Geometry flashcards

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State the sum of the angles on a straight line, around a point, and in a triangle.
Straight line: \(180^\circ\). Around a point: \(360^\circ\). Triangle: \(180^\circ\).
What are the angle facts for parallel lines cut by a transversal?
Corresponding angles are equal (F shape), alternate angles are equal (Z shape), and co-interior angles add to \(180^\circ\) (C shape).
What is the sum of the interior angles of an \(n\)-sided polygon?
\((n-2)\times180^\circ\). E.g. a hexagon: \(4\times180=720^\circ\).
How do you find each exterior and interior angle of a regular \(n\)-gon?
Exterior angle \(=\frac{360^\circ}{n}\); interior angle \(=180^\circ-\) exterior. For a regular octagon: \(45^\circ\) and \(135^\circ\).
How many sides does a regular polygon have if each exterior angle is \(18^\circ\)?
\(n=\frac{360}{18}=20\) sides.
What is special about the base angles of an isosceles triangle?
They are equal, because the two sides opposite them are equal. The vertex angle \(=180^\circ-2\times\)base angle.
List the tests for congruent triangles.
SSS (three sides), SAS (two sides and included angle), ASA or AAS (two angles and a side), and RHS (right angle, hypotenuse, one side).
What is the difference between congruent and similar shapes?
Congruent shapes are identical in shape and size. Similar shapes have equal angles and sides in the same ratio (one is an enlargement of the other).
(Extended) Two similar shapes have scale factor \(k\). How do lengths, areas and volumes compare?
Lengths are in ratio \(k\), areas in ratio \(k^2\), volumes in ratio \(k^3\).
(Extended) Two similar cones have heights 4 cm and 6 cm. The smaller has volume 80 cm³. Find the larger volume.
Length scale factor \(\frac64=1.5\). Volume factor \(1.5^3=3.375\). Larger volume \(=80\times3.375=270\) cm³.
State Pythagoras' theorem and when it applies.
In a right-angled triangle, \(a^2+b^2=c^2\) where \(c\) is the hypotenuse (side opposite the right angle).
A right-angled triangle has shorter sides 5 cm and 12 cm. Find the hypotenuse.
\(\sqrt{5^2+12^2}=\sqrt{169}=13\) cm.
Name the parts of a circle: chord, tangent, arc, sector, segment.
Chord: line joining two points on the circle. Tangent: touches at one point. Arc: part of the circumference. Sector: region between two radii and an arc. Segment: region between a chord and an arc.
What is the angle between a tangent and the radius at the point of contact?
\(90^\circ\). So the radius, the tangent and the line joining the centre to an outside point on the tangent form a right-angled triangle.
(Extended) How does a line from the centre of a circle to a chord's midpoint meet the chord?
At \(90^\circ\): a perpendicular from the centre bisects the chord, and a line from the centre to the midpoint is perpendicular to it. Also, two tangents drawn from the same external point are equal in length.
State the angle in a semicircle theorem.
The angle in a semicircle is \(90^\circ\): if \(AB\) is a diameter and \(C\) is on the circumference, then \(\angle ACB=90^\circ\).
(Extended) State the angle at the centre theorem.
The angle at the centre of a circle is twice the angle at the circumference when both stand on the same arc.
(Extended) What is true of angles in the same segment, and of opposite angles in a cyclic quadrilateral?
Angles in the same segment (on the same chord) are equal. Opposite angles of a cyclic quadrilateral add up to \(180^\circ\).
(Extended) State the alternate segment theorem.
The angle between a tangent and a chord equals the angle in the alternate (opposite) segment.
Describe how to construct the perpendicular bisector of a line segment with compasses.
Set the compasses to more than half the length. Draw arcs from both ends so they cross above and below. Join the two crossing points; this line bisects the segment at \(90^\circ\).
How do you bisect an angle using compasses?
Draw an arc across both arms from the vertex. From each crossing point draw equal arcs that meet inside the angle. Join the vertex to that meeting point.
What are the properties of a parallelogram, a rhombus and a kite?
Parallelogram: opposite sides parallel and equal. Rhombus: all sides equal, diagonals bisect at \(90^\circ\). Kite: two pairs of adjacent equal sides, diagonals cross at \(90^\circ\).
What is the order of rotational symmetry and the number of lines of symmetry of a regular pentagon?
Five lines of symmetry and rotational symmetry of order 5.
What are the bearing rules?
A bearing is measured clockwise from north, written with three figures, e.g. 045°. The back bearing is the bearing \(\pm180^\circ\).
The bearing of \(B\) from \(A\) is \(070^\circ\). Find the bearing of \(A\) from \(B\).
\(070^\circ+180^\circ=250^\circ\).
Find the missing angle: the angles of a triangle are \(x\), \(2x\) and \(3x\).
\(6x=180^\circ\), so \(x=30^\circ\) and the angles are \(30^\circ\), \(60^\circ\), \(90^\circ\).
What are the properties of a net and how many faces, edges and vertices does a cuboid have?
A net is a flat shape that folds into a solid. A cuboid has 6 faces, 12 edges and 8 vertices.
What do you know about the diagonals of a rectangle?
They are equal in length and bisect each other, so the four half-diagonals are equal and each pair of triangles formed is isosceles.

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