Why the two angles are equal here?

It has been given that s and t are the midpoints of PR and QR respectively. My question is how can we say that the angle STR and angle PQR are equal. Is it because ST and PQ are parallel? But it is not given that the ST and PQ are parallel. However, it is given that S and T are the mid points. so, how this conclusion is arrived?

• Can you show that triangles $RST$ and $RPQ$ are similar? It will follow that the angles are equal. Commented Oct 21, 2014 at 9:58
• I can think of SAS - two sides are in proportion and angle is common. However, with this property I can say that triangles are similar but can't say that the angles are equal. Commented Oct 21, 2014 at 10:05
• Similarity is a necessary condition for corresponding angles to be equal.It is like zooming in or out. Commented Oct 21, 2014 at 10:19

Hint: Consider the triangles PQR and STR. Since $S,T$ are the midpoints of sides $PR$ and $QR$ respectively you know that $$\frac{TR}{QR}=\dfrac{SR}{PR}$$ Of course the angle $\angle R$ between these sides is equal in both triangles.

Now use the SAS (side angle side) test to conclude about the similarity of the two triangles.

• Is it necessary that if a triangle is SSS they it has to be AAA and SAS. or I can say that if the triangle is AAA then the triangle has to be SSS and SAS. Or Can I say that if the triangle is SAS then it has to be AAA and SSS. In short, if one is proven then the other two must hold true. Am I correct? Commented Oct 21, 2014 at 10:07
• SAS is enough, see here: mathopenref.com/similarsas.html Commented Oct 21, 2014 at 10:09
• In general, I am asking. Is my assumption correct if one property holds then the other two must hold. Commented Oct 21, 2014 at 10:10
• Yes. Similarity means that all angles are equal and all sides are in the same proportion. So as soon as you have shown that 2 triangles are similar then you know SSS, AAA and SAS all hold true Commented Oct 21, 2014 at 10:17

To prove that $\angle RTS$ and $\angle RQP$ are the same, we need to show that $\Delta STR \sim \Delta PQR$. We know that $PR = 2SR$ and $QR = 2RT$ and that $\angle SRT = \angle PRQ$.

Now we can conclude that $\Delta STR \sim \Delta PQR$ $(SAS)$, this makes $\angle RTS=\angle RQP$. In fact:

$\LARGE\frac{2ST}{PQ}=\frac{2TR}{QR}=\frac{2SR}{PR}$ and $\angle SRT=\angle PRQ$, $\angle RST= \angle RPQ$, $\angle SRT = \angle PRQ$