please help me to solve

Question

In ⊙W, m∠QWS = 80, QS¯¯¯¯¯¯¯ ≅ TV¯¯¯¯¯¯¯, and QR¯¯¯¯¯¯¯¯ ≅ UV¯¯¯¯¯¯¯¯. Complete each statement.

m∠UWV≅∠ -----------

If  = 50, then  = -------------

Segment UT ≅ Segment -----------------------

Answer 1

Hi,

We need to use the following 4 points to solve this problem :

       Pt 1.  Congruent central angles have congruent chords.

       Pt 2.   Congruent chords have congruent arcs.

       Pt 3.   Congruent arcs have congruent central angles.

       Pt 4.   A minor arc measure is the measure of its central angle.

   m ∠ QWS  =   80                       -----   Given Statement 1 (G1)

segment QS  ≅ segment TV         -----  Given Statement 2 (G2)

segment QR ≅ segment UV          -----  Given Statement 3 (G3)

Solution - Part 1 :   ( ∠ UWV ≅ _________ )

segment QR ≅ segment UV           -----  G3

                  arc QR  ≅  arc UV                   ------ Pt 2 

                 ∠ QWR ≅ ∠UWV                     ------ Pt 3

                  ∠ UWV ≅ ∠ QWR  

Solution - Part 2 :   ( If a measure of arc RS =  50 then  measure of arc UV   = ______ )

 

            m∠ RWS = 50                 -----  Pt 4 & measure of arc RS =  50 

           m ∠ QWR + m ∠RWS = m ∠ QWS    ----- Angle addition postulate

            m ∠ QWR + 50  = 80                    -----  Substitute known values

           m ∠ QWR = 30                    ------ Subtract 50 from both sides

            m∠ UWV = 30                     ------ ∠ UWV ≅ ∠ QWR - part 1 answer

           measure of arc UV  = 30                   ------ Pt 4

Solution - Part 3 :   ( Segment UT ≅ Segment ________ )

           segment QS  ≅  segment TV      ----   G2

                   arc QS  ≅ arc TV                ----  Pt 2 

                 ∠ QWS ≅ ∠VWT                ----  Pt 3

             m∠ QWS  = m ∠ VWT = 80 ---- Definition of congruent angles & G1

             m ∠ UWV = 30      ---- From part 2

           m ∠ UWT + m ∠UWV = m ∠ VWT   ------ Angle addition postulate

            m ∠ UWT + 30 = 80                         -----  Substitute known values

            m ∠ UWT  = 50                         ------ Subtract 30 from both sides

            m ∠ UWT = m∠ RWS              ------ m∠ RWS = 50 - given in part 2

            segment UT ≅ segment RS         ------ Pt 1   

Hope this helps!