Answer with explanation:
Given
→Y X is a perpendicular bisector to W X at X between W and Z.
→∠Z W Y= ∠ W Z Y
To Prove
→→ ∠ W X Y ≅ ∠ Z X Y
Proof
In Δ W X Y and Δ Z X Y
1.
→ W Y=Y Z------[Y X is Perpendicular Bisector,so distance from any point on the perpendicular bisector to both sides of ends of line segment is always equal]--------(1)
→∠Y W X= ∠ Y Z X--------[Given]
→ W X=X Z------[Perpendicular bisector divides a line segment into two equal parts]----------(2)
⇒Δ W X Y ≅ Δ Z X Y-------[S AS]
2.
As, X Y is a Perpendicular Bisector.
→∠ Y XW=∠ YX Z=90°
→YW=YZ------Reason same as 1
→WX=XZ-----Reason same as 2
⇒Δ W X Y ≅ Δ Z X Y-------[H L]----If in a right triangle hypotenuse and one side of right triangle is equal to Hypotenuse and other side the two triangles are congruent.]
3.
→Side Y X is common.
As, X Y is a Perpendicular Bisector.
→∠ Y X W=∠ Y X Z=90°
→ ∠ Z WY = ∠ W Z Y-----[Given]
⇒Δ W X Y ≅ Δ Z X Y-------[ A AS]
Since sum of three angles of Triangle is 180°
So, →∠W Y X= ∠ Z Y X as well as →∠ Z WY = ∠ W Z Y and →W Y=Y Z
⇒Δ W X Y ≅ Δ Z X Y-------[ A S A]
4.
→Side Y X is common.
As, X Y is a Perpendicular Bisector.
→YW=YZ------Reason same as 1
→WX=XZ-----Reason same as 2
⇒Δ W X Y ≅ Δ Z X Y-------[ S S S]
Option B:→→ All five can be used to prove ∠ W X Y ≅ ∠ Z X Y,that is by C PCT = Corresponding part of congruent triangles.
