Construct CD so that C is between B and D and CD≅ AC
2.
Ruler Postulate
3.
CD=AC
3.
Definition of congruent segments
4.
∠CAD ≅ ∠D
4.
Isosceles Triangle Theorem
5.
m∠CAD = m∠D
5.
Definition of congruent angles
6.
m∠BAD = m∠BAC + m∠CAD
6.
Angle Addition Postulate
7.
m∠BAD = m∠BAC + m∠D
7.
Substitution
8.
m∠BAD > m∠D
8.
Definition of inequality
9.
BD > AB
9.
Theorem 5.10
10.
BD = BC+CD
10.
Segment Addition Postulate
11.
BD = BC+AC
11.
Substitution
12.
BC+AC > AB
12.
Substitution
Practice makes perfect
We need to prove the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides in a triangle is greater than the length of the third side.
To prove it, we will follow the hint and construct an auxiliary segment CD, so that C is between B and D and also CD≅ AC. Also, we will draw AD.
From the second equation, and by definition of inequality, we get m∠BAD > m ∠CAD. Since m ∠CAD = m ∠D we obtain that m∠BAD > m ∠D.
Next, Theorem 5.10 states that if one angle of a triangle has a greater measure than another angle, then the side opposite to the greater angle is longer than the side opposite the lesser angle. Therefore, BD > AB.
By the Segment Addition Postulate we have that BD = DC + CB, and since DC= AC we obtain BD = AC + CB. By substituting this into the latter inequality we will obtain what we wanted to prove.
cl
BD > AB &
⇓ &
AC + CB > AB & ✓
