Angles, parallel lines and polygonsIB MYP Maths Standard: Revision notes

Section 1

Angles on lines and at a point

Angles on a straight line add up to $180^\circ$ . Angles around a point add up to $360^\circ$ . When two straight lines cross, the angles opposite each other are vertically opposite angles and they are equal. Example: two lines cross and one angle is $(3x+10)^\circ$ next to $(2x+20)^\circ$ on a straight line. Then $5x+30=180$ , so $x=30$ . Always write the reason next to each step, for example 'angles on a straight line add to $180^\circ$ '.

Key termsstraight linevertically oppositeangles at a point

Common mistake

Treating two angles on a straight line as equal. Equal means vertically opposite; adjacent angles on a line add to $180^\circ$ .

Section 2

Angles in triangles

The three interior angles of any triangle add up to $180^\circ$ . If you extend one side, the exterior angle equals the sum of the two opposite interior angles. An isosceles triangle has two equal angles opposite its two equal sides, and an equilateral triangle has three angles of $60^\circ$ . Example: a triangle has angles $x$ and $2x+5$ and the exterior angle is $5x-15$ . Then $5x-15=3x+5$ , so $x=10$ .

Key termsangle sum of a triangleexterior angleisosceles

Exam tip

Exterior angle = sum of the two opposite interior angles. It saves a step, but also say 'angles on a straight line' as a second reason.

Section 3

Parallel lines

Parallel lines never meet. A straight line that crosses them is a transversal. It creates three kinds of equal or supplementary pairs:

Corresponding angles (F-shape) are equal.
Alternate angles (Z-shape) are equal.
Co-interior angles (C or U-shape) add up to $180^\circ$ . Vertically opposite angles are equal too. Always name the rule and mention that the lines are parallel.

Key termscorrespondingalternateco-interiortransversal

Common mistake

Using a parallel-line reason when the lines are not stated to be parallel. The reason only works if the lines are parallel.

Section 4

Interior and exterior angles of polygons

A polygon with $n$ sides can be split into $n-2$ triangles, so the sum of the interior angles is $(n-2)\times180^\circ$ . The exterior angles of any polygon add up to $360^\circ$ . For a regular polygon (equal sides and equal angles): $\text{exterior angle}=\frac{360^\circ}{n},\qquad \text{interior angle}=180^\circ-\text{exterior angle}.$ Example: a regular polygon with exterior angle $15^\circ$ has $360\div15=24$ sides, each interior angle is $165^\circ$ and the interior angles add to $22\times180=3960^\circ$ .

Key termsinterior angleexterior angleregular polygon

Common mistake

Forgetting that interior plus exterior angle at a vertex is $180^\circ$ . Once you have one, find the other by subtracting from $180^\circ$ .

Section 5

Justifying statements with reasons

In MYP maths you must communicate clearly. Each step in an angle problem needs a reason in words, for example 'alternate angles, $PQ\parallel RS$ ' or 'angles in a triangle add to $180^\circ$ '. A good layout is: write the angle, the calculation, then the reason. If you are asked to show or justify something, finish with a clear conclusion that links back to the question, such as 'so the angle is obtuse because it is greater than $90^\circ$ '.

Key termsreasonjustify

Exam tip

Name the geometric rule in words. 'Because they look equal' is never a reason.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Angles, parallel lines and polygons

Straight lines $AB$ and $CD$ cross at the point $O$ . Angle $AOC=(3x+10)^\circ$ and angle $COB=(2x+20)^\circ$ .
Ray $OE$ bisects angle $AOC$ . Find angle $EOB$ .2 marks
Line $PQ$ is parallel to line $RS$ . A straight line crosses $PQ$ at $X$ and $RS$ at $Y$ , and is extended beyond $Y$ to a point $Z$ . Points $P$ and $R$ are on the same side of the crossing line, and $X$ is above $Y$ . Angle $PXY=64^\circ$ .
Find the size of angle $RYZ$ and give a reason for your answer.2 marks
In triangle $ABC$ , side $BC$ is extended to a point $D$ . Angle $BAC=x^\circ$ , angle $ABC=(2x+5)^\circ$ and the exterior angle $ACD=(5x-15)^\circ$ .
Write an equation in $x$ using the exterior angle property of a triangle, and hence find $x$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).