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Angles at a Point & vertically Opposite Angles


The iconic Sydney Harbour Bridge is one of many architectural feats that are built displaying attractive angles

Recall the previous definitions of angles called revolution and straight angle.  These special angles can be used to find short-cuts for solving geometry problems.

Angles at a Point

When we are problem-solving with angles, we can use the revolution fact to create a rule called the angles at a point rule. Angles at a point sum to $360^\circ$360°

We can also use the straight angle fact, to create a rule called angles in a line.  Because a straight angle is $180^\circ$180°, then any number of angles on that line will also sum to $180^\circ$180°.

We call two angles supplementary if they sum to $180^\circ$180°, and we call them complementary if they sum to $90^\circ$90°.

Angle rules and definitions

Angles at a point: Angles at a point sum to $360^\circ$360°

Angles on a line: Adjacent angles on a straight line are supplementary. (add up to $180^\circ$180°)

Supplementary Angles: Sum to $180^\circ$180°

Complementary Angles: Sum to $90^\circ$90°


Example 1

Find the size of the unknown angle $x$x.

Vertically Opposite Angles

A pair of angles are said to be vertically opposite if

  • the angles are made from two intersecting lines (lines that cross)
  • the angles are not adjacent (not next to each other)
  • the two angles share a vertex 

We can use a picture like this to help us remember this rule. 

Vertically Opposite Angles

Vertically opposite angles are equal.  

(To help remember this is sometimes referred to as the X rule).  

This interactive will let us explore different vertically opposite angles in practice, and show us that they are always equal.  

A proof by example

Why are vertically opposite angles equal?

Well.... Lets look some angles like these.



We are given $\angle AHP=108$AHP=108°





Because line AD is a straight line, then the two angles $\angle AHP$AHP and $\angle PHD$PHD must add up to $180^\circ$180°.

So, $\angle PHD=180-108$PHD=180108 $=$= $72^\circ$72°




Similarly, we can see the line VP is a straight line. 

So the two angles $\angle PHD$PHD and $\angle DHV$DHV must add up to $180$180° 

This means that $\angle DHV=180-72$DHV=18072 = $108$108°


And we can now see that $\angle AHP=\angle DHV$AHP=DHV

And, $\angle AHP$AHP is vertically opposite $\angle DHV$DHV

So, vertically opposite angles are equal. 

Here is another worked example.

Example 2

Find $x$x in the figure below.


Solving problems

When solving angle problems in geometry one of the most important components is the reasoning (or rules) you use to solve the problem.  You will mostly be required in geometry problems to not only complete the mathematics associated with calculating angle or side lengths but also to state the reasons you have used.  Read through each of these rules and see if you can describe why and draw a picture to represent it.


Geometry Rules with Angles


Angles at a point sum to 360°
Vertically opposite angles are equal
Adjacent angles forming a right angle are complementary
Adjacent angles in a straight line are supplementary



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