Practice Test 9

Which reasoning supports the angle addition formula?

In the diagram, triangle $ABC$ has point $D$ on segment $AC$. At vertex $A$, ray $AD$ splits $\angle BAC$ into two angles labeled $\theta$ (between $AB$ and $AD$) and $\varphi$ (between $AD$ and $AC$), so $\angle BAC=\theta+\varphi$. Segment $BD$ is drawn. A right-angle marker indicates $BD\perp AC$ at $D$.

Which statement proves an identity for $\sin(\theta+\varphi)$ using this construction?

Which conclusion follows from the diagram?

Triangle $OXY$ is a right triangle with right angle at $X$ (right-angle box at $X$). Ray $OZ$ lies inside $\angle XOY$ and splits it into two angles: $\angle XOZ=\theta$ and $\angle ZOY=\varphi$. From point $Z$ on ray $OZ$, perpendiculars are dropped to $OX$ and to $XY$ meeting them at $M$ and $N$ (right-angle boxes at $M$ and $N$). No other information is marked.

Which statement correctly describes a geometric strategy that can lead to a sum identity (rather than assuming it)?