C4.6 Angles

Learning objectives

C4.6.1Angles at points and in polygons• Calculate unknown angles and give simple explanations using angles at a point, angles on a straight line, vertically opposite angles, and angle sums of triangles and quadrilaterals.C4.6.2Angles in parallel lines• Calculate unknown angles and explain corresponding, alternate and co-interior angle relationships in parallel lines.C4.6.3Regular polygon angles• Know and use interior and exterior angle properties and angle sums of regular polygons.

Unknown angles are found by identifying a complete angle total or an equality, then subtracting known angles and naming the fact used.

Configuration	Geometrical fact
angles at a point	sum to $360^\circ$
adjacent angles on a straight line	sum to $180^\circ$
vertically opposite angles	are equal
angles in a triangle	sum to $180^\circ$
angles in a quadrilateral	sum to $360^\circ$

Mark equal angles first—for example, base angles of an isosceles triangle—then select the smallest shape or line containing the unknown. Write one equation from its total, solve it, and give the precise reason rather than writing only “angles”.

In an isosceles triangle with equal base angles $41^\circ$ , the third angle is $180^\circ-41^\circ-41^\circ=98^\circ$ . If its exterior angle lies on a straight line, that exterior angle is $180^\circ-98^\circ=82^\circ$ .

In $\angle ABC$ , the middle letter $B$ is the vertex. This three-letter notation distinguishes the intended angle when several rays meet at one point.

Do not assume angles are equal because they look equal or because a diagram appears regular. Equality must come from a stated property, such as isosceles sides or vertically opposite angles.

When a transversal crosses two parallel lines, its repeated direction creates equal or supplementary angle pairs.

Relationship	Position cue	Rule
corresponding	same corner at the two intersections	equal
alternate	between the parallel lines, on opposite sides of the transversal	equal
co-interior	between the parallel lines, on the same side of the transversal	sum to $180^\circ$

Start from the given angle and move to the unknown in one justified step when possible. If the required pair is not immediate, first use a straight-line or vertically opposite fact, then apply the named parallel-line relationship.

If an interior angle is $112^\circ$ , its alternate angle is also $112^\circ$ . The co-interior angle on the same side of the transversal is $180^\circ-112^\circ=68^\circ$ .

Equal corresponding or alternate angles can also establish that two lines are parallel; so can co-interior angles whose sum is $180^\circ$ . State the pair used.

These relationships require parallel lines, normally shown by arrow marks or stated in the question. Do not use “corresponding” or “alternate” merely because two angles occupy similar-looking positions.

For a regular $n$ -sided polygon, all sides and interior angles are equal, so one full turn is shared equally among its exterior angles.

$\text{exterior angle}=\frac{360^\circ}{n},\qquad \text{interior angle}=180^\circ-\frac{360^\circ}{n},\qquad \text{interior sum}=(n-2)180^\circ$

For a regular 15-sided polygon, the exterior angle is $360^\circ/15=24^\circ$ , so each interior angle is $180^\circ-24^\circ=156^\circ$ . The interior-angle sum is $13\times180^\circ=2340^\circ$ .

$n=\frac{360^\circ}{\text{exterior angle}}$

If each exterior angle is $20^\circ$ , then $n=360^\circ/20^\circ=18$ . A non-integer result means identical turns of that size cannot close to form a regular polygon.

Interior and exterior angles at one vertex sum to $180^\circ$ . Divide $360^\circ$ by $n$ only for equal exterior angles—that is, for a regular polygon—not for an arbitrary irregular polygon.

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—C4.7 Circle theoremsRead

C4.6 Angles

Learning objectives

Calculate angles using basic geometrical facts

Use angle relationships in parallel lines

Calculate angles and sides of regular polygons

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