G3 - Angle facts and parallel lines

G3 - Angle facts and parallel lines

Use angle sums and equal-angle relationships to work out missing angles. Learn the precise reasons for angles on straight lines, around points and between parallel lines, so you can explain each step without measuring a sketch.

Straight lines and full turns

Two straight railway tracks are parallel. A road crosses them, making an angle of 68∘68^\circ with one track. Can you work out all the angles at both crossings from that one number? Yes: a few angle facts do all the work.

You already know

Perpendicular lines meet at 90∘90^\circ. Parallel lines in the same plane never meet. Use given angle labels and marks, rather than measuring a diagram labelled “Not drawn accurately”.

An angle measures a turn between two rays. A right angle is a quarter-turn; a straight angle is a half-turn. Put two adjacent angles against the same straight line, with no gap or overlap, and together they fill that half-turn.

[DIAGRAM: asset_name: Basic angle relationships; asset_slug: aqa8300_g02_angle_basics; caption: Adjacent angles on a straight line total 180 degrees; angles around a point total 360 degrees; vertically opposite angles are equal; description: Three panels show a straight angle divided into a and b, a full turn divided into a, b and c, and two crossing straight lines with matching opposite angles.]
Adjacent angles on a straight line total 180 degrees

Angles on a line and around a point

Adjacent angles filling a straight angle sum to 180∘180^\circ. Angles filling one full turn around a point sum to 360∘360^\circ.

a+b=180∘p+q+r=360∘a+b=180^\circ\qquad p+q+r=360^\circ

The angles must fill the stated straight angle or full turn without gaps or overlaps.
Not on the formulae sheet: learn it.

You choose the total by looking at what the angles fill, rather than by counting how many labels appear.

Worked example 1: a straight line, non-calculator

Question. Two adjacent angles on a straight line are 127∘127^\circ and x∘x^\circ. Work out xx.

Plan. Subtract the known angle from the straight-angle total.

x+127=180angles on a straight linex=180−127remove the known angle=53subtract\begin{aligned} x+127&=180 &&\text{angles on a straight line}\\ x&=180-127 &&\text{remove the known angle}\\ &=53 &&\text{subtract} \end{aligned}

Why this step? The two angles together make a half-turn, so the correct total is 180∘180^\circ.

Answer. x=53x=53, so the missing angle is 53∘53^\circ.

Check. 127+53=180127+53=180.

The second example changes the total turn: the angles now go all the way around a point.

Worked example 2: a full turn, non-calculator

Question. Three angles around one point are 145∘145^\circ, 96∘96^\circ and y∘y^\circ. Work out yy.

Plan. Add the known angles, then subtract from a full turn.

145+96=241known anglesy=360−241angles around a point=119subtract\begin{aligned} 145+96&=241 &&\text{known angles}\\ y&=360-241 &&\text{angles around a point}\\ &=119 &&\text{subtract} \end{aligned}

Why this step? All three angles fill the whole space around the point, not just one side of a straight line.

Answer. y=119y=119, so the angle is 119∘119^\circ.

Check. 145+96+119=360145+96+119=360.

Two angles adding to 90∘90^\circ are complementary; two adding to 180∘180^\circ are supplementary. Those words describe the total, even if the angles are not next to each other.

Watch out

Wrong: use 180∘180^\circ for every collection of angles. Right: a straight angle totals 180∘180^\circ; a complete turn totals 360∘360^\circ.

A labelled reflex angle is greater than 180∘180^\circ. It can still be found by subtracting the smaller angle from a full turn.

Vertically opposite angles

When two straight lines cross, opposite angles are equal. “Vertically” here means across the vertex; it does not mean pointing towards the top of the page.

Suppose one angle is 70∘70^\circ. Its neighbour is 180∘−70∘=110∘180^\circ-70^\circ=110^\circ. The angle opposite the first is then 180∘−110∘=70∘180^\circ-110^\circ=70^\circ. Subtracting the same neighbouring angle from 180∘180^\circ explains the equality.

Vertically opposite angles

When two straight lines intersect, each pair of vertically opposite angles is equal.

a=cb=da=c\qquad b=d

The two pairs alternate as you go around the intersection.
Not on the formulae sheet: learn it.

The equal angles touch only at their common vertex. Adjacent angles share a ray and generally have different sizes.

Worked example 3: cross the vertex, non-calculator

Question. Two straight lines cross. One angle is 46∘46^\circ. Work out the angle vertically opposite it and either adjacent angle.

Plan. Copy the opposite angle, then use a straight-line total for a neighbour.

opposite angle=46∘vertically oppositeadjacent angle=180∘−46∘straight line=134∘subtract\begin{aligned} \text{opposite angle}&=46^\circ &&\text{vertically opposite}\\ \text{adjacent angle}&=180^\circ-46^\circ &&\text{straight line}\\ &=134^\circ &&\text{subtract} \end{aligned}

Why this step? Equality applies across the vertex, not beside the given angle.

Answer. The opposite angle is 46∘46^\circ; each adjacent angle is 134∘134^\circ.

Check. 46+134+46+134=36046+134+46+134=360.

The second example changes the given angle from acute to obtuse. The neighbouring angles become the acute ones.

Worked example 4: an obtuse angle, non-calculator

Question. Two straight lines intersect. One angle is 118∘118^\circ. Work out the other three angles.

Plan. Use the opposite equality first, then subtract from 180∘180^\circ.

opposite angle=118∘vertically oppositeeach neighbour=180∘−118∘straight line=62∘subtract\begin{aligned} \text{opposite angle}&=118^\circ &&\text{vertically opposite}\\ \text{each neighbour}&=180^\circ-118^\circ &&\text{straight line}\\ &=62^\circ &&\text{subtract} \end{aligned}

Why this step? Crossing lines give two equal pairs, not four equal angles unless the lines are perpendicular.

Answer. The other angles are 62∘62^\circ, 118∘118^\circ and 62∘62^\circ, in order around the point.

Check. 118+62+118+62=360118+62+118+62=360.

An intersection of two straight lines therefore has at most two different angle sizes. At a right-angle crossing, even those two sizes coincide.

Watch out

Wrong: all four angles are equal because the lines cross. Right: opposite angles are equal; neighbouring angles sum to 180∘180^\circ.

Give both the value and the named reason when the question asks for them.

Corresponding angles on parallel lines

A transversal is a straight line which crosses two or more other lines. If those other lines are parallel, matching corner positions at the two crossings have equal angles.

Imagine sliding the top crossing down the transversal without turning it. The direction of the parallel line stays the same. That is why the matching angle stays the same too.

Continue this lesson.

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