The Exterior Angle and AA Similarity

Lesson 2 of 2: Two Results from the Angle Sum

In this lesson:

• Find an exterior angle from the two far interior angles
• Use two matching angles to prove triangles are similar

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Explain why an exterior angle equals the sum of the two non-adjacent interior angles
2. State the Angle-Angle (AA) criterion for triangle similarity
3. Explain why two pairs of equal angles guarantee similarity
4. Apply AA to indirect-measurement problems

Extend One Side of a Triangle

The exterior angle looks bigger than either far angle. Exactly how big?

Two Facts You Already Have

At vertex , the interior and exterior angles make a straight line:

The triangle's angles sum to :

Subtract the Shared Interior Angle

Both equal , so:

Subtract from both sides:

Example: Find the Exterior Angle

In a triangle, and .

Check: interior at .

Work the Rule in Reverse

The exterior angle at is . One far angle, .

A Quick Check on Your Own

A triangle has interior angles and .

What is the exterior angle at the third vertex?

Decide on your own first.

A Surprising Discovery About Two Angles

Same two angles, and — and every side of the larger is exactly double.

Stating the Angle-Angle Similarity Criterion Now

If two angles of one triangle equal two angles of another, the triangles are similar.

That's the whole test — just two angles.

Why Two Angles Are Enough

If and , then:

The third pair is forced to match.

Aligning One Triangle onto the Other

Align one angle with rigid motions, then dilate until the second vertex lands.

Similar, but Not Necessarily Congruent

• AA proves similar — same shape, maybe different size
• Two -- triangles with hypotenuses and match angles but differ in size
• AA works for triangles only — not quadrilaterals

How Tall Is the Building?

Pole m, shadow m. Building shadow m. Scale factor , so height m.

Guided Practice: Are They Similar?

: ,
: ,

Find the third angle in each, then decide.

Spot the Mistake in This Reasoning

A student says: " but , so they're not similar."

Why is this wrong?

Your Turn to Decide and Justify

has angles and .
has angles and .

Are they similar? Justify why two angles were enough.

The Key Takeaways to Remember

✓ Exterior angle sum of the two far interior angles
✓ AA: two matching angles triangles are similar
✓ The angle sum forces the third pair to match

Watch out: AA gives similar, not congruent; match angle sets, not labels

What Is Coming Up Next

This two-angle shortcut is the engine behind a beautiful fact:

Every non-vertical line has one slope, because any two "rise-over-run" triangles on it are AA-similar (8.EE.B.6).