CK-12 Geometry - Second Edition

8.2. Converse of the Pythagorean Theorem http://www.ck12.org

Proof of Theorem 8-3

Given: In 4 ABC,a^2 +b^2 >c^2 , wherecis the longest side.

In 4 LMN,^6 Nis a right angle.

Prove: 4 ABCis an acute triangle. (all angles are less than 90◦)

TABLE8.1:

Statement Reason

1. In 4 ABC,a^2 +b^2 >c^2 , andcis the longest side. In
   4 LMN,^6 Nis a right angle.

Given

2.a^2 +b^2 =h^2 Pythagorean Theorem

3.c^2 <h^2 Transitive PoE

4.c<h Take the square root of both sides

5.^6 Cis the largest angle in 4 ABC. The largest angle is opposite the longest side.
6.m^6 N= 90 ◦ Definition of a right angle
7.m^6 C<m^6 N SSS Inequality Theorem
8.m^6 C< 90 ◦ Transitive PoE
9.^6 Cis an acute angle. Definition of an acute angle
10. 4 ABCis an acute triangle. If the largest angle is less than 90◦, then all the angles
are less than 90◦.

The proof of Theorem 8-4 is very similar and is in the review questions.

Example 2:Determine if the following triangles are acute, right or obtuse.

a)

b)