Angle facts, parallel lines and polygon sums

Geometry and measures • Properties and constructions

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What is the sum of exterior angles of any convex polygon, one at each vertex?

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The sum of exterior angles, one per vertex, for any convex polygon equals 360°.

Key concepts

What you'll likely be quizzed about

Angles at a point and on a straight line

Angles at a point around a single point sum to 360°. Adjacent angles around that point combine to give the full revolution; therefore, the sum of their measures equals 360°. Angles on a straight line sum to 180° because two adjacent angles that share a straight line form a linear pair whose measures add to a straight angle (half a full revolution).

Vertically opposite angles

When two straight lines cross, they form two pairs of vertically opposite angles that are equal. Equality follows because each angle pairs with a supplementary angle on the straight line; the common supplement forces equality. Vertically opposite angle equality provides direct angle values when two lines intersect and one angle measure is known.

Parallel lines: corresponding and alternate angles

When a transversal cuts two parallel lines, corresponding angles occupy the same relative position at each intersection and are equal because parallelism preserves direction across the transversal. Alternate interior angles lie on opposite sides of the transversal inside the parallel lines and are equal for the same reason. Equality of corresponding and alternate angles provides a method to identify parallel lines or to find unknown angles given one intersection angle.

Supplementary and complementary angle relationships

Supplementary angles sum to 180° and arise from linear pairs or pairs of interior angles on the same side of a transversal in parallel-line configurations. Complementary angles sum to 90° and occur mainly in right-angle decompositions. Identification of supplementary or complementary relationships reduces angle systems into single equations that yield unknown measures.

Sum of angles in a triangle

The interior angles of any triangle sum to 180°. Proof by parallel-line construction: draw a line parallel to one side through the opposite vertex; alternate and corresponding angle equalities produce two angles that together equal the alternate interior angles, summing to 180°. Triangle angle-sum allows computation of a missing angle when two angles are known and supports angle-chasing in composite figures.

Polygon interior and exterior angle sums and regular polygons

A polygon can be divided into (n − 2) triangles by drawing non-overlapping diagonals from one vertex, so the sum of interior angles equals (n − 2) × 180°, where n denotes the number of sides. Each interior angle of a regular n-gon equals [(n − 2) × 180°] ÷ n. Each exterior angle of a regular n-gon equals 360° ÷ n because exterior angles sum to a full revolution. These formulae produce side-count or angle values for regular polygons when one measure is known.

Key notes

Important points to keep in mind

Angles around a point sum to 360°; use this to combine multiple adjacent angles.

Angles on a straight line sum to 180°; identify linear pairs to set up equations.

Vertically opposite angles are equal whenever two lines cross.

Corresponding and alternate angles are equal when lines are parallel; use these equalities to transfer angle measures between intersections.

Interior angles on the same side of a transversal with parallel lines are supplementary (sum to 180°).

Triangle interior angles sum to 180°; use this to find a missing triangle angle.

Polygon interior sum equals (n − 2) × 180°; divide to find regular polygon interior angles.

Exterior angles of a polygon sum to 360°; each exterior angle in a regular polygon equals 360° ÷ n.

Check angle-chase answers by verifying that interior and exterior sums match the formulae.

Count diagonals using n(n − 3) ÷ 2 to support polygon partitioning strategies.