Question

21 given: \\(\overline{ab} \cong \overline{ac}\\),\\(\overline{ae} \cong \overline{de} \cong \overline{db} \cong \overline{bc}\\)find: \\(m\angle a\\)

Explanation:

Step1: Let ∠A = x

In △ABE, AE=DE, so ∠ADE=∠A=x.

Step2: Find ∠BED

∠BED is exterior angle of △ADE: ∠BED=∠A+∠ADE=2x.

Step3: Analyze △BDE

DE=DB, so ∠DBE=∠BED=2x.

Step4: Find ∠BDC

∠BDC is exterior angle of △ABD: ∠BDC=∠A+∠DBE=3x.

Step5: Analyze △BDC

DB=BC, so ∠C=∠BDC=3x.

Step6: Analyze △ABC

AB=AC, so ∠ABC=∠C=3x. Sum of angles: x+3x+3x=180°.

Step7: Solve for x

7x=180° → x=180/7≈25.71°.

Answer:

$\frac{180}{7}^\circ$ (or approximately $25.71^\circ$)