Questions: If triangle ABC is congruent to triangle DEC, angle DCE = 55 degrees, angle CDE = 65 degrees, and angle ABC = 2x x = [?]

Transcript text: If $\triangle \mathrm{ABC} \cong \triangle \mathrm{DEC}, \angle \mathrm{DCE}=55^{\circ}$, $\angle C D E=65^{\circ}$, and $\angle A B C=2 x$ \[ x=[?] \]

Solution

Solution Steps

Step 1: Identify Given Information

We are given that:

$ \triangle ABC \cong \triangle DEC $
$ \angle DCE = 55^\circ $
$ \angle CDE = 65^\circ $
$ \angle ABC = 2x $

Step 2: Use Triangle Congruence

Since $ \triangle ABC \cong \triangle DEC $, corresponding angles are equal. Therefore:

$ \angle BAC = \angle EDC $
$ \angle ACB = \angle DCE = 55^\circ $
$ \angle ABC = \angle DEC = 2x $

Step 3: Calculate the Third Angle in $ \triangle DEC $

The sum of the angles in a triangle is $ 180^\circ $. Therefore, in $ \triangle DEC $: \[ \angle DEC + \angle DCE + \angle CDE = 180^\circ \] \[ 2x + 55^\circ + 65^\circ = 180^\circ \] \[ 2x + 120^\circ = 180^\circ \]

Step 4: Solve for $ x $

Subtract $ 120^\circ $ from both sides: \[ 2x = 60^\circ \] Divide by 2: \[ x = 30^\circ \]