E4.6 Angles

Learning objectives

E4.6.1Calculate unknown angles and give simple explanations using the• Calculate unknown angles and give simple explanations using the following geometrical properties: • sum of angles at a point = 360° • sum of angles at a point on a straight line = 180° • vertically opposite angles are equal • angle sum of a triangle = 180° and angle sum of a quadrilateral = 360°.E4.6.2Calculate unknown angles and give geometric explanations for angles• Calculate unknown angles and give geometric explanations for angles formed within parallel lines: • corresponding angles are equal • alternate angles are equal • co-interior angles sum to 180° (supplementary).E4.6.3Know and use angle properties of regular and irregular polygons• Know and use angle properties of regular and irregular polygons. Knowledge of 3-letter notation for angles is required, e.g. angle ABC. Candidates are expected to use the correct geometrical terminology when giving reasons for answers. Includes exterior and interior angles, and angle sum.

An angle chain finds one unknown at a time and states the exact geometric fact that fixes each step.

Configuration	Angle fact	Reason to state
angles around one point	sum to $360^\circ$	angles at a point
adjacent angles on a straight line	sum to $180^\circ$	angles on a straight line
opposite angles where two lines cross	are equal	vertically opposite angles
three interior angles of a triangle	sum to $180^\circ$	angles in a triangle
four interior angles of a quadrilateral	sum to $360^\circ$	angles in a quadrilateral

Mark every known angle, choose the smallest shape or line containing one unknown, write its angle equation, solve it, then transfer that value to the next step. For an isosceles triangle, first mark the equal base angles only after confirming which sides are equal.

In isosceles triangle $ABC$ , let $AB=AC$ and $\angle BAC=38^\circ$ . Then $\angle ABC=\angle BCA=(180^\circ-38^\circ)\div2=71^\circ$ . If $BC$ is extended to $D$ , $\angle ACD=180^\circ-71^\circ=109^\circ$ because angles on a straight line sum to $180^\circ$ .

Do not use a fact because the diagram looks suitable: verify the straight line, intersection, equal sides or closed shape. A numerical answer without the requested geometric reason leaves the angle chain unsupported.

When a transversal crosses parallel lines, its intersections repeat equal-angle positions and create supplementary interior pairs.

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

Confirm the pair of lines is parallel, identify the one transversal that creates both angles, then classify their positions before calculating. Transfer an equal corresponding or alternate angle directly; subtract a co-interior angle from $180^\circ$ . Combine these facts with vertically opposite, straight-line or triangle facts only in separate, reasoned steps.

If parallel lines $AB$ and $CD$ are cut by transversal $EF$ and one acute angle is $38^\circ$ , its alternate and corresponding acute angles are also $38^\circ$ . Each adjacent obtuse angle is $180^\circ-38^\circ=142^\circ$ ; the matching co-interior pair is supplementary.

Corresponding and alternate angles are equal only when the lines are parallel. Co-interior angles are not equal in general; they add to $180^\circ$ . State the relationship by position, not by a memorised letter shape alone.

An $n$ -sided polygon can be split from one vertex into $n-2$ triangles, so its interior-angle sum is controlled by the number of sides.

S_{\text{interior}}=(n-2)\times180^\circ,\qquad e_{\text{regular}}=\frac{360^\circ}{n},\qquad i_{\text{regular}}=180^\circ-e

For an irregular polygon, find the total $(n-2)\times180^\circ$ and subtract the known interior angles. For a regular polygon, equal exterior angles make one full turn, so divide $360^\circ$ by $n$ ; the adjacent interior and exterior angles sum to $180^\circ$ . To recover $n$ , use $n=360^\circ/e$ and check that the answer is a whole number.

A hexagon has interior sum $(6-2)\times180^\circ=720^\circ$ . If five interior angles are each $115^\circ$ , the sixth is $720^\circ-5\times115^\circ=145^\circ$ . A regular 24-gon has exterior angle $360^\circ\div24=15^\circ$ and interior angle $165^\circ$ .

In $\angle ABC$ , the middle letter $B$ is the vertex. Do not divide an irregular polygon's total by $n$ unless all its interior angles are equal, and use precise reasons such as ‘exterior angles of a polygon sum to $360^\circ$ ’.

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—E4.7 Circle theorems IRead

E4.6 Angles

Learning objectives

Build angle chains from basic facts

Use angles in parallel lines

Calculate angles in polygons

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