1. The Converse of the Pythagorean Theorem 9-3
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry

2. Classify each triangle by its angle measures. 1. 2.
9.3 The Converse of the Pythagorean Theorem
Warm Up
Classify each triangle by its angle measures.
3. Simplify
4. If a = 6, b = 7, and c = 12, find a2 + b2 and find c2. Which value is greater?
acute
right 12
85; 144; c2

3. 9.3 The Converse of the Pythagorean Theorem
Objectives
Use the Converse of the Pythagorean Theorem and its converse to solve problem.
Use side lengths to classify triangles by their angle measures.

4. Vocabulary Converse Inequalities
9.3 The Converse of the Pythagorean Theorem
Vocabulary
Converse
Inequalities

5. 9.3 The Converse of the Pythagorean Theorem
Activity

6. 9.3 The Converse of the Pythagorean Theorem
The converse of the Pythagorean Theorem gives you a way to tell if a triangle is a right triangle when you know the side lengths.

7. Example 1: Verifying Right Triangles
9.3 The Converse of the Pythagorean Theorem
Example 1: Verifying Right Triangles
The triangle at the right appears to be a right triangle.
Tell whether it is a right triangle.

8. 9.3 The Converse of the Pythagorean Theorem
Example 1: Continue
We see if the given lengths satisfy the Pythagorean Theorem:

9. Example 2: Verifying Right Triangles
9.3 The Converse of the Pythagorean Theorem
Example 2: Verifying Right Triangles
The triangle at the right appears to be a right triangle.
Tell whether it is a right triangle.

10. 9.3 The Converse of the Pythagorean Theorem
Example 2: Continue
We see if the given lengths satisfy the Pythagorean Theorem:

11. 9.3 The Converse of the Pythagorean Theorem
You can also use side lengths to classify a triangle as acute or obtuse. A B C c b a

12. 9.3 The Converse of the Pythagorean Theorem
To understand why the Pythagorean inequalities are true, consider ∆ABC.

13. 9.3 The Converse of the Pythagorean Theorem
By the Triangle Inequality Theorem, the sum of any two side lengths of a triangle is greater
than the third side length.
Remember!

14. Example 3A: Classifying Triangles
9.3 The Converse of the Pythagorean Theorem
Example 3A: Classifying Triangles
Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.
5, 7, 10
Step 1 Determine if the measures form a triangle.
By the Triangle Inequality Theorem, 5, 7, and 10 can be the side lengths of a triangle.

15. 9.3 The Converse of the Pythagorean Theorem
Example 3A Continued
Step 2 Classify the triangle.
c2 = a2 + b2 ?
Compare c2 to a2 + b2.
102 = ?
Substitute the longest side for c.
100 = ?
Multiply.
100 > 74
Add and compare.
Since c2 > a2 + b2, the triangle is obtuse.

16. Example 3B: Classifying Triangles
9.3 The Converse of the Pythagorean Theorem
Example 3B: Classifying Triangles
Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.
5, 8, 17
Step 1 Determine if the measures form a triangle.
Since = 13 and 13 > 17, these cannot be the side lengths of a triangle.

17. 9.3 The Converse of the Pythagorean Theorem
Check It Out! Example 4a
Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.
7, 12, 16
Step 1 Determine if the measures form a triangle.
By the Triangle Inequality Theorem, 7, 12, and 16 can be the side lengths of a triangle.

18. Check It Out! Example 4a Continued
9.3 The Converse of the Pythagorean Theorem
Check It Out! Example 4a Continued
Step 2 Classify the triangle.
c2 = a2 + b2 ?
Compare c2 to a2 + b2.
162 = ?
Substitute the longest side for c.
256 = ?
Multiply.
256 > 193
Add and compare.
Since c2 > a2 + b2, the triangle is obtuse.

19. 9.3 The Converse of the Pythagorean Theorem
Check It Out! Example 4b
Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.
11, 18, 34
Step 1 Determine if the measures form a triangle.
Since = 29 and 29 > 34, these cannot be the sides of a triangle.

20. 9.3 The Converse of the Pythagorean Theorem
Check It Out! Example 4c
Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.
3.8, 4.1, 5.2
Step 1 Determine if the measures form a triangle.
By the Triangle Inequality Theorem, 3.8, 4.1, and 5.2 can be the side lengths of a triangle.

21. Check It Out! Example 4c Continued
9.3 The Converse of the Pythagorean Theorem
Check It Out! Example 4c Continued
Step 2 Classify the triangle.
c2 = a2 + b2 ?
Compare c2 to a2 + b2.
5.22 = ?
Substitute the longest side for c.
27.04 = ?
Multiply.
27.04 < 31.25
Add and compare.
Since c2 < a2 + b2, the triangle is acute.

22. 9.3 The Converse of the Pythagorean Theorem
Lesson Quiz: Part I
1. Decide if the given is a right triangle.
2. Do the lengths 10, 11, 14 provide a right triangle? If not, decide what type of triangle they can make.
Yes
Yes; Acute  since a2 +b2 > c2

23. 9.3 The Converse of the Pythagorean Theorem
Lesson Quiz: Part II
3. Tell if the measures 7, 11, and 15 can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.
yes; obtuse