Question

ABCDEF and EHG are regular polygons. If mHGJ=220* on the exterior of the polygon, mEGJ is congruent to mGED, and mCDJ=136* on the exterior of the polygon, what is the measure of GJD?

Answer 1

96 deg

Explanation:

Polygon ABCDE is a regular hexagon. The sum of the measures of the interior angles is (n - 2)180 = (6 - 2)180 = 4(180) = 720. Since it's a regular hexagon, each interior angle measures 720/6 = 120 deg.

For the interior angle, m<CDE = 120

On the exterior of the polygon, m<CDJ = 136

m<CDE + m<CDJ + m<EDJ = 360

120 + 136 + m<EDJ = 360

m<EDJ + 256 = 360

m<EDJ = 104 deg

Triangle EHG is regular. The sum of the measures of the angles of a triangle is 180. For a regular triangle, each angle measures 60 deg.

m<EGH = 60

For exterior angle m<HGJ = 220

m<HGJ(exterior) + m<EGH + m<EGJ = 360

220 + 60 + m<EGJ = 360

m<EGJ + 280 = 360

m<EGJ = 80

m<EGJ = m<GED, so

m<GED = 80

Polygon DEGJ is a quadrilateral. The sum of the measures of its interior angles is 360 deg.

m<EGJ + m<GED + m<EDJ + m<GJD = 360

80 + 80 + 104 + m<GJD = 360

m<GJD + 264 = 360

m<GJD = 96 deg

Answer 2

The measure of angle GJD is 96°.

Explanation:

It is given that HGJ=220° on the exterior of the polygon, EGJ is congruent to GED, and CDJ=136° on the exterior of the polygon.

Each side and each interior angle of a regular polygon are same.

It is given that ABCDEF and EHG are regular polygons. It means each interior angle of regular hexagon ABCDEF is 120° and each interior angle of regular triangle EHG is 60°.

`\angle EGH+\angle EGJ+\angle HGJ(exterior)=360^(\circ)`

`60^(\circ)+\angle EGJ+220^(\circ)=360^(\circ)`

`\angle EGJ+280^(\circ)=360^(\circ)`

`\angle EGJ=360^(\circ)-280^(\circ)`

`\angle EGJ=80^(\circ)`

`\angle GED=\angle EGJ=80^(\circ)`

`\angle CDE+\angle EDJ+\angle CDJ(exterior)=360^(\circ)`

`120^(\circ)+\angle EDJ+136^(\circ)=360^(\circ)`

`\angle EDJ+256^(\circ)=360^(\circ)`

`\angle EDJ=360^(\circ)-256^(\circ)`

`\angle EDJ=104^(\circ)`

The sum of all interior angles of a quadrilateral is 360°.

`\angle GED=\angle EGJ+\angle EDJ+\angle GJD=360^(\circ)`

`80^(\circ)+80^(\circ)+104^(\circ)+\angle GJD=360^(\circ)`

`264^(\circ)+\angle GJD=360^(\circ)`

`\angle GJD=360^(\circ)-264^(\circ)`

`\angle GJD=96^(\circ)`

Therefore the measure of angle GJD is 96°.