ANGLES AROUND A POINT

Angles around a Point

Angles around a point will at all times add up to 360 degrees.

The angles above all add to 360°
53° + 80° + 140° + 87° = 360°

Due to this, we can find an unknown angle.

Example:

In the figure above what is angle “c”?

To calculate angle c, we take the sum of the known angles and take that from 360°

Sum of given angles= 110° + 75° + 50° + 63°
= 298°
Angle c= 360° − 298°
= 62°

Recall that
(a) angles on a line add up to 180º
And (b) angles at a point add up to 360º

These are two results which are of a great assistance when finding the size of unknown angles.

Example Questions

In the questions below, you are required to make use of the facts above to calculate the missing angles.

Practice Questions

What does angles around a point means? Angles around a point means that lines from one point form angles around that point.

What is special about angles around a point?
The special thing about angles around a point is that angles around a point add up to 360 degrees.

Does it matter how many angles there are?
No. Any number of angles around a point add up to 360 degrees.

Thus there could be only 2 or 3 angles around a point and they would still add up to 360 degrees?
Yes, so long as the angles go all the way around the point.

Can we do a probable question using the fact that the angles around a point add up to 360 degrees?
Yes, you may be given the size of all except one of the angles around a point and asked to find the size of the unknown angle.

The steps to follow while calculating the unknown angle

STEP 1 —- ADD the known angles together 135 + 90 + 70 = 295

STEP 2 —- SUBTRACT the answer to STEP 1 from 360 360 – 295 = 65

Answer

Angle A = 65 degrees
Angle A = 65°

Pairs of Angles

There are a few special relationships between “pairs” of angles.

Adjacent Angles are two angles that share a common vertex, a common side, and no common interior points. They share a vertex and side, but do not overlap.

∠1 and ∠2 are adjacent angles.
∠ABC and ∠1 are NOT adjacent angles.
(∠ABC overlaps ∠1.)

A Linear Pair is two adjacent angles whose non-common sides form opposite rays.

∠1 and ∠2 form a linear pair.
The line through points A, B and C is a straight line.

∠1 and ∠2 are supplementary.

If two angles form a linear pair, the angles are supplementary. A linear pair forms a straight angle which is made up of 180º, therefore, you have 2 angles whose measures add to 180, which means they are supplementary.

If two congruent angles form a linear pair, the angles are right angles.
If two congruent angles add to 180º, each angle is made up of 90º, forming right angles.

Vertical Angles are two angles whose sides form two pairs of opposite rays (straight lines).

Vertical angles are located across from one another in the corners of the “X” formed by the two straight lines.

∠1 and ∠2 are vertical angles. ∠3 and ∠4 are vertical angles.

Vertical angles are not adjacent. ∠1 and ∠3 are not vertical angles (they are a linear pair).

Vertical angles are always equal in measure.

Vertical angles are congruent. Vertical angles, such as ∠1 and ∠2, form linear pairs with the same angle, ∠4, givingm∠1 + m∠4 = 180 and m∠2 + m∠4 = 180. With substitution, m∠1 = m∠2 and they are congruent.

Complementary Angles are two angles whose sum is 90º.

Complementary angles can be placed in a way that they form perpendicular lines, or they may be two different angles. ∠1 and ∠2 are complementary. ∠P and ∠Q are complementary.

Complements of the same angle, or congruent angles, are congruent. If m∠a is complementary to the m∠b, and m∠c is complementary to m∠b, then m∠a = m∠c. Consider m∠a = 60º, m∠b = 30º and m∠c = 60º.

The acute angles of a right triangle are complementary. The sum of the angles in a triangle add to 180º. After subtracting 90º for the right angle, there are 90º left for the two acute angles, making them complementary.

Supplementary Angles are two angles the sum of whose measures is 180º.

Supplementary angles can be placed so they form a linear pair (straight line), or they may be two separate angles. ∠1 and ∠2 are supplementary. ∠P and ∠Q are supplementary.

The line through points A, B and C is a straight line.

Supplements of the same angle, or congruent angles, are congruent. If m∠a is supplementary to the m∠b, and m∠c is supplementary to m∠b, then m∠a = m∠c. Consider m∠a = 60º, m∠b = 120º and m∠c = 60º.

An angle is the intersection of two lines with a common endpoint.

Remember that an angle is named using three letters, where the middle letter corresponds to the vertex of the angle. The angle at the right is

Angle Addition Postulate

If D lies in the interior of<abc then="" mabd="" _="" mdbc="m

This concept is sometimes stated as “the whole is equal to the sum of its parts”.

Types of Angles (Definitions)

An acute angle is an angle that is less than 90°

• A right angle is an angle that is 90°.
• An obtuse angle is an angle that is more than 90°, but less than 1800
• A straight angle is an angle that is 180°.
• A reflex angle is an angle that is more than 180°.

Pairs of Angles

The definitions above apply to angles when we look at one angle alone, but there are as well special relationships between pairs of angles.

Adjacent Angles are 2 angles that share a common vertex, a common side and no common interior points. They share a vertex and share a side, but do not overlap.

<1 and <2 are adjacent angles.
<1 and<1)=""

Vertical Angles are 2 angles whose sides form two pairs of opposite line (straight lines). Vertical angles are not adjacent. They are situated across from one another in the corners of the “X” produced by the two straight lines. They are at all times equal in measure.

<1 and ❤ are vertical angles.
<2 and <4 are vertical angles.
<1 and <2 are NOT vertical.

THEOREM:

Vertical angles are congruent.

Complementary Angles are 2 angles the sum of whose measures is 90°.
Complementary angles can be placed so that they form perpendicular lines, but do not “have to be” in this configuration.

<1 and <2 are complementary.<1="" are="" not="" complementary.="" (the="" rays="" perpendicular)=""

THEOREM:

Complements of the same angle, or congruent angles, are congruent.

Supplementary Angles are 2 angles the sum of whose measures is 180°. Supplementary angles can be placed so that they form a straight line (a linear pair), but they do not “have to be” in this configuration.

<1 and <2 are supplementary. The line passing through points A, B, and C is a straight line.

THEOREM:

Supplements of the same angle, or congruent angles, are congruent.
A Linear Pair is 2 adjacent angles whose non-common sides form opposite rays. The angles MUST be adjacent.

<1 and <2 form a linear pair.
The line passing through points A, B, andC is a straight line.
<1 and <2 are supplementary.

THEOREM:

If two angles form a linear pair, they are supplementary.

THEOREM:

If two congruent angles form a linear pair, they are right angles.

An angle of 150 degrees as well has a reference angle of 30 degrees (180 – 150). An angle of 750 degrees has a reference angle of 30 degrees (750 – 720)

Vertical and adjacent angle pairs

Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles.

When two straight lines intersect at a point, four angles are formed. Pairwise these angles are named with respect to their location in relation to each other.

A pair of angles opposite each other, formed by two intersecting straight lines that form an “X”-like shape, are known as vertical angles or opposite angles or vertically opposite angles. They are abbreviated as vert. opp. ∠s.

The equality of vertically opposite angles is referred to as the vertical angle theorem. The suggestion showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measurement.

Thales concluded that one could prove that all vertical angles are equal if one accepted a few common notionslike: all straight angles are equal, equals added to equals are equal, and equals subtracted from equals are equal.

Assuming the measure of Angle A = x. When two adjacent angles form a straight line, they are supplementary. Thus, the measure of Angle C = 180 − x. In the same way, the measure of Angle D = 180 − x. Both Angle C and Angle D have measures equal to 180 − x and are congruent. Since Angle B is supplementary to both Angles C and D, either of these angle sizes may be used to calculate the size of Angle B. With the use of the size of either Angle C or Angle D we calculate the size of Angle B = 180 − (180 − x) = 180 − 180 + x = x. Thus, both Angle A and Angle B have sizes equal to x and are equal in size.

Angles A and B are adjacent.

Adjacent angles, often abbreviated as adj. ∠s, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles that are side by side, or adjacent, sharing an “arm”. Adjacent angles sum up to a right angle, straight angle or full angle are special and are in that order referred to as complementary, supplementary and explementary angles

A transversal is a line that intersects a pair of (frequently parallel) lines and is connected with alternate interior angles, corresponding angles, interior angles, and exterior angles.

Combining angle pairs

There are three special angle pairs which involve the summation of angles:

Complementary angles are angle pairs whose sizes sum to one right angle (1/4 turn, 90°, or π / 2 radians). If the two complementary angles are adjacent their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary, due to the fact that the sum of internal angles of a triangle is 180 degrees, and the right angle itself accounts for ninety degrees.

An acute angle is “filled up” by its complement to form a right angle.

The difference between an angle and a right angle is known as the complement of the angle.

If angles A and B are complementary, the following relationships hold:

The tangent of an angle equals the cotangent of its complement and its secant equals the cosecant of its complement. The prefix “co-” in the names of a few trigonometric ratios refers to the word “complementary”.

Two angles that sum to a straight angle (1/2 turn, 180°, or π radians) are known as supplementary angles. If the two supplementary angles are adjacent (i.e. have a common vertex and share only one side), their non-shared sides form a straight line. Angles like that are known as a linear pair of angles. Nevertheless, supplementary angles do not have to be on the same line, and can be alienated in space. For instance, adjacent angles of a parallelogram are supplementary, and opposite angles of a cyclic quadrilateral (one whose vertices all fall on a single circle) are supplementary.

If a point P is exterior to a circle with center O, and if the tangent lines from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.

The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.

In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third, due to the fact that the sum of internal angles of a triangle is a straight angle.

Two angles that sum to a complete angle (1 turn, 360°, or 2π radians) are known as explementary angles or conjugate angles.

The difference between an angle and a complete angle is known as the explement of the angle or conjugate of an angle.