Use the Exterior Angle Inequality Theorem. Also, remember that if an angle of a triangle is greater than a second angle, then the side opposite to the greater angle is longer than the side opposite to the lesser angle.
Statements
Reasons
1.
FG⊥ l, FH is any nonperpendicular segment from F to l
1.
Given
2.
∠ 1 and ∠ 2 are right angles
2.
Perpendicular lines form right angles
3.
∠ 1 ≅ ∠ 2
3.
All right angles are congruent
4.
m∠ 1 = m∠ 2
4.
Definition of congruent angles
5.
m∠ 1 > m∠ 3
5.
Exterior Angle Inequality Theorem
6.
m∠ 2 > m∠ 3
6.
Substitution
7.
FH > FG
7.
Theorem 5.10
Practice makes perfect
Let's consider a a line l and FG perpendicular to l. Also, we will consider FH any nonperpendicular segment from F to l.
Since l ⊥ FG, we have that ∠ 1 and ∠ 2 are right angles. Therefore, they are congruent. Additionally, by the Exterior Angle Inequality Theorem we get that m∠ 1 > m∠ 3.
In consequence, m∠ 2> m∠ 3. Now, if an angle of a triangle is greater than a second angle, then the side opposite to the greater angle is longer that the side opposite to the lesser angle (Theorem 5.10).
Two-Column Proof Table
Given:& FG⊥ l, FH is any nonperpendicular
& segment from F to l
Prove:& FH > FG
In the following table, we will summarize the proof we did above.
Statements
Reasons
1.
FG⊥ l, FH is any nonperpendicular segment from F to l
