Problems — 8.15 Parallel Lines and Polygons

1

**Angles in parallel lines with algebra.** Two parallel lines are crossed by a transversal. One marked angle is $(3x + 20)°$ and the co-interior angle on the same side is $(2x + 40)°$. (a) Set up and solve an equation in $x$. (b) State the size of both angles and verify they are co-interior. (c) Find the corresponding angle of the $(3x + 20)°$ angle on the other parallel line.

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2

**Interior-angle polygon problem.** A regular polygon has interior angle 162°. (a) Find the exterior angle. (b) How many sides? (c) Find the sum of interior angles.

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3

**Star polygon problem.** A regular pentagon has all its diagonals drawn, forming a 5-pointed star. (a) Find the interior angle of the regular pentagon. (b) Find the angle at each star tip. (c) Show that the five tip angles sum to 180°.

Working space

4

**Two-step parallel-line problem.** In a diagram with two parallel lines $\ell_1$ and $\ell_2$, and a transversal: - The acute angle at $\ell_1$ is $(2x + 10)°$. - The vertically opposite angle is $(3x - 20)°$. (a) Use vertically-opposite-angles property to find $x$. (b) Find both angles. (c) Find the obtuse co-interior angle on $\ell_2$.

Working space

5

**Mixed polygon angle problem.** A pentagon and a triangle share an edge. (a) Find the interior angles of each (regular). (b) The shared edge: what is the sum of the two interior angles at one endpoint? (c) Can a regular pentagon and a regular triangle tile a plane around a vertex?

Working space

6

**Sum-of-angles identity.** Prove that the sum of the exterior angles of any convex polygon equals 360°.

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7

**Regular hexagon investigation.** A regular hexagon has side $a$. (a) Find the interior angle. (b) The hexagon can be split into 6 equilateral triangles. Use this to find the area in terms of $a$. (c) Find the perimeter.

Working space

8

**Mixed angles in a complex figure.** In a figure: two parallel lines crossed by a transversal create angles labelled $a, b, c, d$ at the upper intersection and $e, f, g, h$ at the lower. Given $a = 60°$: (a) Find $b, c, d$ at the upper intersection. (b) Find $e, f, g, h$ at the lower intersection. (c) Identify which are alternate, corresponding, co-interior, vertically opposite.

Working space

9

**Irregular polygon angles.** A pentagon has angles 100°, 110°, 95°, 130°, $x$. (a) Find $x$. (b) Classify the polygon (convex or concave). (c) If $x > 180°$, what would that mean?

Working space

10

**Tiling investigation.** A regular polygon tiles the plane if its interior angle divides 360° exactly. (a) Verify that equilateral triangles, squares, and hexagons can tile. (b) Show that regular pentagons cannot tile. (c) Suggest a non-regular pentagon that could.

Working space

11

**Parallel-line angle proof.** In a diagram, parallel lines $\ell_1$ and $\ell_2$ are crossed by transversal $t$. Show that the sum of the two co-interior angles is 180°.

Working space

12

**Polygon side count.** A regular polygon has interior angle 144°. (a) Find the number of sides. (b) Find the sum of all interior angles. (c) Find each exterior angle. (d) Show that the interior angle of a polygon with $n + 5$ sides where this polygon has $n$ sides is bigger.

Working space