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SPI Mathematics · Triangles

Angles, Sides and Pythagoras Theorem

The three angles of a triangle add to 180°, an exterior angle equals the two far interior angles, any two sides add to more than the third, and in a right triangle the square on the hypotenuse equals the squares on the other two sides.

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Why this matters

Eight questions in the bank, seven of them EASY: angles in a ratio, an exterior angle, a triangle that cannot be drawn, two ladders against a wall and a 7, 24, 25 triangle. Every later page of this chapter leans on these four facts, so learn them first.

Concept 1 of 4: The angle sum of a triangle

Tear the three corners off a paper triangle and put them side by side: they make a straight line. That is why the three angles always add to 180°, whatever the shape of the triangle.

Definition

  • The three angles of every triangle add to 180°.
  • Angles in a ratio a:b:ca : b : c: write them as ak,bk,ckak, bk, ck, add to 180° and find kk.
  • In an isosceles triangle the angles opposite the two equal sides are equal.
  • In an equilateral triangle every angle is 60°.

Angle sum

∠A+∠B+∠C=180∘ak+bk+ck=180∘\angle A + \angle B + \angle C = 180^\circ \qquad ak + bk + ck = 180^\circ

Worked example

The angles of a triangle are in the ratio 2 : 3 : 4. Find them.
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The same idea in a practice question:

Practice questionEasy

Example 1 · Triangles · Angles, Sides and Pythagoras Theorem

The three angles of a triangle are in the ratio 5:4:1. Find all the angles of the triangle.

Dividing 180 by the wrong number

For the ratio 2 : 3 : 4 you divide 180° by the SUM of the parts, 9, to get kk. Dividing by 3 (the number of angles) or by the largest part gives angles that do not add to 180°.

Which angles of an isosceles triangle are equal

The two equal angles are the ones opposite the equal sides, not the angle between them. If the angle between the equal sides is 40∘40^\circ, the other two are 70∘70^\circ each.
Drill 2 more on the angle sum of a triangle

Concept 2 of 4: The exterior angle of a triangle

Stretch one side of a triangle past a corner and you make an outside angle. It fills the gap the two far corners leave, so it equals their sum.

Definition

  • An exterior angle is made by one side and the next side extended.
  • It equals the sum of the two interior opposite angles (the two corners away from it).
  • It and the interior angle next to it make a straight line, so they add to 180°.

Exterior angle property

exterior angle=sum of the two interior opposite angles\text{exterior angle} = \text{sum of the two interior opposite angles}

Worked example

An exterior angle of a triangle is 120∘120^\circ and one interior opposite angle is 50∘50^\circ. Find the other interior opposite angle.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Triangles · Angles, Sides and Pythagoras Theorem

An exterior angle of a triangle is of measure 90∘90^\circ and one of its interior opposite angles is of measure 45∘45^\circ. Find the measure of the other interior opposite angle.

Taking 180° away instead of the given angle

180° minus the exterior angle gives the interior angle NEXT to it. The far interior angle you are asked for is the exterior angle minus the other far angle.

Concept 3 of 4: Any two sides are longer than the third

Walking straight from one corner to another is always shorter than going round through the third corner. So the two other sides together must be longer than the side they go round.

Definition

  • In every triangle the sum of any two sides is greater than the third side.
  • Three lengths make a triangle only if the two shorter ones add to more than the longest.
  • If they add to exactly the longest, the three points lie on one line and there is no triangle.
  • So the third side lies strictly between the difference and the sum of the other two.

Triangle inequality

a+b>c∣a−b∣<c<a+ba + b > c \qquad |a - b| < c < a + b

Worked example

Can a triangle have sides 3 cm, 4 cm and 8 cm?
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Triangles · Angles, Sides and Pythagoras Theorem

It is not possible to construct a △ABC\triangle ABC with BC = 5 cm, ∠\angle B = 75∘75^\circ and AB + AC equal to ____.

Equal is not enough

Sides 2, 3 and 5 do not make a triangle: 2+32 + 3 equals 5 instead of being greater, so the points fall on one straight line.

Concept 4 of 4: Pythagoras theorem and its converse

In a right triangle the longest side, the hypotenuse, sits opposite the right angle. The square built on it has exactly the area of the squares on the other two sides put together.

Definition

  • In a triangle right-angled at B: AC2=AB2+BC2AC^2 = AB^2 + BC^2, where AC is the hypotenuse.
  • To find a leg, subtract: leg2=hypotenuse2−other leg2\text{leg}^2 = \text{hypotenuse}^2 - \text{other leg}^2.
  • Converse: if the square of the longest side equals the sum of the squares of the other two, the triangle is right-angled.
  • If the longest side's square is more than that sum, the triangle is obtuse; if less, it is acute.
  • Triples worth knowing: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25 (and their multiples, such as 6, 8, 10).

Pythagoras theorem

hypotenuse2=base2+height2\text{hypotenuse}^2 = \text{base}^2 + \text{height}^2

Worked example

A ladder 25 m long leans against a wall and its top reaches a window 24 m above the ground. How far is the foot of the ladder from the wall?
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Triangles · Angles, Sides and Pythagoras Theorem

A 34 m long ladder reached a window 16 m from the ground on placing it against a wall. Find the distance of the foot of the ladder from the wall.

Adding when you should subtract

When the hypotenuse is given and a leg is asked, subtract the squares. A 25 m ladder with its top 24 m up stands 7 m out, not 252+242\sqrt{25^2 + 24^2} m.

Testing the wrong side as the hypotenuse

In the converse, square the LONGEST side and compare it with the other two. For 7, 24 and 25, check 72+242=2527^2 + 24^2 = 25^2, not 72+2527^2 + 25^2.
Drill 2 more on pythagoras theorem and its converse

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

  • The angle sum of a triangle

    Angle sum

    ∠A+∠B+∠C=180∘ak+bk+ck=180∘\angle A + \angle B + \angle C = 180^\circ \qquad ak + bk + ck = 180^\circ
  • The exterior angle of a triangle

    Exterior angle property

    exterior angle=sum of the two interior opposite angles\text{exterior angle} = \text{sum of the two interior opposite angles}
  • Any two sides are longer than the third

    Triangle inequality

    a+b>c∣a−b∣<c<a+ba + b > c \qquad |a - b| < c < a + b
  • Pythagoras theorem and its converse

    Pythagoras theorem

    hypotenuse2=base2+height2\text{hypotenuse}^2 = \text{base}^2 + \text{height}^2

Watch out for (6)

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