AE}{EC}$.

AAA (Angle-Angle-Angle) Criterion

If in two triangles, corresponding angles are equal, then their corresponding sides are in the same ratio, making the triangles similar. If in $\triangle ABC$ and $\triangle XYZ$, $\angle A = \angle X$, $\angle B = \angle Y$, and $\angle C = \angle Z$, then $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX}$, and $\triangle ABC \sim \triangle XYZ$.

AA Similarity Criterion

If two angles of one triangle are respectively equal to two angles of another triangle, then the two triangles are similar. For triangles $\triangle ABC$ and $\triangle XYZ$, if $\angle A = \angle X$ and $\angle B = \angle Y$, then $\triangle ABC \sim \triangle XYZ$.

SSS (Side-Side-Side) Similarity Criterion

If in two triangles, the sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal, and therefore the triangles are similar. For triangles $\triangle ABC$ and $\triangle XYZ$, if $\frac{AB}{XY} = \frac{BC}{YZ} = \frac{CA}{ZX}$, then $\triangle ABC \sim \triangle XYZ$.

SAS (Side-Angle-Side) Similarity Criterion

If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the two triangles are similar. For triangles $\triangle ABC$ and $\triangle XYZ$, if $\angle A = \angle X$ and $\frac{AB}{XY} = \frac{AC}{XZ}$, then $\triangle ABC \sim \triangle XYZ$.