IMAT Mathematics · Geometry

Angles, Triangles and Polygons

Angle facts for crossing and parallel lines, the angle sum and Pythagoras for triangles, similarity, and the angle sums of polygons.

All Geometry notes

Why this matters

Past questions used Pythagoras on a right triangle of given area (2016), similar triangles cut off by a parallel line (2021) and the area of a symmetric pentagon (2018). These facts also sit underneath most circle and coordinate questions.

Concept 1 of 4: Angle facts for crossing lines and parallel lines

A straight line is a half turn, 180°, and a full turn round a point is 360°. When a third line (a transversal) crosses two parallel lines, it meets both at the same slope, so the angles at the two crossings copy each other.

Definition

A transversal is a line crossing two or more other lines.
Corresponding angles sit in the same position at each crossing (an F shape).
Alternate angles lie between the parallels on opposite sides of the transversal (a Z shape).
Co-interior angles lie between the parallels on the same side of the transversal (a C shape).

Situation	Fact
Angles on a straight line	Add up to $180^\circ$
Angles around a point	Add up to $360^\circ$
Vertically opposite angles, where two lines cross	Are equal
Corresponding angles on parallel lines	Are equal
Alternate angles on parallel lines	Are equal
Co-interior angles on parallel lines	Add up to $180^\circ$ Co-interior angles are supplementary, not equal.

Practice this conceptself-check · 4 quick reps

Only parallel lines give equal alternate and corresponding angles

The equal-angle facts need the two lines to be parallel. If a diagram does not say so (arrows on the lines or words in the question), you cannot assume it, even if the lines look parallel.

Concept 2 of 4: Triangles: angle sum, Pythagoras and special right triangles

Draw a line through one corner parallel to the opposite side and the three angles of the triangle fit together on it, so they add up to 180°. In a right triangle the squares on the two short sides together equal the square on the longest side. Two right triangles come up so often that their side ratios are worth knowing by heart.

Definition

The angles of a triangle add up to $180^\circ$ . An exterior angle equals the sum of the two interior angles not next to it.
Area $= \tfrac{1}{2} \times \text{base} \times \text{perpendicular height}$ .
Pythagoras: in a right triangle with legs $a, b$ and hypotenuse $c$ (opposite the right angle), $a^2 + b^2 = c^2$ .
Isosceles: two equal sides and two equal base angles; the height from the apex bisects the base.
45-45-90: sides $1 : 1 : \sqrt{2}$ . 30-60-90: sides $1 : \sqrt{3} : 2$ (shortest side faces 30°).
Equilateral with side $s$ : height $\dfrac{\sqrt{3}}{2}s$ , area $\dfrac{\sqrt{3}}{4}s^2$ .

Pythagoras and triangle area

a^2 + b^2 = c^2 \qquad A = \tfrac{1}{2}bh

$a, b$ the two legs, which meet at the right angle
$c$ the hypotenuse
$b, h$ in the area formula: a base and the height perpendicular to it

Worked example

An isosceles triangle has two sides of 13 cm and a base of 10 cm. Find its height and its area.

Practice this conceptself-check · 4 quick reps

The hypotenuse is the side opposite the right angle

a^2 + b^2 = c^2

c

must be the longest side, facing the right angle. Adding the squares of a leg and the hypotenuse gives nonsense; when the hypotenuse is known, subtract:

a^2 = c^2 - b^2

Concept 3 of 4: Congruent and similar triangles

Similar triangles are enlargements of each other: same angles, every side multiplied by the same scale factor. A line drawn parallel to one side of a triangle cuts off a smaller copy of it, because the parallel line makes equal corresponding angles. Areas grow faster than lengths: by the square of the scale factor.

Definition

Congruent triangles are identical in shape and size. Tests: SSS (three sides), SAS (two sides and the angle between), ASA (two angles and a side), RHS (right angle, hypotenuse, side).
Similar triangles have equal angles; their corresponding sides are in a fixed ratio $k$ , the scale factor. Two equal angles are enough to prove similarity.
A line parallel to one side of a triangle cuts off a triangle similar to the whole.
If lengths scale by $k$ , areas scale by $k^2$ .

Similar triangles

\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c} = k \qquad \frac{\text{Area}'}{\text{Area}} = k^2

$a, b, c$ sides of one triangle
$a', b', c'$ the matching sides of the other
$k$ scale factor

Worked example

In triangle ABC, point D is on AB and point E is on AC, with DE parallel to BC. AD = 4 cm, DB = 6 cm and DE = 5 cm. Find BC, and the ratio of the areas of triangles ADE and ABC.

Practice this conceptself-check · 4 quick reps

Match the sides through the angles, not through their position on the page

In the cut-off triangle, AD matches AB (the whole side), not DB. Use

AD/AB

, not

AD/DB

, for the scale factor; the second ratio gives a wrong answer that is usually among the options.

Concept 4 of 4: Polygons: interior and exterior angles

From one corner of an n-sided polygon you can draw diagonals that split it into

n - 2

triangles, each holding 180°. Walking round any convex polygon you turn through one full turn in total, so the exterior angles always add up to 360°.

Definition

The interior angles of an n-sided polygon add up to $(n - 2) \times 180^\circ$ .
The exterior angles (one at each corner, between a side and the next side extended) add up to $360^\circ$ .
At each corner, interior + exterior $= 180^\circ$ .
In a regular polygon (all sides and angles equal), each exterior angle is $360^\circ / n$ .

Polygon angles

S_{\text{int}} = (n - 2) \times 180^\circ \qquad \text{regular: each exterior} = \frac{360^\circ}{n}

$n$ number of sides

Worked example

Find the sum of the interior angles of an octagon and the size of each interior angle of a regular octagon.

Practice this conceptself-check · 4 quick reps

Exterior angles total 360° for every polygon

The interior sum grows with the number of sides, but the exterior angles of any convex polygon add up to

360^\circ

. For a regular polygon, start from the exterior angle: it is the quickest route to

n

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 (3)

Triangles: angle sum, Pythagoras and special right triangles
Pythagoras and triangle area
$a^2 + b^2 = c^2 \qquad A = \tfrac{1}{2}bh$
Congruent and similar triangles
Similar triangles
$\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c} = k \qquad \frac{\text{Area}'}{\text{Area}} = k^2$
Polygons: interior and exterior angles
Polygon angles
$S_{\text{int}} = (n - 2) \times 180^\circ \qquad \text{regular: each exterior} = \frac{360^\circ}{n}$

Reference tables (1)

Angle facts for crossing lines and parallel lines6 rows

Situation	Fact
Angles on a straight line	Add up to $180^\circ$
Angles around a point	Add up to $360^\circ$
Vertically opposite angles, where two lines cross	Are equal
Corresponding angles on parallel lines	Are equal
Alternate angles on parallel lines	Are equal
Co-interior angles on parallel lines	Add up to $180^\circ$ Co-interior angles are supplementary, not equal.

Watch out for (4)

Only parallel lines give equal alternate and corresponding angles
→ Angle facts for crossing lines and parallel lines
The hypotenuse is the side opposite the right angle
→ Triangles: angle sum, Pythagoras and special right triangles
Match the sides through the angles, not through their position on the page
→ Congruent and similar triangles
Exterior angles total 360° for every polygon
→ Polygons: interior and exterior angles

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