Angle chasing

Algebra Level 2

In right $\triangle ACB$ , $\angle ACB=90^\circ$ , $\angle BAC=25^\circ$ , $CF$ is the median to hypotenuse, $CE$ is the bisector of $\angle ACB$ and $CD$ is the altitude to $AB$ . Which is larger, $\angle DCE$ or $\angle ECF$ ?

They are equal.

\angle DCE

\angle ECF

1 solution

A Former Brilliant Member
Oct 17, 2017

Consider the diagram on the left.

$CF=\dfrac{1}{2}AB=FA$ .

Therefore, $\triangle CFA$ is isosceles with $\angle FCA=\angle FAC=25^\circ$ . Since $CE$ is an angle bisector, we have that $\angle ECA=\angle ECB=45^\circ$ . Then,

$\angle DCE=\angle ECB-25=45-25=20^\circ$

and

$\angle ECF=\angle ECA-25=45-25=20^\circ$ .

They are equal.

Angle chasing

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