Try to prove that △ XWZ and △ XQY are congruent. To do that, use the definition of the midpoint of a segment and look for a common angle between the triangles.
See solution.
Practice makes perfect
We begin by separating △ XWZ and △ XQY from the given diagram. Also, we will mark the congruent segments XY and XZ.
Notice that ∠ X is a common angle for both triangles, and by the Reflexive Property of Congruent Segments we get ∠ X ≅ ∠ X. By the definition of the midpoint of a segment, we obtain two important equations.
XW = 1/2XY and XQ = 1/2XZ
Since XY ≅ XZ, we get XY=XZ. Let's substitute this into the left-hand side equation.
XW = 1/2XZ and XQ = 1/2XZ
⇓
XW = XQ
From the above information, we conclude that XW ≅ XQ.
Given: & XY ≅ XZ, W is the midpoint of XY
& Q is the midpoint of XZ
Prove: & WZ ≅ QY
Proof: Since XY ≅ XZ then XY=ZX. Also, XW = 12XY and XQ= 12XZ by the definition of a midpoint. Substituting XY=ZX, we get XW = 12XZ=XQ and therefore XW ≅ XQ. By the Reflexive Property of Congruent Angles we get ∠ X ≅ ∠ X.
