VR≅RT and m∠ SRV > m∠ QRT and R is the midpoint of SQ
1.
Given
2.
SR ≅ QR
2.
Definition of midpoint
3.
SV > QT
3.
Hinge Theorem
4.
VW+SV > VW+QT
4.
Adding VW to both sides
5.
VW ≅ WT
5.
Given
6.
VW = WT
6.
Definition of congruent segments
7.
VW+SV > WT+QT
7.
Substitution
8.
WS > WQ
8.
Segment Addition Postulate
Practice makes perfect
Let's start by highlighting the congruent parts and the given information in the given diagram.
Since R is the midpoint of SQ we have that SR ≅ QR. Additionally, we have that m ∠ SRV > m ∠ QRT. Next, we apply the Hinge Theorem to △ RSV and △ QRT.
By adding VW to both sides of the latter inequality we get VW + SV > QT + VW, but since VW= WT we get VW + SV > WT + QT. By applying the Segment Addition Postulate we obtain the required inequality.
VW + SV > WT + QT
⇒
WS > WQ ✓
Two-Column Proof
In the following table we summarize the proof we did before.
Statements
Reasons
1.
VR≅RT and m∠ SRV > m∠ QRT and R is the midpoint of SQ
