Asked by Toxic
Line segment W Y is an altitude in triangle WXZ.
Triangle W X Z is shown. An altitude is drawn from point W to point Y on side Z X, forming a right angle. Angles Z W Y and W X Y are congruent.
If ΔYWZ ~ ΔYXW, what is true about AngleXWZ?
AngleXWZ is an obtuse angle.
AngleXWZ is a right angle.
AngleXWZ is congruent to AngleWXY.
AngleXWZ is congruent to AngleXZW.
All Answers 1
Answered by
GPT-5 mini
AI
Answer: Angle XWZ is a right angle.
Reason: Y is the foot of the altitude, so Y lies on ZX and WY ⟂ ZX; hence ∠Y in both ΔYWZ and ΔYXW is 90°. From the similarity ΔYWZ ~ ΔYXW we get
∠YWZ = ∠WXY = ∠X (big triangle) and ∠YZW = ∠YWX = ∠Z (big triangle).
Thus ∠XWZ (the angle at W) = ∠X + ∠Z. But in triangle WXZ, ∠W + ∠X + ∠Z = 180°, so
∠W = ∠X + ∠Z = 180° − ∠W, giving 2∠W = 180° and ∠W = 90°.
Reason: Y is the foot of the altitude, so Y lies on ZX and WY ⟂ ZX; hence ∠Y in both ΔYWZ and ΔYXW is 90°. From the similarity ΔYWZ ~ ΔYXW we get
∠YWZ = ∠WXY = ∠X (big triangle) and ∠YZW = ∠YWX = ∠Z (big triangle).
Thus ∠XWZ (the angle at W) = ∠X + ∠Z. But in triangle WXZ, ∠W + ∠X + ∠Z = 180°, so
∠W = ∠X + ∠Z = 180° − ∠W, giving 2∠W = 180° and ∠W = 90°.
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