Similar Triangles
Two triangles are similar if either one of the following conditions is true: 1. Their corresponding angles are equal. 2. Their corresponding sides are proportional. Triangles have a special shortcut for similarity. You only need to prove one of the two conditions. If their angles match, their sides must be proportional. If their sides are proportional, their angles must match. This makes them easier to work with. For example: One triangle has angles 30^\circ and 70^\circ. Another has angles 70^\circ and 80^\circ. The third angle in the first.... This skill is part of Grade 8 math in Yoshiwara Core Math.
Key Concepts
Property Two triangles are similar if either one of the following conditions is true: 1. Their corresponding angles are equal. 2. Their corresponding sides are proportional.
Examples One triangle has angles $30^\circ$ and $70^\circ$. Another has angles $70^\circ$ and $80^\circ$. The third angle in the first is $80^\circ$ and in the second is $30^\circ$. Since all three corresponding angles are equal, the triangles are similar. A triangle has sides of length 5, 12, and 13. A second triangle has sides 10, 24, and 26. The ratios of corresponding sides are all equal to 2 ($\frac{10}{5} = \frac{24}{12} = \frac{26}{13} = 2$), so they are similar. A large triangle is formed by a 20 foot flagpole and its 15 foot shadow. A person who is 6 feet tall stands nearby. Their shadow forms a smaller, similar triangle. The length of the person's shadow, $s$, can be found by the proportion $\frac{6}{s} = \frac{20}{15}$, so $s=4.5$ feet.
Explanation Triangles have a special shortcut for similarity. You only need to prove one of the two conditions. If their angles match, their sides must be proportional. If their sides are proportional, their angles must match. This makes them easier to work with.
Common Questions
What is Similar Triangles?
Two triangles are similar if either one of the following conditions is true: 1. Their corresponding angles are equal. 2.
How does Similar Triangles work?
Example: One triangle has angles 30^\circ and 70^\circ. Another has angles 70^\circ and 80^\circ. The third angle in the first is 80^\circ and in the second is 30^\circ. Since all three corresponding angles are equal, the triangles are similar.
Give an example of Similar Triangles.
A triangle has sides of length 5, 12, and 13. A second triangle has sides 10, 24, and 26. The ratios of corresponding sides are all equal to 2 (\frac{10}{5} = \frac{24}{12} = \frac{26}{13} = 2), so they are similar.
Why is Similar Triangles important in math?
Triangles have a special shortcut for similarity. You only need to prove one of the two conditions.
What grade level covers Similar Triangles?
Similar Triangles is a Grade 8 math topic covered in Yoshiwara Core Math in Chapter 6: Core Concepts. Students at this level study the concept as part of their grade-level standards and are expected to explain, analyze, and apply what they have learned.
What are the key rules for Similar Triangles?
Their corresponding angles are equal. 2. Their corresponding sides are proportional..
What are typical Similar Triangles problems?
One triangle has angles 30^\circ and 70^\circ. Another has angles 70^\circ and 80^\circ. The third angle in the first is 80^\circ and in the second is 30^\circ. Since all three corresponding angles are equal, the triangles are similar.; A triangle has sides of length 5, 12, and 13. A second triangle has sides 10, 24, and 26. The ratios of corresponding sides are all equal to 2 (\frac{10}{5} = \frac{24}{12} = \frac{26}{13} = 2), so they are simila