Problem
GEO-B1-M01-P017 Angles 36 and 72
#17
★★★☆☆ Level 3 of 5
In isosceles triangle \(ABC\), \(AB=AC\) and \(\angle A=36^\circ\). Point \(D\) lies on side \(AC\), and \(BD=BC\). Find \(\angle ABD\).
First find the base angles of triangle \(ABC\), then consider isosceles triangle \(BCD\).
Since \(AB=AC\), the base angles are equal: \(\angle B=\angle C=\frac{180^\circ-36^\circ}{2}=72^\circ\). In triangle \(BCD\), sides \(BD\) and \(BC\) are equal, so \(\angle BDC=\angle BCD\). Since \(D\) lies on \(AC\), \(\angle BCD=72^\circ\), hence \(\angle BDC=72^\circ\). Therefore \(\angle DBC=180^\circ-72^\circ-72^\circ=36^\circ\). Thus \(\angle ABD=\angle ABC-\angle DBC=72^\circ-36^\circ=36^\circ\).
Useful for developing careful decomposition of an angle into parts.