Exercise 46

In the illustration, the sum of angles \( \angle AOB \) and \( \angle COD \) is \( 55^{\circ} \), and the sum of angles \( \angle BOD \) and \( \angle AOC \) is \( 95^{\circ} \). What is the measure of angle \( \angle AOD \)?

  1. \(55^{\circ} \)

  2. \(65^{\circ} \)

  3. \(75^{\circ} \)

  4. \(85^{\circ} \)

  5. \(95^{\circ} \)

Try to solve it before checking the answer.

  1. \(75^{\circ} \)

Indeed; let be:

\( \alpha + \beta = 55 \hspace{4em} \boldsymbol{(1)} \)

\( \theta + \phi = 95 \hspace{4em} \boldsymbol{(2)} \)

\( \theta = \alpha + x \hspace{4em} \boldsymbol{(3)} \)

\( \phi = x + \beta \hspace{4em} \boldsymbol{(4)} \)

Replacing (3) and (4) in (2):

\[ (\alpha + x ) + ( x + \beta ) = 95 \]

then,

\( 2x + ( \alpha + \beta ) = 95 \hspace{4em} \boldsymbol{(5)} \)

Replacing (1) in (5):

\[ \begin{aligned} 2x + 55 = 95 &\Rightarrow 2x = 40 \\[.5em] &\Rightarrow x = 20 \end{aligned} \]

Finally,

\[ \begin{aligned} \angle AOD = \alpha + x + \beta &= (\alpha + \beta) + x \\[.5em] &= 55 + 20 \\[.5em] &= \boldsymbol{75} \end{aligned} \]

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