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find the measure of arc vw and arc zu. 1 0/1 point numeric arc vw= ___°…

Arc VW= $132^\circ$ Arc ZU= $132^\circ$

Category: geometry Updated: 2026-02-09

Question

find the measure of arc vw and arc zu.
1 0/1 point numeric
arc vw= ___°
× 48
2 0/1 point numeric
arc zu= ___°
× 66

Solution Steps

  1. Understand the question

    find the measure of arc vw and arc zu.
    1 0/1 point numeric
    arc vw= ___°
    × 48
    2 0/1 point numeric
    arc zu= ___°
    × 66

  2. Explanation

    Step1: Identify congruent chords

    Chords $VZ$ and $WU$ are congruent (marked with $\times$), so their corresponding arcs $\overset{\frown}{VZ}$ and $\overset{\frown}{WU}$ are congruent.

    Step2: Use vertical angle property

    $\angle VTZ$ and $\angle WTU$ are vertical angles, so $m\angle VTZ = m\angle WTU = 48^\circ$.

    Step3: Relate central angle to arc

    The measure of a central angle equals its intercepted arc. So $m\overset{\frown}{VZ} = m\angle VTZ = 48^\circ$, thus $m\overset{\frown}{WU} = 48^\circ$.

    Step4: Find arc VW

    $\overset{\frown}{VW}$ and $\overset{\frown}{ZU}$ are intercepted by the same central angle pair? No, use the fact that $\overset{\frown}{VW}$ is supplementary to $\overset{\frown}{VZ}$? No, wait: $VU$ and $ZW$ are diameters, so $\overset{\frown}{VZW}$ is a semicircle ($180^\circ$).
    $m\overset{\frown}{VW} = 180^\circ - m\overset{\frown}{WU} = 180^\circ - 48^\circ = 132^\circ$.

    Step5: Find arc ZU

    $\overset{\frown}{ZU}$ is supplementary to $\overset{\frown}{VZ}$ in the semicircle $\overset{\frown}{VZU}$ ($180^\circ$):
    $m\overset{\frown}{ZU} = 180^\circ - m\overset{\frown}{VZ} = 180^\circ - 48^\circ = 132^\circ$? No, wait, correct: The central angle for $\overset{\frown}{ZU}$ is supplementary to $\angle VTZ$? No, $\angle VZU$ is an inscribed angle? No, $VU$ is a diameter, so $\angle VZU$ is a right angle? No, chords $VZ=WU$, so triangles $VTZ$ and $WTU$ are congruent. The arc opposite the central angle: $\overset{\frown}{VW}$ is intercepted by $\angle VTW$, which is supplementary to $\angle VTZ$:
    $m\angle VTW = 180^\circ - 48^\circ = 132^\circ$, so $m\overset{\frown}{VW} = 132^\circ$.
    For $\overset{\frown}{ZU}$, the central angle is $\angle ZTU = 180^\circ - 48^\circ = 132^\circ$, so $m\overset{\frown}{ZU} = 132^\circ$. Wait no, no: $\angle ZTU$ is a straight angle? No, $VU$ and $ZW$ are intersecting diameters, so $\angle VTZ = 48^\circ$, $\angle ZTU = 180^\circ - 48^\circ = 132^\circ$, so $\overset{\frown}{ZU} = 132^\circ$, and $\angle VTW = 132^\circ$, so $\overset{\frown}{VW} = 132^\circ$.

  3. Final answer

    Arc VW= $132^\circ$
    Arc ZU= $132^\circ$

Answer

Explanation

Step1: Identify congruent chords

Chords $VZ$ and $WU$ are congruent (marked with $\times$), so their corresponding arcs $\overset{\frown}{VZ}$ and $\overset{\frown}{WU}$ are congruent.

Step2: Use vertical angle property

$\angle VTZ$ and $\angle WTU$ are vertical angles, so $m\angle VTZ = m\angle WTU = 48^\circ$.

Step3: Relate central angle to arc

The measure of a central angle equals its intercepted arc. So $m\overset{\frown}{VZ} = m\angle VTZ = 48^\circ$, thus $m\overset{\frown}{WU} = 48^\circ$.

Step4: Find arc VW

$\overset{\frown}{VW}$ and $\overset{\frown}{ZU}$ are intercepted by the same central angle pair? No, use the fact that $\overset{\frown}{VW}$ is supplementary to $\overset{\frown}{VZ}$? No, wait: $VU$ and $ZW$ are diameters, so $\overset{\frown}{VZW}$ is a semicircle ($180^\circ$).
$m\overset{\frown}{VW} = 180^\circ - m\overset{\frown}{WU} = 180^\circ - 48^\circ = 132^\circ$.

Step5: Find arc ZU

$\overset{\frown}{ZU}$ is supplementary to $\overset{\frown}{VZ}$ in the semicircle $\overset{\frown}{VZU}$ ($180^\circ$):
$m\overset{\frown}{ZU} = 180^\circ - m\overset{\frown}{VZ} = 180^\circ - 48^\circ = 132^\circ$? No, wait, correct: The central angle for $\overset{\frown}{ZU}$ is supplementary to $\angle VTZ$? No, $\angle VZU$ is an inscribed angle? No, $VU$ is a diameter, so $\angle VZU$ is a right angle? No, chords $VZ=WU$, so triangles $VTZ$ and $WTU$ are congruent. The arc opposite the central angle: $\overset{\frown}{VW}$ is intercepted by $\angle VTW$, which is supplementary to $\angle VTZ$:
$m\angle VTW = 180^\circ - 48^\circ = 132^\circ$, so $m\overset{\frown}{VW} = 132^\circ$.
For $\overset{\frown}{ZU}$, the central angle is $\angle ZTU = 180^\circ - 48^\circ = 132^\circ$, so $m\overset{\frown}{ZU} = 132^\circ$. Wait no, no: $\angle ZTU$ is a straight angle? No, $VU$ and $ZW$ are intersecting diameters, so $\angle VTZ = 48^\circ$, $\angle ZTU = 180^\circ - 48^\circ = 132^\circ$, so $\overset{\frown}{ZU} = 132^\circ$, and $\angle VTW = 132^\circ$, so $\overset{\frown}{VW} = 132^\circ$.

Answer

Arc VW= $132^\circ$
Arc ZU= $132^\circ$

Question Image

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Question Analysis

Subject mathematics
Sub Subject geometry
Education Level high school
Difficulty unspecified
Question Type with image, calculation
Multi Question No
Question Count 1
Analysis Status completed
Analyzed At 2026-02-09T20:31:46

OCR Text

Show OCR extraction 
find the measure of arc vw and arc zu.
1 0/1 point numeric
arc vw= ___°
× 48
2 0/1 point numeric
arc zu= ___°
× 66

Related Topics

mathematicsgeometrywith image, calculationhigh schoolturns-1

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