FG3, HG3 - Angle facts and parallel lines

FG3, HG3 - Angle facts and parallel lines

Find missing angles without measuring the drawing. FG3 and HG3 supply the facts about turns, intersecting lines and parallel lines; use those facts to build equations and explain each step.

Basic angle facts

Two straight roads cross. One angle is 67∘67^\circ: can you find the other three without a protractor? A drawing gives the arrangement, while angle facts give exact answers.

An angle measures a turn. One full turn is 360∘360^\circ, half a turn is 180∘180^\circ, and a quarter turn is 90∘90^\circ. Angles around a point add to 360∘360^\circ when they fill the whole turn with no gap or overlap. Angles on a straight line add to 180∘180^\circ when they fill one side of that line.

Vertically opposite angles

When two straight lines cross, the pair of angles opposite one another are vertically opposite angles. They are equal. The word “vertically” does not mean that the lines must be upright.

Here is why they are equal. If neighbouring angles are aa and bb, then a+b=180∘a+b=180^\circ. If the angle opposite aa is cc, then b+c=180∘b+c=180^\circ. Subtracting the same bb gives a=ca=c.

[DIAGRAM: asset_name: Turns and intersecting lines; asset_slug: angle-basics; caption: A straight angle is half a full turn; opposite angles at a crossing are equal.; description: Three panels show three sectors completing a straight angle, three sectors around a point completing a full turn, and two crossing straight lines with equal opposite sectors labelled a and equal opposite sectors labelled b.]
A straight angle is half a full turn

Worked example 1: a straight line

Question. Three adjacent angles on one side of a straight line are 46∘46^\circ, 79∘79^\circ and x∘x^\circ. Work out xx.

Plan. Subtract the two known angles from a half turn.

46+79+x=180angles on a straight linex=180−125=55\begin{aligned} 46+79+x&=180 &&\text{angles on a straight line}\\ x&=180-125=55 \end{aligned}

Answer. x=55x=55, so the angle is 55∘55^\circ.

Check. 46+79+55=18046+79+55=180, as required.

The next example changes the available turn: the angles now surround the point completely.

Worked example 2: a full turn and a crossing

Question. Three angles around a point are 112∘112^\circ, 153∘153^\circ and y∘y^\circ. Separately, two straight lines cross with one angle 67∘67^\circ. Find yy and the other three crossing angles.

Plan. Use a full turn for yy; use opposite angles and straight lines at the crossing.

y=360−112−153=95angles around a pointopposite angle=67∘vertically opposite angleseach neighbouring angle=180∘−67∘=113∘\begin{aligned} y&=360-112-153=95 &&\text{angles around a point}\\ \text{opposite angle}&=67^\circ &&\text{vertically opposite angles}\\ \text{each neighbouring angle}&=180^\circ-67^\circ=113^\circ \end{aligned}

The two neighbours are also a vertically opposite pair.

Answer. y=95y=95; the other crossing angles are 113∘,67∘,113∘113^\circ,67^\circ,113^\circ in order around the point.

Check. 112+153+95=360112+153+95=360 and 67+113+67+113=36067+113+67+113=360.

Name the complete turn before subtracting.

Watch out

“Angles touching each other add to 180∘180^\circ” is incomplete. They must fill a straight angle. Angles filling the whole turn add to 360∘360^\circ.

Apply the crossing facts to new values.

Parallel lines

A line crossing two other lines is called a transversal. When the two lines are parallel, the transversal creates equal angles at different intersections.

Follow the same side of the transversal at each crossing. Angles in the same relative position are corresponding angles. Inside the parallel lines, on opposite sides of the transversal, the angles form an alternate pair.

Parallel-line angle facts

For parallel lines cut by a transversal, corresponding angles are equal and alternate angles are equal. Interior angles on the same side of the transversal, also called co-interior angles, add to 180∘180^\circ.

Co-interior angles add to 180∘180^\circ because one equals an alternate angle adjacent to the other on a straight line. Do not decide from an apparent letter shape alone: identify the lines, the transversal and the angle positions. Parallel arrow marks or a stated fact establish that the lines are parallel.

[DIAGRAM: asset_name: Angle pairs on parallel lines; asset_slug: parallel-angle-pairs; caption: Corresponding and alternate pairs are equal; co-interior angles add to a straight angle.; description: Three panels show two horizontal parallel lines cut by an oblique transversal. Corresponding angles occupy matching positions, alternate angles lie inside on opposite sides, and co-interior angles lie inside on the same side.]
Corresponding and alternate pairs are equal

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