Can you find any similar triangles in the picture below?
If two triangles are similar, then their corresponding angles are congruent and their corresponding sides are proportional. There are many theorems about triangles that you can prove using similar triangles.
- Triangle Proportionality Theorem: A line parallel to one side of a triangle divides the other two sides of the triangle proportionally. This theorem and its converse will be explored and proved in Example A, Example B, and the practice exercises.
- Triangle Angle Bisector Theorem: The angle bisector of one angle of a triangle divides the opposite side of the triangle into segments proportional to the lengths of the other two sides of the triangle. This theorem will be explored and proved in Example C.
- Pythagorean Theorem: For a right triangle with legs and and hypotenuse , . This theorem will be explored and proved in the Guided Practice problems.
Prove that .
Use your result from Example A to prove that . Then, use algebra to show that .
Solution: which means that corresponding sides are proportional. Therefore, . Now, you can use algebra to show that the second proportion must be true. Remember that and .
You have now proved the triangle proportionality theorem: a line parallel to one side of a triangle divides the other two sides of the triangle proportionally.
Consider with the angle bisector of and point constructed so that . Prove that .
Solution: By the triangle proportionality theorem, . Multiply both sides of this proportion by .
Now all you need to show is that in order to prove the desired result.
- Because is the angle bisector of , .
- Because , (corresponding angles).
- Because , (alternate interior angles).
- Thus, by the transitive property.
Therefore, is isosceles because its base angles are congruent and it must be true that . This means that . Therefore:
This proves the triangle angle bisector theorem: the angle bisector of one angle of a triangle divides the opposite side of the triangle into segments proportional to the lengths of the other two sides of the triangle.
Concept Problem Revisited
There are three triangles in this picture: , , . All three triangles are right triangles so they have one set of congruent angles (the right angle). and share , so by . Similarly, and share , so by . By the transitive property, all three triangles must be similar to one another.
Two figures are similar if a similarity transformation will carry one figure to the other. Similar figures will always have corresponding angles congruent and corresponding sides proportional.
The Pythagorean Theorem states that for a right triangle with legs and and hypotenuse , .
An angle bisector divides an angle into two congruent angles.
The large triangle above has sides , , and . Side has been divided into two parts: and . In the Concept Problem Revisited you showed that the three triangles in this picture are similar.
1. Explain why .
2. Explain why .
3. Use the results from #1 and #2 to show that .
1. When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you see that .
2. When triangles are similar, corresponding sides are proportional. Carefully match corresponding sides and you see that .
3. Cross multiply to rewrite each equation. Then, add the two equations together.
You have just proved the Pythagorean Theorem using similar triangles.
Solve for in each problem.
Use the picture below for #8-#10.
8. Solve for .
9. Solve for .
10. Solve for .
Use the picture below for #11-#13.
11. Assume that . Use algebra to show that .
12. Prove that
13. Prove that
14. Prove that a segment that connects the midpoints of two sides of a triangle will be parallel to the third side of the triangle.
15. Prove the Pythagorean Theorem using the picture below.