Asked by kj
Point Z is equidistant from the vertices of ΔTUV.
Point Z is equidistant from the vertices of triangle T U V. Lines are drawn from point Z to the sides of the triangle to form right angles and line segments Z A, Z B, and Z C.
Which must be true?
Line segment T A is-congruent-to line segment T B
Line segment A Z is-congruent-to line segment B Z
AngleBTZ Is-congruent-to AngleBUZ
AngleTZA Is-congruent-to AngleTZB
All Answers 1
Answered by
GPT-5 mini
AI
Answer: Angle BTZ is congruent to Angle BUZ.
Reason: Z is the circumcenter, so its perpendiculars to the sides meet each side at its midpoint. Thus B is the midpoint of segment TV, so TB = UB. Also TZ = UZ (Z is equidistant from the vertices) and BZ is common. Therefore triangles TBZ and UBZ are congruent (SSS), so corresponding angles BTZ and BUZ are equal.
(The other statements need not hold in a general triangle.)
Reason: Z is the circumcenter, so its perpendiculars to the sides meet each side at its midpoint. Thus B is the midpoint of segment TV, so TB = UB. Also TZ = UZ (Z is equidistant from the vertices) and BZ is common. Therefore triangles TBZ and UBZ are congruent (SSS), so corresponding angles BTZ and BUZ are equal.
(The other statements need not hold in a general triangle.)
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