Knowing that the big angle is 90, how many acute angles we have in this shape?
I know acute angle is less than 90, so we have 4 acute angles between the inner lines. Also we have 3 more acute angles combining the above angles. So the total will be 7 acute angles.
But the answer says its 9 acute angles, what am I missing?
4 Answers
You have 9 different angles in that figure:
Call the five rays $A,B,C,D,E$, going clockwise, so that $A$ is horizontal and $E$ is vertical. Then acute angles correspond to $$ AB,\quad AC,\quad AD, \quad BC,\quad BD,\quad BE,\quad CD, \quad CE, \quad DE $$ That's nine acute angles.
Let $O$ be the common vertex, and $A,B,C,D,E$ be the other 5 vertices in clockwise order. Observe that, we have the following 9 acute angles - $\angle AOB$, $\angle AOC$, $\angle AOD$, $\angle BOC$, $\angle BOD$, $\angle BOE$, $\angle COD$, $\angle COE$, $\angle DOE$.
There are ${5 \choose 2} - 1 = 9 $ acute angles because you can select any two of the rays except the outermost pair, which are at right angles.
