Two triangles are similar if and only if their corresponding angles are congruent and their corresponding sides are proportional.
See solution.
Practice makes perfect
We will investigate whether a circumscribed triangle about a circle and an inscribed triangle in the same circle are similar. Let's begin by circumscribing a triangle about a circle.
Now that we have △ ABC circumscribed about ⊙ O, we will next inscribe △ DEF in ⊙ O using the points of tangency.
The measure of an inscribed angle is one-half the measure of its intercepted arc.
Using this theorem we can determine the interior angles of △ DEF as follows.
m∠ D& = 1/2 EF
m∠ E& = 1/2 DF
m∠ F& = 1/2 DE
Next, to determine the angles of △ ABC we will use the Circumscribed Angle Theorem (Theorem 10.17).
By this theorem and knowing that the measure of a central angle is the measure of its intercepted arc, we can write the interior angles of △ ABC in the terms that follow.
m∠ A& = 180^(∘)-DE
m∠ B& = 180^(∘)-EF
m∠ C& = 180^(∘)-DF
Since we do not have further information about the triangles, we cannot know which angles of the triangles are corresponding. Additionally, no matter how we pair the angles they do not have to be congruent. Therefore, the triangles are not necessarily similar.
