Hi can someone help me with this problenm?

In triangle PQR, P= Q+R. Of all three exterior angles of triangle what is the least exterior angle, in degrees?

Thanks!

Guest Mar 21, 2020

#2

#3**+2 **

The rule is that exterior angles of any polygon ALWAYS add up to 360 degrees.

CalTheGreat
Mar 21, 2020

#4**+2 **

Following up on CalTheGreat's tidbit on the exterior angle sum(**the sum is important, as exterior angle theorem is something different**) theorem, if you don't understand it, here's a proof:

Start by realizing that the interior angles of an n-gon have an angle sum of 180(n-2). This is derived from the fact that an n-gon can be divided into n-2 triangles, which all have an angle sum of 180 degrees(there are other derivations of it, but I think I'll use this one for now). Now if you look at an exterior angle, it's pair(the interior angle) forms a 180 degree angle, so the total sum of these pairs would be 180n(the exterior angles + the interior angles). Realize that because the interior angle sum is 180(n-2), the exterior angles sum is just:

\(180n-180(n-2) = 180n-180n+360 = 360\). That's just a short proof of why the exterior angles of a polygon ALWAYS add up to 360.

jfan17 Mar 21, 2020

#5**+1 **

Yes, jfan17! That was what I was trying to say, even though I don't know the theorem that well. Thanks!

CalTheGreat
Mar 21, 2020

#7**+1 **

The vertex that has the smallest experior angle is the one that has the largest interior angle.

So, you need to determine which angle, P, Q, or R must be the largest.

Then, to find the size of the exterior angle, subtract this value from 180°.

geno3141 Mar 21, 2020

#8**+1 **

Assuming the P,Q and R are angles we have that

P = Q + R (1)

P + Q + R = 180 (2)

Sub (1) into (2) and we have that

P + P = 180

2P = 180

P = 90

This will be the largest angle in the triangle......so.....it will have the smallest associated exterior angle.....and its associated exterior angle will be supplemental = 90°

CPhill Mar 21, 2020

#9**0 **

Hi can someone help me with this problenm?

In triangle PQR, P= Q+R. Of all three exterior angles of triangle what is the least exterior angle, in degrees?

Exterior angle theorem

**The exterior angle of a triangle is equal to the sum of the 2 opposite interior angles.**

Enterior angles: ∠P = ∠Q + ∠R ( ∠P MUST be 90° )

∠P = 90° ∠Q + ∠R = 90°

The sum of angles Q and R is equal to the exterior angle at P vertex.

**The least exterior angle is at P vertex, and it's 90°. **

Guest Mar 22, 2020

edited by
Guest
Mar 22, 2020

edited by Guest Mar 22, 2020

edited by Guest Mar 22, 2020