Welcome to our lesson on similar triangles. Similar triangles are triangles that have the same shape but different sizes, just like scaled maps or photographs. Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. Let's explore this concept with visual examples.
The first criterion for triangle similarity is AA, or Angle-Angle similarity. If two angles of one triangle are equal to two angles of another triangle, then the triangles are similar. This works because if two angles are equal, the third angle must also be equal since all angles in a triangle sum to one hundred eighty degrees. Let's see this with an example where angle A equals angle A prime and angle B equals angle B prime.
The second criterion is SAS, or Side-Angle-Side similarity. If two sides of one triangle are proportional to two sides of another triangle, and the included angles are equal, then the triangles are similar. This means the ratio of corresponding sides AB to A prime B prime equals the ratio of AC to A prime C prime, and angle A equals angle A prime. The included angle is the angle between the two proportional sides.
The third criterion is SSS, or Side-Side-Side similarity. If all three corresponding sides of two triangles are proportional, then the triangles are similar. This means the ratio of AB to A prime B prime equals the ratio of BC to B prime C prime, which equals the ratio of AC to A prime C prime. This constant ratio is called the scale factor. In our example, the scale factor is two, as each side of the larger triangle is twice the length of the corresponding side in the smaller triangle.
To summarize what we have learned about similar triangles: Similar triangles have equal corresponding angles and proportional corresponding sides. We can prove similarity using three criteria: AA similarity with two equal angles, SAS similarity with two proportional sides and an equal included angle, or SSS similarity with all three sides proportional. These concepts are essential tools in geometry and have many real-world applications including indirect measurement and scaling problems.