Math Tests - mathematics practice questions

Converting degrees and minutes

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Degrees and minutes

In a full circle there are 360 degrees. A degree is denoted by °, e.g. 360°.

Each degree is split up into 60 parts called minutes (of arc), each part being 1/60 of a degree. A minute is denoted by ', e.g. 1°= 60'.

Each minute is split up into 60 parts, each part being 1/60 of a minute called seconds (of arc). A second is denoted by '', e.g. 1°= 60×60 = 3600 seconds = 3600''.

Sometimes it is needed to convert the measure of the angle that is written as a decimal number into degrees and minutes, e.g 40.5 degrees. This is 40°+0.5×60' = 40° 30'

Radians

The circumference of a circle is 2πr. Based on this we set that 2π(rad) is equivalent to 360° and one π(rad) is equivalent to 180°. A radian is about $360/{2π}$ or 57.3 degrees.

Converting degrees to radians

$${α(°)×π}/{180°} = α(rad)$$
$$45°={45°×π}/180°=π/4 rad$$

Converting radians to degrees

$$απ(rad) =α× 180°$$
$$5/6 π rad = 5/6×180°=360°/3=150°$$
360°=2π270°=3π/2180°= π90°=π/2


Converting degrees to radians in terms of π

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Radians

The circumference of a circle is 2πr. Based on this we set that 2π(rad) is equivalent to 360° and one π(rad) is equivalent to 180°. A radian is about $360/{2π}$ or 57.3 degrees.

Converting degrees to radians

$${α(°)×π}/{180°} =α(rad)$$
$$45°={45°×π}/180°=π/4 rad$$

Converting radians to degrees

$$απ(rad) =α× 180°$$
$$5/6 π rad = 5/6×180°=360°/3=150°$$
360°=2π270°=3π/2180°= π90°=π/2


Definition of angle types

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Types of angles

Acute angle is less than 90° (green).
Right angle is exactly 90° (blue).
Obtuse angle is greater than 90° and less than 180° (brown).

162°90°45°



Straight angle is exactly 180° (green).
Reflex angle is greater than 180° and less than 360° (blue).
Full angle is exactly 360° (brown).
360°225°180°

And null angle is exactly 0°.



Estimating the measurement of angle (in degrees)

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Angle measure

An angle is a fraction of a full rotation (circle) which is 360°. Half the circle (straight angle) is 180°. A right angle is a quarter of a circle and is 90°.

360°270°180° 90°


Linear pair, adjacent and vertical angles

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Angle relationships

Two angles are complementary if the sum of their angles equals 90° (right angle).

Two angles are supplementary if the sum of their angles equals 180° (straight line).

Two angles are adjacent when they have a common side and a common vertex and don't overlap (on the picture below it is γ and δ, δ and η, η and α, etc.)

A linear pair is a pair of adjacent angles formed when two lines intersect (they always share the same vertex). The two angles of a linear pair are always supplementary (their sum equals 180°). On the picture below it is only α and β, β and γ.

αβγδη

Vertical angles (or vertically opposite) are the angles opposite each other when two lines cross. They are congruent (of the same measure). On the picture it is only γ and α.



Measurement of angles - parallel lines

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Pairs of angles

Two parallel lines crossed by another line (called transversal) form a special pairs of angles:

Alternate interior angles are formed on opposite sides of the transversal (green pair, blue pair). When the pair of lines are parallel, the resulting alternate interior angles are congruent (equal to each other). Each green and blue angle are supplementary (green+blue=180°)

Alternate exterior angles are formed on opposite sides of the transversal on the outside (green pair, blue pair). When the pair of lines are parallel, the resulting alternate interior angles are congruent (equal to each other). Again, each green and blue angle are supplementary (green+blue=180°)

Corresponding angles are the angles in matching corners (four pairs of different color). When the pair of lines are parallel, the corresponding angles are congruent (equal to each other).



Triangles and lines

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Angles of polygons

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Angles of polygons

Angles of a triangle add up to 180°. Angles of quadrilaterals have 360° altogether.

To calculate the sum of interior angles you can use formula (n−2)×180° where n is the number of sides. It doesn't matter whether the polygon is regular (each angle has the same size) or not.



Circle - angles at the centre and circumference (easy)

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Angles in circles

A chord of a circle is a straight line segment whose endpoints both lie on the circle (CD).

A tangent is a line which touches a circle at just one point. The radius to the point of tangency (A, B) is always perpendicular to the tangent line. Conversely, the perpendicular to a radius through the same endpoint is a tangent line.

OABCD

An inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle. Here, the circle with center O has the inscribed angle ∠ADB. A central angle in a circle is formed by lines from two points on the circle’s circumference to the center of the circle. The vertex is located at the center of a circle. Here it is angle ∠AOB. In a circle, the measure of a central angle is twice the measure of the inscribed angle with the same intercepted arc:

αOABD
OBAαD

In a circle, two inscribed angles with the same intercepted arc are congruent (they have the same angle measure):

∠ADB≅∠AEB≅∠AFB
OABDEF

Thales' theorem: Any angle inscribed in a semi-circle is a right angle.

OAB

The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

OADEFGZ

Angles of the same color have the same angle measure.

OABC

A cyclic quadrilateral has every vertex on a circle's circumference. A cyclic quadrilateral's opposite angles add to 180°:

αO180°−αRSQP180°−ββ


Circle - inscribed quadrilaterals

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Cyclic/inscribed quadrilaterals

A cyclic quadrilateral has every vertex on a circle's circumference. A cyclic quadrilateral's opposite angles add to 180°:

αO180°−αRSQP180°−ββ


Circle - triangles and tangents

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Angles in circles

A chord of a circle is a straight line segment whose endpoints both lie on the circle (CD).

A tangent is a line which touches a circle at just one point. The radius to the point of tangency (A, B) is always perpendicular to the tangent line. Conversely, the perpendicular to a radius through the same endpoint is a tangent line.

OABCD

An inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle. Here, the circle with center O has the inscribed angle ∠ADB. A central angle in a circle is formed by lines from two points on the circle’s circumference to the center of the circle. The vertex is located at the center of a circle. Here it is angle ∠AOB. In a circle, the measure of a central angle is twice the measure of the inscribed angle with the same intercepted arc:

αOABD
OBAαD

In a circle, two inscribed angles with the same intercepted arc are congruent (they have the same angle measure):

∠ADB≅∠AEB≅∠AFB
OABDEF

Thales' theorem: Any angle inscribed in a semi-circle is a right angle.

OAB

The angle between a tangent and a chord through the point of contact is equal to an angle in the alternate segment.

OADEFGZ

Angles of the same color have the same angle measure.

OABC

A cyclic quadrilateral has every vertex on a circle's circumference. A cyclic quadrilateral's opposite angles add to 180°:

αO180°−αRSQP180°−ββ


   
   

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