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Math Olympiad PRMO

Measure of Angle | PRMO-2018 | Problem No-29

Try this beautiful Problem on Trigonometry from PRMO -2018.You may use sequential hints to solve the problem.

Try this beautiful Trigonometry Problem based on Measure of Angle from PRMO -2018.

Measure of Angle – PRMO 2018- Problem 29


Let $D$ be an interior point of the side $B C$ of a triangle ABC. Let $l_{1}$ and $l_{2}$ be the incentres of triangles $A B D$ and $A C D$ respectively. Let $A l_{1}$ and $A l_{2}$ meet $B C$ in $E$ and $F$ respectively. If $\angle B l_{1} E=60^{\circ},$ what is the measure of $\angle C l_{2} F$ in degrees?

,

  • \(25\)
  • \(20\)
  • \(35\)
  • \(30\)
  • \(45\)

Key Concepts


Trigonometry

Triangle

Angle

Suggested Book | Source | Answer


Suggested Reading

Pre College Mathematics

Source of the problem

Prmo-2018, Problem-29

Check the answer here, but try the problem first

\(30\)

Try with Hints


First Hint

Measure of Angle

According to the questations at first we draw the picture . We have to find out the value of

$\angle C l_{2} F$. Now at first find out \(\angle AED\) and \(\angle AFD\) which are the exterioe angles of \(\triangle BEL_1\) and \(\triangle CL_2F\). Now sum of the angles is $180^{\circ}$

Now can you finish the problem?

Second Hint

$\angle E A D+\angle F A D=\angle E A F=\frac{A}{2}$
$\angle A E D=60^{\circ}+\frac{B}{2}$
$\angle A F D=\theta+\frac{C}{2}$
Therefore $\quad \ln \Delta A E F: \frac{A}{2}+60^{\circ}+\frac{B}{2}+\theta+\frac{C}{2}=180^{\circ}$
$90^{\circ}+60^{\circ}+\theta=180^{\circ}$ (as sum of the angles of a Triangle is $180^{\circ}$
Therefore $\quad \theta=30^{\circ}$

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