Triangles — SPI Mathematics Formulas and Traps

Angles, Sides and Pythagoras Theorem

The angle sum of a triangle

Angle sum

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

The exterior angle of a triangle

Exterior angle property

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

Any two sides are longer than the third

Triangle inequality

a + b > c \qquad |a - b| < c < a + b

Pythagoras theorem and its converse

Pythagoras theorem

\text{hypotenuse}^2 = \text{base}^2 + \text{height}^2

Common traps

Dividing 180 by the wrong number

For the ratio 2 : 3 : 4 you divide 180° by the SUM of the parts, 9, to get

k

. 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^\circ

, the other two are

70^\circ

each.

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.

Equal is not enough

Sides 2, 3 and 5 do not make a triangle:

2 + 3

equals 5 instead of being greater, so the points fall on one straight line.

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

\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

7^2 + 24^2 = 25^2

, not

7^2 + 25^2

.

Congruence of Triangles

Learn this subtopic in the notes

Congruent triangles and the order of the letters

Matching parts

\triangle ABC \cong \triangle PQR \Rightarrow AB = PQ,\ BC = QR,\ CA = RP,\ \angle A = \angle P,\ \angle B = \angle Q,\ \angle C = \angle R

The congruence rules SSS, SAS, ASA, AAS and RHS

Rule	Parts that must be equal	Note
SSS	All three pairs of sides	No angle is needed
SAS	Two sides and the angle they make	The angle must sit between the two sides
ASA	Two angles and the side joining their corners	The side must sit between the two angles
AAS	Two angles and a side not between them	Works because the third angles are then equal too
RHS	Right angle, hypotenuse, one other side	Only for right triangles
AAA	Three angles	Not a congruence rule Equal angles give similar triangles, which may differ in size.
SSA	Two sides and an angle not between them	Not a congruence rule The right-angled case of this pattern is RHS, which does work.

The letters name the parts in order round the triangle: S for a side, A for an angle.

Common traps

Matching letters in alphabetical order

\triangle ABC \cong \triangle PQR

is only right if A really matches P. When AB = QR, B and R are not paired by alphabet but by the sides they share. Work out each corner before writing the statement.

Equal angles alone are not congruence

Two triangles with the same three angles can be different sizes. They are similar, not congruent, until a pair of matching sides is also equal.

Calling RHS when the side given is not the hypotenuse

RHS needs the HYPOTENUSE of each right triangle to be equal, plus one more side. Two equal legs with the right angle between them is SAS, not RHS.

An angle that is not between the sides

Two equal sides and an equal angle at the wrong corner is SSA, and it does not prove congruence. Check that the angle's letter is the letter the two sides share.

Similar Triangles and Proportional Sides

Learn this subtopic in the notes

Corresponding sides are in one ratio

Ratio of corresponding sides

\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \qquad \frac{\text{height}_1}{\text{shadow}_1} = \frac{\text{height}_2}{\text{shadow}_2}

Basic proportionality theorem

DE \parallel BC \Rightarrow \frac{AD}{DB} = \frac{AE}{EC} \quad \text{and} \quad \frac{AD}{AB} = \frac{AE}{AC}

The angle bisector divides the opposite side

Angle bisector theorem

\frac{BD}{DC} = \frac{AB}{AC}

When two triangles are similar

Criterion	What you must show	Example
AA	Two pairs of equal angles	Angles 40°, 60° in one and 40°, 60° in the other
SSS	Three pairs of sides in one ratio	Sides 3, 4, 5 and 6, 8, 10 (ratio 1 : 2)
SAS	Two pairs of sides in one ratio, included angles equal	4 and 6 with 50° between; 6 and 9 with 50° between
Parallel line	A line parallel to one side cuts off a small triangle	Gives equal corresponding angles, so AA
Every pair of equilateral triangles	Nothing more: all angles are 60°	Always similar Congruent triangles are similar too, with ratio 1 : 1.

Similar means the same shape; congruent means the same shape and the same size.

Common traps

Finding the angle at the wrong corner

In

\triangle ABC \sim \triangle ADE

, the angle at D matches the angle at B because both are in second place. Find the angle at B first, then copy it to D.

Pairing sides by their position on the page

In

\triangle ACB \sim \triangle APQ

, C matches P and B matches Q, so CB pairs with PQ and AB pairs with AQ. Read the pairs from the statement, never from how the figure looks.

Turning the ratio upside down

If the second triangle is the bigger one, its sides are the small sides TIMES the factor. Check the answer is bigger when it should be: a 2 m stick with a 3 m shadow means the tower is shorter than its shadow.

Mixing a part with a whole

\frac{AD}{DB} = \frac{AE}{EC}

compares part with part;

\frac{AD}{AB} = \frac{AE}{AC}

compares part with whole.

\frac{AD}{AB} = \frac{AE}{EC}

mixes the two and is wrong.

Joining B to the wrong point

To divide AB in the ratio 4 : 7, B is joined to

A_{11}

, the last of

4 + 7

steps. The parallel line then goes through

A_4

.

Pairing a part with the far side

BD sits next to AB, so

\frac{BD}{DC} = \frac{AB}{AC}

. Writing

\frac{BD}{DC} = \frac{AC}{AB}

puts the longer part next to the shorter side.

Areas of Triangles and Similar Triangles

Learn this subtopic in the notes

Triangles with the same height

Area of a triangle

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \qquad \frac{\text{ar}_1}{\text{ar}_2} = \frac{\text{base}_1}{\text{base}_2} \text{ (same height)}

Areas of similar triangles

Area ratio of similar triangles

\frac{\text{ar}(\triangle ABC)}{\text{ar}(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{\text{height}_1}{\text{height}_2}\right)^2

Common traps

Squaring a ratio that should not be squared

Triangles that only share a height are not similar, so their areas go as the bases, not as the squares of the bases. Bases 6 and 9 give areas 2 : 3, not 4 : 9.

Giving the side ratio as the area ratio

Sides in the ratio 2 : 3 give areas 4 : 9. Leaving the ratio as 2 : 3 is the most common wrong option.

Forgetting the square root on the way back

Areas 64 and 121 give sides (and heights) 8 : 11. Writing 64 : 121 as the side ratio skips the square root.

Stopping at the whole triangle

When a median splits the similar triangle, find the whole triangle's area first, then halve it. The whole area is usually printed as a wrong option.

Triangles formulas

Angles, Sides and Pythagoras Theorem

The angle sum of a triangle

The exterior angle of a triangle

Any two sides are longer than the third

Pythagoras theorem and its converse

Common traps

Congruence of Triangles

Congruent triangles and the order of the letters

The congruence rules SSS, SAS, ASA, AAS and RHS

Common traps

Similar Triangles and Proportional Sides

Corresponding sides are in one ratio

The angle bisector divides the opposite side

When two triangles are similar

Common traps

Areas of Triangles and Similar Triangles

Triangles with the same height

Areas of similar triangles

Common traps

More SPI Mathematics formula sheets