[G.CO.9] Congruence #9

Objective

Common Core Text:

  • [G.CO.9] Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Said Differently:

    • Prove the following theorems about lines and angles

      • Vertical Angles Theorem

      • Alternate Interior Angles Theorem

      • Corresponding Angles Theorem

      • Perpendicular Bisector Theorem

Example

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Explanation

Properties of Equality

These are useful to know for theorems. That way we can just say "because of the transitive property of equality" in our proofs, rather than listing a bunch of extra things.

Same-Side Interior Angles Postulate

(Postulate means you can believe it's true just by looking at it)

What are same-side interior angles?

Same side interior angles come up when two parallel lines are intersected by a transversal. They are two angles that are

  • on the inside (interior) of the two parallel lines

    • in this picture, 3, 4, 5, and 6 are interior because they are between between the two parallel lines

    • on the same side of the transversal

      • 3 and 5 are on the same side. 4 and 6 are on the same side.

Example

Which angle is a same-side interior angle with 3 in the picture above?

    • Well, 4, 5, and 6 are my other interior angles

    • 5 is on the same side of the transversal

  • Therefore, 3 and 5 are same-side interior angles

So in this picture, we have 2 pairs of same-side interior angles

    • ∠3 and ∠5 are same-side interior angles

    • ∠4 and ∠6 are same-side interior angles

Why are same-side interior angles special? Because they are always supplementary (add to 180°).

Linear Pair Theorem

Before we can prove any of these theorems, we need to understand the linear pair theorem. That way we can just say "because of the linear pair theorem" in our proofs, rather than listing a bunch of extra things.

What is a linear pair?

Linear Pair

    • Two angles on the same side of a straight line, created by an intersecting line.

Sounds complicated, but just remember

    • two angles on the

    • same side of a

  • straight line

So in this picture, we have 8 linear pairs

    • ∠1 and ∠2 are a linear pair

    • ∠2 and ∠4 are a linear pair

    • ∠4 and ∠3 are a linear pair

    • ∠3 and ∠1 are a linear pair

    • ∠5 and ∠6 are a linear pair

    • ∠6 and ∠8 are a linear pair

    • ∠8 and ∠7 are a linear pair

    • ∠7 and ∠5 are a linear pair

Why are linear pairs special? Since a straight line is a 180° angle, and a linear pairs are the two angles on the same side of a straight line

    • linear pairs always add to be 180°

See what this means by interacting with this GeoGebra applet

Remember this, and proofs will be much easier

Vertical Angles Theorem

What are vertical angles?

Vertical Angles

    • When two lines intersect, vertical angles are the angles across from each other

So in this picture, we have 4 pairs of vertical angles

    • ∠1 and ∠4 are vertical angles

    • ∠2 and ∠3 are vertical angles

    • ∠5 and ∠8 are vertical angles

    • ∠7 and ∠6 are vertical angles

Why are vertical angles special? Because vertical angles are always congruent.

Here's a proof using the picture above. Notice in the proof that we put an m in front of the angle if we are comparing it to a number with a degree (in this case, 180°)

~~~~~

Prove that ∠1 and ∠4 are congruent

Since this proof works for any vertical angles, we know that all vertical angles are always congruent

Alternate Interior Angles Theorem

What are alternate interior angles?

Alternate interior angles come up when two parallel lines are intersected by a transversal. They are two angles that are

  • on the inside (interior) of the two parallel lines

    • in this picture, 3, 4, 5, and 6 are interior because they are between between the two parallel lines

    • on opposite (alternate) parallel lines

      • 5 and 6 are opposite from 3 and 4

    • on the opposite (alternate) side of the transversal

      • 3 and 5 are opposite from 4 and 6

Whew that's a lot to take it. Really, its easiest to just look at some examples and remember what they look like

Example

Which angle is an alternate interior angle with 3 in the picture above?

    • Well, 4, 5, and 6 are my other interior angles

    • 5 and 6 are on the opposite parallel line

      • (4 is on the same parallel line, so 4 can't be an alternate interior angle)

    • 4 and 6 are on the opposite side of the transversal

      • (5 is on the same side of the transveral, so 5 can't be an alternate interior angle)

    • but 6 is the only angle that is both on the opposite parallel line AND on the opposite side of the transversal, so 6 is the alternate interior angle of 3

      • (since our other interior angles are 4, 5, and 6, but 4 and 5 don't work, we know the answer is 6)

So in this picture, we have 2 pairs of alternate interior angles

    • ∠3 and ∠6 are alternate interior angles

    • ∠4 and ∠5 are alternate interior angles

Why are alternate interior angles special? Because alternate interior angles are always congruent.

Here's a proof using the picture above

~~~~~

Prove that ∠3 and ∠6 are congruent

Since this proof works for any same-side interior angles, we know that all same-side interior angles are always congruent

Corresponding Angles Theorem

What are corresponding angles?

Corresponding angles come up when two parallel lines are intersected by a transversal. The easiest way to know which angles correspond is by imagining quadrants.

    • The top left angle in the bottom intersection corresponds with the top left angle in the top intersection

    • The top right angle in the bottom intersection corresponds with the top right angle in the top intersection

    • The bottom left angle in the bottom intersection corresponds with the bottom left angle in the top intersection

    • The bottom right angle in the bottom intersection corresponds with the bottom right angle in the top intersection

Watch this

So in this picture, we have 4 pairs of corresponding angles

    • ∠1 and ∠5 are corresponding angles

    • ∠2 and ∠6 are corresponding angles

    • ∠3 and ∠7 are corresponding angles

    • ∠4 and ∠8 are corresponding angles

Why are corresponding angles special? Because corresponding angles are always congruent.

Here's a proof using the picture above

~~~~~

Prove that ∠1 and ∠5 are congruent

Since this proof works for any corresponding angles, we know that all corresponding angles are always congruent

Perpendicular Bisector Theorem

What is a perpendicular bisector?

Here we have

Notice

    • They are perpendicular to each other (we know because of the right angle symbol at the intersection).

    • Line segment AB is being bisected (cut exactly in half) by line CD (we know this because both halves have a tick on them).

      • If you were wondering the reverse isn't true. AB does not bisect CD. This is because CD is a line which goes on forever, and there's no middle to something that goes forever.

Since line CD is perpendicular to line segment AB, and line CD bisects line segment AB, we say line CD is a perpendicular bisector to line segment AB.

The perpendicular bisector theorem says that:

  • points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

In other words, looking at our picture above, any point on the perpendicular bisector will be the same distance away from A as it is from B

Let's see if it's true. I'll draw a random point E on the perpendicular bisector

Next, I'll draw a dashed line from point E to the endpoints of AB. This creates line segment EA and line segment EB.

If we can prove that line segment EA and line segment EB are the same length, that would mean that E is the same distance away from A as it is from B, which would prove the theorem.

Here's a proof using the picture above

~~~~~

If is a perpendicular bisector of line segment ,

Statement

Reason

Pythagorean Theorem

+ =

    • (remember A2 + B2 = C2 ?)

+ =

Line CD bisects line segment AB (given)

Substitution Property of Equality

≅

+ =

    • (since AD = BD, we can replace BD with AD in the second equation)

+ =

Transitive Property of Equality

Exponential Property of Equality

=

=

Q.E.D (Quod Erat Demonstrandum)

≅

Since this proof works for any point on the perpendicular bisector, we know that:

  • points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

So why is this special? It's really not. But like so many other theorems, it lets us just say "by perpendicular bisector theorem" in future proofs instead of writing these 6 steps every time.