True or False? The diagonals of a kite bisect each other.

False. The diagonals of a kite cross at right angles , but only the main diagonal bisects the other one. The main diagonal is the kite's line of symmetry, and it is not itself cut in half.

What is special about the diagonals of a rhombus?

The diagonals of a rhombus bisect each other at right angles , so each is cut exactly in half where they cross. A rhombus is a parallelogram with four equal sides, and its two diagonals are not equal in length.

What makes a trapezium different from a parallelogram?

A trapezium has only one pair of parallel sides, and those two sides are not of equal length. A parallelogram has two pairs, which is why equal parallel sides would make the shape a parallelogram instead.

What extra properties does an isosceles trapezium have?

An isosceles trapezium has a line of symmetry, which gives it two pairs of equal angles. Its two non-parallel sides are equal in length, and so are its two diagonals.

Define a chord of a circle.

A chord is a straight line joining two points on the circumference of a circle. A diameter is the special case of a chord that passes through the centre.

Define a tangent to a circle.

A tangent is a straight line that touches the circumference at exactly one point. Where it touches, the tangent is perpendicular to the radius drawn to that point.

What is the difference between a segment and a sector ?

A segment is the region between a chord and an arc, whereas a sector is bounded by two radii and an arc. So a segment is cut off by a straight line across the circle, and a sector by two lines drawn from the centre.

How do you spot the chord and perpendicular bisector property on a diagram?

Look for a radius or a diameter that crosses a chord, or for a chord where you could usefully draw one in. Drawing that line is what creates the right-angled triangle the rest of the question needs.

National 5MathsSQAFlashcardsGeometric SkillsProperties of Shapes & Angles

Properties of Shapes & Angles (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontTriangles & Quadrilaterals
Complete the two angle sums by filling in the missing numbers of degrees.
$the angles in any triangle add to______$
$the angles in any quadrilateral add to______$

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Cards in this collection (23)

Complete the two angle sums by filling in the missing numbers of degrees.
$the angles in any triangle add to______$
$the angles in any quadrilateral add to______$
The completed sums are:
$the angles in any triangle add to 180 °$
$the angles in any quadrilateral add to 360 °$
Both hold for any such shape, however irregular it looks.
How do you find a missing angle in a triangle or a quadrilateral?
Subtract all the other angles from the total for that shape.
In a triangle where two angles are $32 °$ and $42 °$ the third is $180 - 32 - 42 = 106 °$ .
How can angles on a straight line help you find an angle in a quadrilateral?
Angles on a straight line add to $180 °$ , so an angle found in one shape gives you the angle beside it in another.
An angle of $106 °$ in a triangle leaves $74 °$ next to it, which may be the fourth angle you need.
What are the four types of triangle and what makes each one?
An equilateral triangle has three equal sides and three equal angles, and an isosceles one has two of each.
A right-angled triangle has one $90 °$ angle, and a scalene triangle has three sides all of different lengths.
True or False?
The diagonals of a kite bisect each other.
False.
The diagonals of a kite cross at right angles, but only the main diagonal bisects the other one.
The main diagonal is the kite's line of symmetry, and it is not itself cut in half.
What is special about the diagonals of a rhombus?
The diagonals of a rhombus bisect each other at right angles, so each is cut exactly in half where they cross.
A rhombus is a parallelogram with four equal sides, and its two diagonals are not equal in length.
What makes a trapezium different from a parallelogram?
A trapezium has only one pair of parallel sides, and those two sides are not of equal length.
A parallelogram has two pairs, which is why equal parallel sides would make the shape a parallelogram instead.
What extra properties does an isosceles trapezium have?
An isosceles trapezium has a line of symmetry, which gives it two pairs of equal angles.
Its two non-parallel sides are equal in length, and so are its two diagonals.
Define a regular polygon.
A polygon is a 2D shape with $n$ straight sides, and a regular polygon is one whose sides are all the same length and whose angles are all the same size.
A regular polygon with three sides is an equilateral triangle, and one with four sides is a square.
How many sides do a pentagon, a hexagon, an octagon and a decagon have?
A pentagon has $5$ , a hexagon $6$ , an octagon $8$ and a decagon $10$ .
The ones in between are a heptagon with $7$ and a nonagon with $9$ .
How are the interior and exterior angles at one corner related?
The interior and exterior angles at a corner add up to $180 °$ , because together they make a straight line.
The exterior angle is the one needed to continue the side, not the angle that completes a full turn.
Complete the rule for the sum of the interior angles of a polygon with $n$ sides.
$sum of interior angles = 180 ° \times (______)$
The completed rule is:
$sum of interior angles = 180 ° \times (n - 2)$
The $n - 2$ counts the triangles the polygon can be split into, and each of them contributes $180 °$ .
True or False?
The exterior angles of a polygon add up to $360 °$ , however many sides it has.
True.
The exterior sum is always $360 °$ , unlike the interior sum, which grows as sides are added.
A pentagon and a decagon have interior sums of $540 °$ and $1440 °$ , yet both have exterior angles adding to $360 °$ .
What is the quickest way to find an interior angle of a regular polygon?
Find the exterior angle first with $\frac{360}{n}$ , then subtract it from $180 °$ .
For a pentagon that gives $\frac{360}{5} = 72 °$ and then $180 - 72 = 108 °$ , with no extra formula to remember.
How do you find the number of sides from a regular polygon's interior angle?
Subtract the interior angle from $180 °$ to get the exterior angle, then set that equal to $\frac{360}{n}$ and solve.
An interior angle of $135 °$ gives an exterior angle of $45 °$ , and $\frac{360}{45} = 8$ sides.
How do you find a missing angle in an irregular polygon?
Work out the total with $180 ° \times (n - 2)$ , then subtract all the angles you already know.
The $\frac{360}{n}$ shortcut does not apply, because the angles of an irregular polygon are not all equal.
Define a chord of a circle.
A chord is a straight line joining two points on the circumference of a circle.
A diameter is the special case of a chord that passes through the centre.
Define a tangent to a circle.
A tangent is a straight line that touches the circumference at exactly one point.
Where it touches, the tangent is perpendicular to the radius drawn to that point.
What is the difference between a segment and a sector?
A segment is the region between a chord and an arc, whereas a sector is bounded by two radii and an arc.
So a segment is cut off by a straight line across the circle, and a sector by two lines drawn from the centre.
True or False?
A radius that passes through the midpoint of a chord meets that chord at right angles.
True.
The same property states equally well the other way round, that the perpendicular bisector of a chord passes through the centre.
Both wordings describe the one line through the centre that cuts the chord in half at $90 °$ .
How do you spot the chord and perpendicular bisector property on a diagram?
Look for a radius or a diameter that crosses a chord, or for a chord where you could usefully draw one in.
Drawing that line is what creates the right-angled triangle the rest of the question needs.
How does bisecting a chord let you use Pythagoras to find a length?
Half the chord, the distance from the centre to the chord, and a radius together form a right-angled triangle.
For a chord of $2$ cm and a radius of $2.8$ cm that gives $x^{2} + 1^{2} = 2 . 8^{2}$ , so $x = 2.62$ cm to three significant figures.
How are the diameter and the radius of a circle related?
The diameter is twice the radius, so $d = 2 r$ .
The diameter is also the circle's line of symmetry, and it is the longest chord a circle has.

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