Euclidian geometry/Angle congruence

Notice: Incomplete

What is an Angle?

The angle is, in some sense, how far away two rays with a common starting point are. One way is to go one unit away from the common starting point and draw an arc between the rays. The angle is just simply the length of the arc. This notion gives us radians, the 'unitless' angle measure. $\pi$ is defined such that the angle of an entire circle is $2\pi$ (as opposed to $\pi$ itself for some reason). Another way is to draw a circle, divide it into 360 parts for some reason called 'degrees', then divide each degree into 60 smaller parts called 'arcminutes', then divide each arcminute into 60 even smaller parts called 'arcseconds', and see how much of the circle the space between the rays takes up.

Terminology

If the two rays form a line, then as the space between them is half a circle, the angle is $\pi$ , 180 degrees (180 being half of 360), or a straight angle (as it's the angle of a straight line). Angles larger than this are called reflex (the space between the rays isn't measured in this case, but rather, the distance outside the rays), unless it's an entire circle, in which case it's called a revolution. Half a straight angle is a right angle. If two things are at right angles to each other, they're normal, or perpendicular. Angles smaller than this are called acute angles. Angles between a right and straight angle are called obtuse angles.


Type of angle	Radians	Degrees
Acute angle	$0<\theta <{\frac {\pi }{2}}$	$0^{\circ }<\theta <90^{\circ }$
Right angle	$\theta ={\frac {\pi }{2}}$	$\theta =90^{\circ }$
Obtuse angle	${\frac {\pi }{2}}<\theta <\pi$	$90^{\circ }<\theta <180^{\circ }$
Straight angle	$\theta =\pi$	$\theta =180^{\circ }$
Reflex angle	$\pi <\theta <2\pi$	$180^{\circ }<\theta <360^{\circ }$
Revolution	$\theta =2\pi$	$\theta =360^{\circ }$

If two angles sum to $\pi$ or $180^{\circ }$ , they're supplementary. If two angles sum to ${\frac {\pi }{2}}$ or $90^{\circ }$ , they're complementary.

Parallel lines

Parallel lines are lines which don't intersect (and are on the same plane), that is, lines without common points. In flat space, there is only one parallel line through any given point. On a sphere, there are no parallel lines, and on a saddle shape, there are infinitely many through any given point. But this is meant to be about Euclidian geometry, so that's for another time.

We can look at what angles are like around transversals (lines that are not parallel to a set of parallel lines). Everything stated about them here only applies to flat space, the subject of this collection of resources.

Vertically Opposite Angles

Vertically opposite angles are the same size

I'll take a detour from parallel lines for a second and consider pairs of distinct lines with a common point. Angles opposite the common point are always the same, no matter what common point you choose, what pairs of lines you choose, or even what surface you're using (though we're talking about Euclidian geometry here)!

Activity

Ask your students to do the following:

Draw two distinct lines that intersect in the middle of the sheet of paper.
Draw an arc between the lines on one side, then on its opposite.
Use tracing paper to copy what was on the sheet of paper.
Place a pin through the line-intersections on the normal paper and tracing paper.
Rotate the tracing paper $\pi$ or $180^{\circ }$
Observe that the arcs have the same angle as they've been mapped perfectly onto each other!

Corresponding Angles

The cyan angles are the same

Given two parallel lines and a transversal (see image), the angles on the same side of both the parallel lines and the transversal are necessarily the same. This can be seen by sliding the angle labeled a along the transversal until it lines up on b.

Activity

Ask your students to do the following:

Draw two parallel lines
Cut out a wedge of a circle such that the angle is less than $\pi$ or $180^{\circ }$
Place one side of the wedge on a parallel line
Place a ruler directly against the other side of the wedge
Slide the wedge along the ruler until the wedge's corner is on the second parallel line
Notice that the entire side of the wedge is on the second parallel line and not just the corner

Alternate Angles

Alternate Interior Angles

Alternate Exterior Angles

Alternate interior angles are angles formed by a Z shape. The angles on the 'Z' may be acute, right, or obtuse.

Alternate Exterior Angles

Alternate exterior angles are angles vertically opposite from alternate interior angles.

Alternate angles are congruent

Ask your students if alternate angles are congruent or not and to justify their reasoning.

In either case, if you start with an angle, take the angle vertically opposite to it, then take the corresponding angle (or vice versa), you get the alternate angle. Demonstrate this by drawing the steps listed above.

If:

An angle is the same as the one vertically opposite to it
The vertically opposite angle is the same as the one corresponding to said vertically opposite angle
The angle corresponding to the vertically opposite angle is the alternate angle

Then:

4. The vertically opposite angle is the same as the alternate angle (from 2 and 3; substituting a definition with the word it defines)

5. An angle is the same as the alternate angle (from 1 and 4; If something equals something else, which in turn equals a third thing, then the first equals the third)

Demonstrate this by, again, drawing out each step

Co-Interior angles

X and Y are co-interior

Co-interior angles are angles formed by a box (see image)

Co-Interior angles are supplementary

Ask your students if co-interior angles are supplementary or not and to justify their reasoning.

If you start with an angle, the corresponding and co-interior angles sum to $\pi$ or $180^{\circ }$ as they form a straight line.

As an angle is the same as the corresponding angle, the original and co-interior angle also sum to $\pi$ or $180^{\circ }$

Co-Exterior angles

Co-exterior angles are...

Triangles

The internal angles of a triangle sum to $\pi$ or $180^{\circ }$ ...