Geometry › Polygons

Angles in pentagons

Solve the first problem 10 problems

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1. Arithmetic sequence of pentagon angles ●○○○○ 10 ptsLet $v$, $w$, $x$, $y$, and $z$ be the degree measures of the five angles of a pentagon. Suppose $v<w<x<y<z$ and $v$, $w$, $x$, $y$, $z$ form an arithmetic sequence. Find the value of $x$ in degrees.
2. Interior angle of a regular pentagon ●○○○○ 10 ptsFind the measure, in degrees, of an interior angle of a regular pentagon.
3. Missing angles of a heptagon ●○○○○ 10 ptsFive angles of a heptagon have measures $80^\circ$, $85^\circ$, $175^\circ$, $90^\circ$ and $140^\circ$. The other two angles are congruent. Find the measure of each of the other two angles, in degrees.
4. Angle in a regular pentagon ●●○○○ 20 ptsIn a regular pentagon $PQRST$, find the measure of $\angle PRS$ in degrees.
5. Angle at vertex of clock star polygon ●●○○○ 20 ptsA star polygon is drawn on a clock face by drawing a chord from each number to the fifth number clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. Find the degree measure of the angle at each vertex of the star polygon.
6. Angle in pentagon with inner triangle ●●○○○ 20 ptsIn a regular pentagon $ABCDE$, an equilateral triangle $ABM$ is drawn so that $M$ lies inside the pentagon. Determine the size of the angle $BCM$ in degrees.
7. Angle in a pentagon with interior point ●●●●○ 100 ptsLet $P$ be a point inside regular pentagon $ABCDE$ such that $\angle PAB=48^{\circ}$ and $\angle PDC=42^{\circ}$. Find $\angle BPC$, in degrees.
8. Cosine squared in a heptagon ●●●●○ 100 ptsLet $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of triangle $ABD$. Find $\cos^2 \angle GOM$. Give your answer as a decimal.

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