Answer to Question #140173 in Geometry for Miriam

Question

A rhombus ABCD is divided into two equilateral triangles, along the diagonal BD. Points P and Q are in segments AD and CD respectively and are such that ∠P BQ = 60◦ Determine the measurements of the Remaining angles of the triangle P BQ.

Answer 1

Let $\angle DBQ=x\degree$, so $\angle PBD=60\degree-x\degree$.

$\angle ABP=60\degree-\angle PBD=60\degree-(60\degree-x\degree)=x\degree.$

If $\angle ABP=\angle DBQ$, $\angle BAP=\angle BDQ$, $AB=BD$, so $\triangle ABP=\triangle DQB$

and $BP=BQ$.

In $\triangle PBQ: \angle PBQ=60\degree, PB=BQ$, so $\angle QPB=\angle PQB=\frac{180\degree-\angle PBQ}{2}=60\degree$.