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Outside the Box Geometry

# Angle Hunting Axioms

Find the measure of $$\angle c.$$

Our Notation:

The red square in the diagram indicates that the vertical line intersects the lower horizontal line at a right angle $$($$at an angle of exactly $$90^ \circ)$$. This notation will be used throughout this exploration: either a solid square or a square outline indicates a right angle.

Additionally, the matching "<" marks on the horizontal lines indicate that those two lines are perfectly parallel (they are always the same distance apart and will never touch). Note that matching double arrow marks on lines in a diagram, "<<", mean the same thing.

If all of the unknown angles are the same measure, $$a^ \circ,$$ what is the value of $$a?$$

Now for the three fundamental axioms! Every strategy for angle hunting breaks down to, somehow, creatively applying these axiomatic principles!

Axiom 1. Angles that form a straight line have a total angle sum of $$180 ^ \circ.$$ The blue and pink angles below form a linear pair. In addition, any angles that sum to $$180^\circ$$ are said to be supplementary.

Axiom 2. When two lines intersect, the vertical angle measures are equivalent.

Axiom 3. When parallel lines are intersected by a line, the following will be true:

1. Corresponding angles are congruent.
2. Alternate interior angles are congruent.
3. Alternate exterior angles are congruent.

Here's one more general tip for performing angle hunts: remember that the internal angles of any triangle always add up to $$180 ^ \circ.$$

This fact wasn't added to the list of axioms because you can prove it instead of needing to assume it!

Consider the following image. If we were to place all the interior angles of the triangle together, we can see that they form a straight line.

We can also prove that the angles sum to $$180^\circ$$ using the parallel lines in our diagram. The green angles are congruent, as are the pink angles, because they are alternate interior angles.

What is the measure of $$\angle X\,?$$

The five orange triangles in this image are all congruent, right triangles.

What is the measure of the green angle, $$g?$$

Notation and Definitions

If two or more figures in a problem are said to be"congruent," then they are all identical copies of the same shape, although these copies might be rotated, reflected or moved anywhere in the figure.

Additionally, in a diagram, a small dash (or a set of several small dashes) drawn across two or more line segments indicates that all of those line segments are the same length. In the figure above, because the five orange triangles are congruent, all of the short triangle legs that have dashes across them are the same length.

What is the measure of $$\angle Z?$$

Solve for $$x^\circ.$$

The three horizontal lines in the figure below are parallel. What is the measure of $$\angle B?$$

Be careful, lots of people get this one wrong!

Note: The diagram is not drawn perfectly to scale, so don't just guess or try to measure it with a protractor.

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