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Answer:

  • ∠ADB = 47°
  • ∠ADC = 137°

Step-by-step explanation:

The angle addition theorem tells you an angle is the sum of its parts:

  ∠ADC = ∠ADB +∠BDC

Angle BDC is marked as a right angle, so has a measure of 90°. Filling known values into the above equation gives ...

  (16x -55)° = (5x -13)° +90°

  11x = 132 . . . . . . . . . divide by °, add 55-5x

  x = 12 . . . . . . . . . . divide by 11

Then the measures of the angles are ...

  ∠ADB = (5x -13)° = (5(12) -13)° = 47°

  ∠ADC = (16x -55)° = (16(12) -55)° = 137° . . . . . . same as 47° +90°

Here <BDC is a right angled triangle i.e (90°) and rest of the two angles are <ADC and <ADB.

Here,

<ADC = (16x - 55)°

<ADB = (5x - 13)°

<BDC = 90°

Now by applying the addition theorem property of triangle we get,

[tex]:\implies\rm{16x - 55 = 5x - 13 + 90}[/tex]

[tex]:\implies\rm{16x - 5x - 55 + 13 = 90}[/tex]

[tex]:\implies\rm{11x - 42 = 90}[/tex]

[tex]:\implies\rm{11x = 90 + 42}[/tex]

[tex]:\implies\rm{11x = 132}[/tex]

[tex]:\implies\rm{x = \frac{132}{11} }[/tex]

[tex]:\implies\rm{x = 12}[/tex]

For angle <ADC = (16x - 55)°

[tex]:\implies\rm{16\times 12 - 55}[/tex]

[tex]:\implies\rm{192 - 55}[/tex]

[tex]:\implies\rm{137}[/tex]

For angle <ADB = (5x - 13)°

[tex]:\implies\rm{5 \times 12 - 13}[/tex]

[tex]:\implies\rm{60 - 13}[/tex]

[tex]:\implies\rm{47}[/tex]

  • The measure of angles are 137° and 47°.

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