What this lesson covers
to memorise here. Draw every diagonal from one corner and the pentagon splits
into three triangles, with no angle created or lost in the cut. Three triangles
at 180 degrees each is 540, and that is the sum for any pentagon, regular or not.
Then subtract what they handed you.
- Building the 540 from triangles instead of memorising it
- The rule underneath: (n - 2) triangles, times 180
- The running sum: 100, 212, 342, 437, then 540 - 437 = 103
- Why 108 is sitting in the choices, and why it is wrong here
- The three traps and the check that catches all of them
- Diagonals from one vertex cut an n-sided polygon into (n - 2) triangles
- So the interior angles sum to (n - 2) x 180
- A pentagon is 540, regular or not
- 540 / 5 = 108 is the angle of a REGULAR pentagon only
- The exterior angle is 180 minus the interior angle, and it is not the answer
Chapters
- 0:00Intro
- 0:04Four angles, find the fifth
- 0:14The item, and four believable answers
- 0:33Cut it into three triangles
- 0:57Add the four, subtract from 540
- 1:25Three traps
- 1:59Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
0:14The item, and four believable answers
A pentagon. Four of its interior angles are given. One hundred, one hundred twelve, one hundred thirty, and ninety five. Find the fifth. There is nothing to memorize here, because the sum of the angles is something you can build out of triangles you already know.
0:33Cut it into three triangles
Pick one corner and draw every diagonal from it. A pentagon splits into three triangles, and no angle is created or lost in the cut. Three triangles at one hundred eighty degrees each is five hundred forty. That is the sum for any pentagon, regular or not. The rule underneath it. n minus two triangles, times one hundred eighty. Now add the four you were handed.
0:57Add the four, subtract from 540
One hundred plus one hundred twelve is two hundred twelve. Plus one hundred thirty is three hundred forty two. Plus ninety five is four hundred thirty seven. Five hundred forty minus four hundred thirty seven is one hundred three. One hundred three degrees. And notice one hundred eight sitting in the choices. That is five hundred forty split five ways, the angle of a regular pentagon, and this pentagon is not regular.
1:25Three traps
Three traps. One, multiplying by n instead of n minus two. Five triangles would be nine hundred degrees, and a pentagon does not hold that. Two, answering with the regular pentagon angle. One hundred eight only applies when all five angles match, and here they do not. Three, stopping at the exterior angle. Seventy seven is one hundred eighty minus one hundred three, which is the turn at that corner, not the angle inside it. The fast check. Your missing angle plus the four given has to land on five hundred forty exactly.
1:59Recap
n minus two triangles, times one hundred eighty. Then subtract what they gave you.
Read the written version: the Lines, angles & triangles strategy guide, then try 25 hard Lines, angles & triangles questions with full explanations. Both are free, no account needed.