Year 7GeometryShapesYear 7 Maths

Classifying triangles by angles and sides

What this lesson covers

  • Triangles can be categorised by their angles. An acute triangle has three acute angles. A right triangle has one right angle. An obtuse triangle has one obtuse angle.
  • Triangles can also be categorised by their sides. An equilateral triangle has three equal-length sides. An isosceles triangle has at least two equal-length sides. A scalene triangle has different measures for each side length.
  • All angles in an equilateral triangle are 60°.
  • The base angles (opposite the equal sides) of an isosceles triangle are equal.
Key idea

This triangle has two names

  • One right angle, so it is a right triangle.
  • Two equal sides, so it is also isosceles.
  • Both at once, and neither one cancels the other.
This triangle has two names
Key idea

Two separate questions

  • What are its angles like?
  • What are its sides like?
  • You answer both, every time.
QuestionPossible answers
About the anglesacute, right, obtuse
About the sidesequilateral, isosceles, scalene
Key idea

The three angle names

  • Acute: every angle under 90°.
  • Right: exactly one angle of 90°.
  • Obtuse: one angle over 90°.
NameMeans
Acuteall three angles under 90°
Rightexactly one angle of 90°
Obtuseone angle over 90°
Key idea

Only one angle can be 90 or more

  • The three angles have to total 180°.
  • Two 90s would use all of it, leaving nothing.
  • So right and obtuse are never both true.
Only one angle can be 90 or more
Key idea

Acute: all three under 90

  • 50, 60 and 70: every one of them under 90.
  • They still add to 180.
  • So this is an acute triangle.
Acute: all three under 90
Key idea

The three side names

  • Equilateral: all three sides equal.
  • Isosceles: at least two equal.
  • Scalene: all three different.
NameMeans
Equilateralall three sides equal
Isoscelesat least two sides equal
Scaleneall three sides different
Key idea

Equilateral

  • Three equal sides, marked with matching ticks.
  • And every angle is 60°.
  • Because 180 shared equally three ways is 60.
Equilateral
Key idea

Isosceles, and its equal base angles

  • Two equal sides, marked with ticks.
  • The angles opposite them are equal too.
  • 55 and 55, with 70 at the top.
Isosceles, and its equal base angles
Key idea

The same triangle, marked for angles

  • Here are its angles: 50, 60, 70.
  • All under 90, so it is acute.
  • That is the angle answer.
The same triangle, marked for angles
Key idea

And now marked for sides

  • Same triangle. No ticks anywhere.
  • Because no two sides are the same length.
  • Scalene. So the full name is acute scalene.
And now marked for sides
Key idea

Three by three

  • Three angle names across, three side names down.
  • Nine possible combinations.
  • Seven of them exist.
EquilateralIsoscelesScalene
Acuteyes (60/60/60)yes (55/55/70)yes (50/60/70)
RightIMPOSSIBLEyes (90/45/45)yes (90/60/30)
ObtuseIMPOSSIBLEyes (100/40/40)yes (100/50/30)
Key idea

Two boxes are empty

  • There is no right equilateral triangle.
  • And no obtuse equilateral triangle either.
  • Not rare. Impossible.
EquilateralIsoscelesScalene
Acuteyes (60/60/60)yes (55/55/70)yes (50/60/70)
RightIMPOSSIBLEyes (90/45/45)yes (90/60/30)
ObtuseIMPOSSIBLEyes (100/40/40)yes (100/50/30)
Worked example

Why they cannot exist

  • Three equal sides force three equal angles.
  • They must total 180, so each is 60.
  • 60 is under 90, so it is always acute.
BecauseIt follows
Three equal sidesmeans three equal angles
They must total180°
So each is180 ÷ 360°
60 is under 90so always acute
Worked example

Name this one

  • One angle of 100, over 90, so obtuse.
  • Three different angles mean three different sides.
  • Obtuse scalene.
Name this one
Worked example

And this one

  • A 90° angle, so right.
  • Two ticks, so two equal sides. Isosceles.
  • Right isosceles, the one we started with.
And this one
Your turn

Your turn: four questions

  • Pause here and try all four.
  • Give both names wherever the question asks for them.
  • Question 3 wants a reason, not just yes or no.
Question
1A triangle has sides 5, 5 and 8 cm. What is its side name?
2A triangle has angles 90°, 45°, 45°. Give both its names.
3Can a triangle be obtuse and equilateral? Say why.
4A triangle has angles 100°, 50°, 30°. Give both its names.
Show answer

1. Isosceles. Two of its sides are equal. 2. Right isosceles. One 90° angle, and two equal angles mean two equal sides. 3. No. Equilateral means three equal sides, so three equal angles, so 180 ÷ 3 = 60° each. Every angle is 60, so none can be over 90. 4. Obtuse scalene. One angle over 90, and three different angles mean three different sides.

Worked example

The answers

  • 5, 5, 8 is isosceles. 90, 45, 45 is right isosceles.
  • No obtuse equilateral: every angle would have to be 60.
  • 100, 50, 30 is obtuse scalene.
Answer
1isosceles
2right isosceles
3no, equilateral is always 60/60/60
4obtuse scalene
Summary

What to take away

  • Ask two questions: about the angles, about the sides.
  • Equal sides always mean equal opposite angles.
  • Equilateral is 60/60/60, so it can only ever be acute.
Remember
Two questionswhat are the angles like, what are the sides like
By angleacute, right, or obtuse
By sideequilateral, isosceles, or scalene
Equilateralis always 60/60/60, so always acute
Isoscelesbase angles are equal

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