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# Geometry Problem

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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!

Mar 21, 2020

#1
+1956
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Hint:

We know that P+Q+R=360 (exterior angles theorem)

Mar 21, 2020
#2
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I don't understand how they add up to 360... Could you elaborate? Thanks.

Mar 21, 2020
#3
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The rule is that exterior angles of any polygon ALWAYS add up to 360 degrees.

CalTheGreat  Mar 21, 2020
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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.

Mar 21, 2020
#5
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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
#6
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Oh ,thank you.

Guest Mar 21, 2020
#7
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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°.

Mar 21, 2020
#8
+111321
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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°

Mar 21, 2020
#9
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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°.

Mar 22, 2020
edited by Guest  Mar 22, 2020
edited by Guest  Mar 22, 2020