Year 7GeometrySpatial reasoning and pathwaysYear 7 Maths

Angle rules, parallel lines and directions

What this lesson covers

  • The interior angle sum of a triangle is 180°; for a quadrilateral, it is 360°.
  • The interior angle sum of any polygon can be found using the formula 180(n−2)°, where n represents the total number of sides.
Key idea

Eight angle rules, or two?

  • Supplementary, complementary, vertically opposite...
  • Corresponding, alternate, co-interior.
  • They all come out of two facts.
ThisMakes
Along a straight line180°
All the way round a point360°
Key idea

One: a straight line is 180°

  • Half a full turn.
  • So two angles sitting on a line add to 180.
  • They are called supplementary.
One: a straight line is 180°
Key idea

Two: all the way round is 360°

  • A full turn.
  • So angles meeting at a point add to 360.
  • That is the whole list of facts.
ThisMakes
Along a straight line180°
All the way round a point360°
Key idea

Everything else is one of those two

  • Complementary is 90, half a straight line.
  • Angles at a point is the full turn.
  • And vertically opposite comes from the straight line.
NameSaysComes from
Supplementaryadd to 180°straight line
Complementaryadd to 90°half a straight line
Vertically oppositeequalboth on a straight line with a third
Angles at a pointadd to 360°full turn
Worked example

Why vertically opposite angles are equal

  • a and b sit on a line, so a + b = 180.
  • b and c sit on a line, so b + c = 180.
  • Both a and c equal 180 − b. So a = c.
BecauseWhich means
a and bon a straight linea + b = 180
b and con a straight lineb + c = 180
Soa and c both equal 180 − ba = c
Key idea

So you do not have to remember it

  • You can rebuild it in three lines.
  • The same is true of every rule coming up.
  • Which is why there are only two to learn.
NameSaysComes from
Supplementaryadd to 180°straight line
Complementaryadd to 90°half a straight line
Vertically oppositeequalboth on a straight line with a third
Angles at a pointadd to 360°full turn
Key idea

The triangle sum

  • Every triangle, however it is shaped, has angles adding to 180.
  • You have used this since Year 6.
  • Now it is going to build something.
The triangle sum
Key idea

A quadrilateral is two triangles

  • Draw one diagonal from a corner.
  • Two triangles, 180 each.
  • So a quadrilateral is 360.
A quadrilateral is two triangles
Key idea

And any polygon at all

  • A pentagon splits into three triangles. 540.
  • A hexagon into four. 720.
  • n sides gives n − 2 triangles: 180(n − 2).
ShapeSplits intoAngle sum
Triangle1 triangle180°
Quadrilateral2 triangles360°
Pentagon3 triangles540°
Hexagon4 triangles720°
n sidesn − 2 triangles180(n − 2)
Key idea

Check it against last lesson

  • A regular pentagon: 540 ÷ 5 = 108° each.
  • Last lesson we got 180 − 72 = 108.
  • Two different routes, one answer.
RouteWorkingAnswer
Angle sum540 ÷ 5108°
Exterior180 − 72108°
Key idea

Two parallel lines, and one crossing them

  • The crossing line is called a transversal.
  • It makes eight angles.
  • And there are only two different sizes among them.
Two parallel lines, and one crossing them
Key idea

Corresponding angles are equal

  • Matching positions at each crossing.
  • Because the two lines are parallel...
  • ...the second crossing is a copy of the first.
Corresponding angles are equal
Key idea

Alternate angles are equal

  • Opposite sides of the transversal, both inside.
  • Also equal, and you can get there two ways.
  • Corresponding, then vertically opposite.
Alternate angles are equal
Key idea

Co-interior angles add to 180

  • Same side of the transversal, both inside.
  • These two are NOT equal.
  • They sit on a straight line, so 180.
Co-interior angles add to 180
Key idea

The three names, in one place

  • Corresponding and alternate: equal.
  • Co-interior: add to 180.
  • If you forget which, work it out from the straight line.
NameWhereRelationship
Correspondingmatching positionsequal
Alternateopposite sides, insideequal
Co-interiorsame side, insideadd to 180°
Worked example

Setting up an equation

  • Two angles on a straight line: x + x + 40 and nothing else.
  • So 2x + 40 = 180.
  • Take 40 off, divide by 2. x = 70.
StepWorkingWhy
Set it up2x + 40 = 180a straight line
Take 40 off both2x = 140
Divide both by 2x = 7070°
Check140 + 40 = 180true
Key idea

Eight directions, 45° apart

  • North, east, south, west: 90° apart.
  • Then the halfway ones between them.
  • NE is 45° east of north.
PointClockwise from N
N0°
NE45°
E90°
SE135°
S180°
SW225°
W270°
NW315°
Worked example

Describing a route

  • Start at B2. Go 2 squares east to B4.
  • Then 2 squares north to D4.
  • Squares, and a direction, for each leg.
Describing a route
Your turn

Your turn: four questions

  • Pause here and try all four.
  • On question 1, count the triangles.
  • On question 3, remember co-interior is the 180 one.
Question
1Find the interior angle sum of a heptagon (7 sides).
2Two angles lie on a straight line. One is 115°. Find the other.
3Parallel lines are cut by a transversal. One angle is 73°. Find its co-interior partner.
4Which compass direction is 135° clockwise from north?
Show answer

1. 900°. 180 × (7 − 2) = 180 × 5. 2. 65°. They are supplementary, so 180 − 115. 3. 107°. Co-interior angles add to 180, so 180 − 73. 4. South-east. Each of the eight points is 45° apart, and 135 is three of those from north.

Worked example

The answers

  • A heptagon is 5 triangles: 900°.
  • 180 − 115 = 65°. And 180 − 73 = 107°.
  • 135° clockwise from north is south-east.
Answer
1900°
265°
3107°
4south-east
Summary

What to take away

  • Two facts: a straight line is 180, a full turn is 360.
  • Every other angle rule is one of those, applied.
  • A polygon splits into n − 2 triangles, so 180(n − 2).
Remember
Straight line180°: this is half of everything
Round a point360°: this is the other half
Polygon sum180(n − 2), because n − 2 triangles fit
Parallel linescorresponding and alternate equal, co-interior 180
Compasseight points, 45° apart

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