Question

YX is a perpendicular bisector to WX at X between W and Z. Angle ZWY = Angle WZY. By which of the five congruence statements, HL, AAS, ASA, SAS, and SSS, can you conclude that angle WXY = ZXY
A. HL and ASA
B. All five
C. HL and AAS
D. HL, AAS, and ASA

Answer 1

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.

Answer 2

The answer is A, by process of elimination.