Geometry 02 Evaluate

1. Definition of Similar Triangles

Two triangles are similar if their corresponding angles are congruent and their corresponding sides are in proportion.

  • Congruent Angles: All three pairs of corresponding angles are equal in measure.

    • If \triangle ABC \sim \triangle DEF, then \angle A = \angle D, \angle B = \angle E, and \angle C = \angle F.

  • Proportional Sides: The ratio of the lengths of corresponding sides is constant.

    • If \triangle ABC \sim \triangle DEF, then \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = k, where k is the scale factor.

2. Conditions for Triangle Similarity

To prove that two triangles are similar, you don't need to show that all six conditions (three angles and three side ratios) are met. Specific postulates provide shortcuts.

2.1. AA (Angle-Angle) Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

  • If \angle A = \angle D and \angle B = \angle E, then \triangle ABC \sim \triangle DEF.

2.2. SAS (Side-Angle-Side) Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

  • If \angle A = \angle D and \frac{AB}{DE} = \frac{CA}{FD}, then \triangle ABC \sim \triangle DEF.

2.3. SSS (Side-Side-Side) Similarity Theorem

If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

  • If \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}, then \triangle ABC \sim \triangle DEF.

3. Properties and Applications

  • Scale Factor (k): The ratio of corresponding side lengths of similar triangles. It can be greater than 1, less than 1, or equal to 1. If k=1, the triangles are congruent.

  • Perimeters: The ratio of the perimeters of two similar triangles is equal to the scale factor (k).

    • \frac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of } \triangle DEF} = k

  • Areas: The ratio of the areas of two similar triangles is equal to the square of the scale factor (k^2).

    • \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = k^2

  • Finding Missing Sides: Once similarity is established, the proportionality of sides allows for finding unknown side lengths by setting up and solving proportions.

1. Definition of Similar Triangles

Two triangles are similar if their corresponding angles are congruent and their corresponding sides are in proportion.

  • Congruent Angles: All three pairs of corresponding angles are equal in measure.

    • If \triangle ABC \sim \triangle DEF, then \angle A = \angle D, \angle B = \angle E, and \angle C = \angle F.

  • Proportional Sides: The ratio of the lengths of corresponding sides is constant.

    • If \triangle ABC \sim \triangle DEF, then \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = k, where k is the scale factor.

2. Conditions for Triangle Similarity

To prove that two triangles are similar, you don't need to show that all six conditions (three angles and three side ratios) are met. Specific postulates provide shortcuts.

2.1. AA (Angle-Angle) Similarity Postulate

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

  • If \angle A = \angle D and \angle B = \angle E, then \triangle ABC \sim \triangle DEF.

2.2. SAS (Side-Angle-Side) Similarity Theorem

If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides including these angles are proportional, then the triangles are similar.

  • If \angle A = \angle D and \frac{AB}{DE} = \frac{CA}{FD}, then \triangle ABC \sim \triangle DEF.

2.3. SSS (Side-Side-Side) Similarity Theorem

If the corresponding side lengths of two triangles are proportional, then the triangles are similar.

  • If \frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD}, then \triangle ABC \sim \triangle DEF.

3. Properties and Applications

  • Scale Factor (k): The ratio of corresponding side lengths of similar triangles. It can be greater than 1, less than 1, or equal to 1. If k=1, the triangles are congruent.

  • Perimeters: The ratio of the perimeters of two similar triangles is equal to the scale factor (k).

    • \frac{\text{Perimeter of } \triangle ABC}{\text{Perimeter of } \triangle DEF} = k

  • Areas: The ratio of the areas of two similar triangles is equal to the square of the scale factor (k^2).

    • \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle DEF} = k^2

  • Finding Missing Sides: Once similarity is established, the proportionality of sides allows for finding unknown side lengths by setting up and solving proportions.

4. Example: Proving Similarity with Parallel Lines (AA Similarity)

Consider a large triangle, which we can call \triangle ABC. A line segment, \text{let's label it } DE, is drawn inside \triangle ABC such that point D is on side AB and point E is on side AC. If the segment DE is parallel to the base BC (DE \parallel BC), this configuration creates a smaller triangle, \triangle ADE, at the top vertex.

To prove that \triangle ADE is similar to \triangle ABC:

  • Shared Angle: Both triangles, \triangle ADE and \triangle ABC, share the angle at the top vertex, \angle A. By the Reflexive Property, \angle A \cong \angle A.

  • Corresponding Angles from Parallel Lines: Since the line segment DE is parallel to the base BC (DE \parallel BC), and lines AB and AC act as transversals intersecting these parallel lines:

    • The angle \angle ADE (in the smaller triangle) is a corresponding angle to \angle ABC (in the larger triangle). Therefore, \angle ADE \cong \angle ABC.

    • Similarly, the angle \angle AED (in the smaller triangle) is a corresponding angle to \angle ACB (in the larger triangle). Therefore, \angle AED \cong \angle ACB.

Because we have identified two pairs of congruent angles (e.g., \angle A \cong \angle A and \angle ADE \cong \angle ABC), we can conclude that \triangle ADE \sim \triangle ABC by the AA (Angle-Angle) Similarity Postulate. The specific side lengths provided in such a diagram (e.g., AD=4, AE=6, DE=8, BC=12) can then be used to establish the scale factor (k = \frac{DE}{BC} = \frac{8}{12} = \frac{2}{3}) and find missing side lengths, but the initial proof of similarity relies on the angular relationships due to parallel