8.15 Parallel Lines and Polygons

Two lines cross. One of the four angles is 47°. State the other three angles and name the relationship for each.

A transversal crosses two parallel lines. One alternate (Z) angle is 75°. Find the other.

A transversal crosses two parallel lines. A corresponding (F) angle is 62°. Find the other.

Two co-interior (allied / "C") angles between parallel lines are 110° and $x$. Find $x$.

Find the sum of the interior angles of an $n$-sided polygon with $n = 6$.

Find one interior angle of a regular 6-gon.

Find the exterior angle of a regular 6-gon.

In a triangle, two angles are 80° and 60°. Find the third.

A pentagon has angles 100°, 110°, 95°, 105° and $x$. Find $x$.

A regular octagon has 8 interior angles. State the size of each.

Two parallel lines crossed by a transversal: the acute angle on the upper line is 58°. Find the corresponding angle on the lower line.

Two parallel lines have a transversal. One angle is 70°; another (alternate) is $(x + 15)$°. Find $x$.

The exterior angle of a triangle is 110°. The two non-adjacent interior angles are equal. Find each.

A regular polygon has exterior angle 30°. How many sides?

A regular polygon has interior angle 144°. How many sides?

Two parallel lines, transversal. One angle is 75°. Find all six other angles labelled around the two intersection points.

The interior angles of an octagon are 135°, 130°, 140°, 125°, 145°, 130°, 140°, $x$. Find $x$.

A regular nonagon (9 sides) has interior angle?

Two parallel lines have angles labelled $a = 65°$ on the upper and $b$ co-interior on the lower. Find $b$, then find the alternate angle to $b$ on the upper line.

Find the angle marked $x$ in a Z-shape: two parallel lines crossed by a transversal, with $x$ alternate to $(2a + 30)°$ where $a = 25°$.

In a parallel-lines diagram, one angle is $(4x - 12)°$ and a co-interior angle is $(2x + 30)°$. Find $x$.

A regular polygon has interior angle 150°. How many sides? Find the sum of interior angles.

In the diagram, two parallel lines crossed by a transversal: acute angle on the upper line is 65°. Find the obtuse co-interior on the lower; then the acute on the lower.

An irregular pentagon has angles 90°, 110°, 130°, $x$, $x + 20$. Find $x$.

The angles around a point on a transversal between two parallel lines are $a, b, c, d$. Given $a = 70°$, find $b, c, d$.

A regular polygon has interior angle 156°. Find the number of sides and verify with the sum formula.

A pentagon has interior angles in the ratio 3:4:5:6:7. Find each angle.

In a regular dodecagon (12 sides), find (a) interior angle, (b) exterior angle, (c) sum of interior angles.

A regular hexagon and a regular triangle share a side. Find the angle at the join.

The bisector of the exterior angle of a regular polygon meets the polygon's side at angle 75°. Find the number of sides.

Two parallel lines $\ell_1$ and $\ell_2$ are crossed by a transversal. Let $a$ be an angle on $\ell_1$ and $b$ the angle alternate to $a$ on $\ell_2$. Prove (using corresponding-angle property) that $a = b$.

A regular pentagon has its interior diagonals drawn, forming a five-pointed star. Find the angle at each "point" of the star.

The interior angles of an octagon are five angles each $x°$ and three angles each $(x + 10)°$. Find $x$.

Find $x$ in a quadrilateral with angles $(x + 30)°, (2x - 10)°, (3x)°, (x + 10)°$.

A regular $n$-gon has interior angle equal to $\dfrac{180(n-2)}{n}$. (a) Show that as $n \to \infty$ the interior approaches 180°. (b) For which $n$ is the interior > 150°?

Two parallel lines are cut by two transversals forming a quadrilateral region. The quadrilateral has two right angles. Use angle properties to prove the remaining two angles are supplementary.

A regular polygon's interior angle is 3 times its exterior angle. Find the number of sides.

Three regular polygons meet at a point with no gap and no overlap. Each interior angle of each polygon must satisfy a constraint. Find a valid combination.

In a quadrilateral, the angles in order are $(x)°, (2x)°, (x - 10)°, (3x + 10)°$. Find $x$ and the angles.

A pentagon has 4 interior angles each $x°$ and a 5th of $(2x + 20)°$. Find $x$.