[G.CO.7] Congruence #7

Objective

Common Core Text:

  • [G.CO.7] Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Said Differently:

    • Know why CPCTC (Correspinding Parts of Congruent Triangles are Congruent) and how to use CPCTC

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Explanation

CPCTC. One of the cornerstones of proofs. You'll see this in most proofs from here on out, so make sure you understand it well.

The idea isn't too complicated. Lets run through it

    1. For two triangles to be congruent, one should be able to be moved onto the other through rigid motions (translation, rotation, reflection).

    2. Rigid motions don't change the shape (angles) or size (sides) of a shape.

    3. Therefore, if two triangles are congruent, all of their corresponding angles and sides will be congruent.

    4. In other words, corresponding parts (parts are angles and sides) of congruent trianges are congruent. CPCTC.

Said in a sentence: If two triangles are congruent, that means they have the same shape and size, so all their corresponding sides and angles must have the same shape and size too.

CPCTC works both ways

    1. If we know that all the corresponding parts of two triangles are congruent, then we know the two triangles are congruent.

    2. If we know that two triangles are congruent, then we know that all of the corresponding parts of the two triangles are congruent.

In this objective, we'll focus on part 2. In the next objective we'll incorporate part 1.

Example

Question 1:

In this picture, the two triangles are congruent. Which sides are congruent? Which angles are congruent?

Answer:

We just don't know.

We could theoretically use a ruler and a protractor, but what if we don't have one? Or more importantly, what if the triangles aren't drawn to scale? It's actually almost impossible to draw scalene triangles to scale without a computer, you should try it if you don't believe me.

We know the triangles are congruent because the question told us, but we don't know which parts correspond. So we can't answer the question.

Question 2:

In this picture, ΔABC ≅ ΔXYZ. Which sides are congruent? Which angles are congruent?

Now we're cooking. The statement ΔABC ≅ ΔXYZ tells us everything we need to know. Here's how it works

  • ΔABC ≅ ΔXYZ

    • The first letters correspond

    • The second letters correspond

    • The third letters correspond

Lets start with angles. Just make sure the letters correspond.

  • ∠A ≅ ∠X

  • ∠B ≅ ∠Y

  • ∠C ≅ ∠Z

Next, the sides. Just make sure both letters of the sides correspond.

And that would be our answer. 3 angles, 3 sides

Answer:

  • ∠A ≅ ∠X

  • ∠B ≅ ∠Y

  • ∠C ≅ ∠Z

  • ≅

  • ≅

  • ≅

Let's try another

Example 2

If ΔBAC ≅ ΔZYX, how long is ?

First thing we need to do is figure out which side XY corresponds to. Let's see. In ΔZYX

  • X is the 3rd letter

  • Y is the 2nd letter

So, I need the 3rd and 2nd letter of ΔBAC

  • C is the 3rd letter

  • A is the 2nd letter

This means

  • ≅

And since the picture shows that CA = 5.1, then we get

Answer:

= 5.1