4.1 Angles, lines and triangles

Learning objectives

4.1A Distinguish between acute, obtuse, reflex and rightdistinguish between acute, obtuse, reflex and right angles 4.1B Angle properties of intersecting lines, parallel linesuse angle properties of intersecting lines, parallel lines and angles on a straight line Angles at a point, vertically opposite angles, alternate angles, corresponding angles, allied angles 4.1C The exterior angle of a triangle property and the angleunderstand the exterior angle of a triangle property and the angle sum of a triangle property 4.1D The terms ‘isosceles’, ‘equilateral’ and ‘right-angledunderstand the terms ‘isosceles’, ‘equilateral’ and ‘right-angled triangles’ and the angle properties of these triangles

An angle measures the turn between two rays meeting at a vertex. Classify it from its degree size, not from how wide it appears in a sketch.

Angle type	Size
acute	$0^\circ<\theta<90^\circ$
right	$\theta=90^\circ$
obtuse	$90^\circ<\theta<180^\circ$
straight	$\theta=180^\circ$
reflex	$180^\circ<\theta<360^\circ$

A small square marks a right angle. An arc normally marks the intended angle; for a reflex angle, follow the larger turn around the vertex.

The endpoints are exact: $90^\circ$ is right, not acute or obtuse, and $180^\circ$ is straight, not reflex.

Angle facts form a chain of justified equalities or sums. Mark parallel lines and identify the transversal before choosing a parallel-line rule.

Configuration	Fact
angles on a straight line	sum to $180^\circ$
angles around a point	sum to $360^\circ$
vertically opposite angles	equal
corresponding angles, parallel lines	equal
alternate angles, parallel lines	equal
allied/co-interior angles, parallel lines	sum to $180^\circ$

Write one equation at a time and name the fact used. Transfer an angle through equalities first, then use a straight-line or point sum where needed.

Corresponding, alternate and allied facts require parallel lines. Similar-looking angles are not enough without parallel markings or a stated condition.

The three interior angles of a triangle sum to $180^\circ$ . An exterior angle equals the sum of the two opposite interior angles.

Task	Equation
missing interior angle	$180^\circ-$ the other two
algebraic angles	add all three expressions and set equal to $180^\circ$
exterior angle	add the two remote interior angles
interior beside exterior	subtract exterior from $180^\circ$

If a triangle has angles $30^\circ$ , $(4x+10)^\circ$ and $(x+20)^\circ$ , then $30+4x+10+x+20=180$ , so $x=24$ .

An exterior angle does not equal either adjacent interior angle. It equals the sum of the two non-adjacent interior angles.

Side markings reveal angle facts: equal sides face equal angles, and a right-angle square fixes one angle at $90^\circ$ .

Triangle	Defining property	Angle consequence
isosceles	two equal sides	opposite base angles equal
equilateral	three equal sides	all angles $60^\circ$
right-angled	one right angle	other two angles sum to $90^\circ$

If the equal base angles of an isosceles triangle are each $(x+52)^\circ$ and another expression for one is $(3x+10)^\circ$ , equate them first, then use the triangle sum to find the apex angle.

Equal angles also face equal sides. This converse can prove that a triangle is isosceles when side equality is not given.

Equal-angle conclusions follow the side tick marks, not visual symmetry. A diagram marked ‘not accurately drawn’ must never be measured.

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—4.2 PolygonsRead

4.1 Angles, lines and triangles

Learning objectives

Classify angles by size

Use angle facts for lines and intersections

Use triangle angle sums and exterior angles

Use special-triangle angle properties

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