2D Shapes & Angles
<p>In the 11+ exam, you need to recognise shapes, know their properties, and work with angles. This guide covers the essential facts and shows you how to use them to solve problems.</p>
<h2>Types of Angles</h2>
<ul>
<li><strong>Acute angle:</strong> less than 90°</li>
<li><strong>Right angle:</strong> exactly 90° (shown by a small square)</li>
<li><strong>Obtuse angle:</strong> between 90° and 180°</li>
<li><strong>Straight line:</strong> exactly 180°</li>
<li><strong>Reflex angle:</strong> between 180° and 360°</li>
</ul>
<h2>Angle Rules</h2>
<ul>
<li>Angles on a straight line add up to <strong>180°</strong></li>
<li>Angles around a point add up to <strong>360°</strong></li>
<li>Vertically opposite angles are <strong>equal</strong></li>
<li>Angles in a triangle add up to <strong>180°</strong></li>
<li>Angles in a quadrilateral add up to <strong>360°</strong></li>
</ul>
<p><strong>Worked Example:</strong> Two angles in a triangle are 65° and 48°. Find the third angle.</p>
<p>Third angle = 180° − 65° − 48° = 67°.</p>
<h2>Triangles</h2>
<ul>
<li><strong>Equilateral:</strong> all sides equal, all angles 60°</li>
<li><strong>Isosceles:</strong> two sides equal, two base angles equal</li>
<li><strong>Scalene:</strong> no sides or angles equal</li>
<li><strong>Right-angled:</strong> one angle is 90°</li>
</ul>
<p><strong>Worked Example:</strong> An isosceles triangle has one angle of 40°. Find the other angles.</p>
<p>If 40° is the odd angle, the two base angles are each (180° − 40°) ÷ 2 = 70°. If 40° is one of the equal angles, the third angle is 180° − 40° − 40° = 100°. Both solutions are valid — read the question carefully for clues about which it is.</p>
<h2>Quadrilaterals</h2>
<ul>
<li><strong>Square:</strong> 4 equal sides, 4 right angles</li>
<li><strong>Rectangle:</strong> opposite sides equal, 4 right angles</li>
<li><strong>Parallelogram:</strong> opposite sides parallel and equal, opposite angles equal</li>
<li><strong>Rhombus:</strong> 4 equal sides (a tilted square), opposite angles equal</li>
<li><strong>Trapezium:</strong> exactly one pair of parallel sides</li>
<li><strong>Kite:</strong> two pairs of adjacent sides equal, one pair of opposite angles equal</li>
</ul>
<h2>Regular Polygons</h2>
<p>A regular polygon has all sides and all angles equal. The interior angle of a regular polygon with n sides is (n − 2) × 180° ÷ n.</p>
<ul>
<li>Regular pentagon (5 sides): interior angle = 108°</li>
<li>Regular hexagon (6 sides): interior angle = 120°</li>
<li>Regular octagon (8 sides): interior angle = 135°</li>
</ul>
<h2>Lines of Symmetry and Rotational Symmetry</h2>
<p>A line of symmetry divides a shape into two mirror-image halves. A square has 4 lines of symmetry; an equilateral triangle has 3; a rectangle has 2. Rotational symmetry is the number of positions a shape looks the same when rotated — a square has rotational symmetry of order 4.</p>
<h2>Practice Tips</h2>
<ul>
<li>Learn the angle sum for triangles (180°) and quadrilaterals (360°) — they come up constantly</li>
<li>Always mark known angles on the diagram and look for angle rules to find the rest</li>
<li>If a question says "not drawn to scale," do NOT measure with a protractor — calculate</li>
<li>Practise drawing lines of symmetry on printed shapes with a mirror or by folding</li>
</ul>