Problem

Draw axis to split into congruent halves

An eight-sided figure is line-symmetric about the horizontal axis that runs through point O on the left and vertex E on the right. Going around it the vertices are S and F across the top, E at the right end, D and C along the bottom, B at the lower-left, and A at the upper-left. The figure marks a 140 degree interior angle at D, a 45 degree angle just outside vertex B, and an 80 degree angle just outside vertex E. We must find the marked angle at the top vertex S.

GeometryMeasurement & data

Your answer

How to solve

Strategy Draw a Diagram — I draw in the axis of symmetry, which cuts the eight-sided figure into two congruent halves and makes the matching (corresponding) angles easy to see. The top half is a five-sided figure A-S-F-E-O, so the problem breaks into small subproblems: turn each marked outside angle into its interior angle, copy the corresponding angles across the axis, and finally use the fact that the five angles of a pentagon add to 540 degrees to peel off the one unknown angle at S.

1STEP 1

Draw the axis and split off the top pentagon

The axis through O and E splits the figure into mirror halves; the top half is pentagon O-A-S-F-E.

top half = pentagon O-A-S-F-E

2STEP 2

Find the angle at E using the 80 degree mark

The 80 degree mark sits outside, so the interior angle is 100, and the axis halves it to 50.

180^° - 80^° = 100^°, 100^° ÷ 2 = 50^°

3STEP 3

Find the angle at A using the 45 degree mark

The 45 degree mark is outside too, so B's interior angle is 135 and its mirror A matches at 135.

180^° - 45^° = 135^°

4STEP 4

Find the angle at F by corresponding angles

F mirrors D, so it carries the same 140 degrees.

∠ SFE = ∠ CDE = 140^°

5STEP 5

Use the right angle on the axis at O

AB joins mirror points, so it meets the axis square-on and angle A-O-E is 90 degrees.

∠ AOE = 90^°

6STEP 6

Subtract from the pentagon's 540 degrees

A pentagon's angles total 540, so removing the four known ones leaves 125 degrees at S.

540^° - 90^° - 135^° - 140^° - 50^° = 125^°

Answer

125 degrees

The four known pentagon angles total 90 + 135 + 140 + 50 = 415 degrees, leaving 540 - 415 = 125 degrees, a single positive angle under 180 degrees as a polygon corner should be. The answer 125 degrees is in degrees, the right unit for an angle, and looks sensible for the wide top corner shown.

Takeaway

Fold the symmetric shape on its axis: matching corners share the same angle, and once you know all but one angle of the half-shape, the pentagon's 540 degree total hands you the last one.

Draw the axis and split off the top pentagon
Find the angle at E using the 80 degree mark
Find the angle at A using the 45 degree mark
Find the angle at F by corresponding angles
Use the right angle on the axis at O
Subtract from the pentagon's 540 degrees

Where next?

Another one like thissuggested

12 practice problems

Step back and solidifysuggested

Even number of flips returns original Flip across a line creates symmetry

3 reasoning problems grow out of this type — the bridge toward competition →

▶ Practice — 12 problems