LK≅JK and m∠ SKL > m∠ QKJ and K is the midpoint of QS
1.
Given
2.
QK ≅ KS
2.
Definition of midpoint
3.
LS > QJ
3.
Hinge Theorem
4.
LS +RL > QJ +RL
4.
Adding RL to both sides
5.
RL ≅ RJ
5.
Given
6.
RL = RJ
6.
Definition of congruent segments
7.
LS +RL > QJ +RJ
7.
Substitution
8.
RS > QR
8.
Segment Addition Postulate
Practice makes perfect
Let's start by highlighting the congruent parts and the given information in the given diagram.
Since K is the midpoint of QS, we have that QK ≅ KS. Remember that we are told that m ∠ SKL > m ∠ QKJ. Next, we apply the Hinge Theorem to △ KSL and △ KJQ.
By adding RL to both sides of the latter inequality we get RL + LS > QJ + RL. However, since RL= RJ we get RL + LS > RJ + QJ. By applying the Segment Addition Postulate we obtain the required inequality.
RL + LS > RJ + QJ
⇒
RS > RQ ✓
Two-Column Proof
In the following table we summarize the proof we did before.
Statements
Reasons
1.
LK≅JK and m∠ SKL > m∠ QKJ and K is the midpoint of QS
