Question

in the figure, ∆ABC is is isosceles, ∆ADC is equilateral, ∆AEC is isosceles, and measures of <9, <1, and <3are all equal. Find the measure of the nine numbered angles.

Answer 1

Given,

$\begin{gathered} \Delta\text{ABC is a isosceles triangle.} \\ \Delta ADC\text{ is a equilateral triangle.} \\ \Delta AEC\text{ is isosceles triangle.} \\ \angle9,\text{ }\angle1,\text{ }\angle3\text{ have equal measure.} \end{gathered}$

From the given figure,

The measure of side AB is equal to side BC. As ABC is a isoscles triangle.

Then,

$\begin{gathered} \angle BAC=\angle BCA \\ \angle1+\angle2+\angle3=\angle4+\angle5+\angle6\ldots\ldots\ldots\ldots\ldots..(i) \end{gathered}$

From the given figure,

The measure of side AD , DC and AC is equal. As ADC is a equilateral triangle.

Then,

$\begin{gathered} \angle DAC=\angle DCA \\ \angle2+\angle3=\angle4+\angle5\ldots\ldots\ldots..(ii) \end{gathered}$

Also,

$\angle2+\angle3+\angle4+\angle6+\angle8=180^{\circ}$