Triangle Angle Sum and Isosceles Proofs

Lesson 1 of 2

In this lesson:

• Prove that triangle angles sum to 180°
• Prove that base angles of isosceles triangles are congruent
• Apply both theorems to solve geometric problems

Lesson Goals for Triangle Proofs Today

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

1. Prove triangle interior angles sum to 180° using a parallel line construction
2. Prove isosceles base angles are congruent via SAS or reflection
3. Apply both theorems to find unknown angles

Measuring Angles — A Surprising Pattern

You've measured angles in triangles before. Consider this:

• Take any triangle — acute, right, or obtuse
• Measure all three interior angles
• Add them up

The sum is always close to 180°.

But close is not exact. Is it always exactly 180°? Can we prove it?

The Triangle Angle Sum Theorem

Theorem: The measures of the three interior angles of any triangle sum to exactly 180°.

Why it matters:

• Used in virtually every area of geometry
• Essential for finding unknown angles
• Distinguishes Euclidean from non-Euclidean geometry

The Key Construction: A Parallel Line

Given: Triangle

Construction: Draw line through vertex , parallel to side .

This line exists by the Parallel Postulate — the cornerstone of Euclidean geometry.

Identifying the Alternate Interior Angles

With line :

• — alternate interior angles, transversal
• — alternate interior angles, transversal

This uses the Alternate Interior Angles Theorem from HSG.CO.C.9.

Two-Column Proof: Angle Sum Theorem

Given: , line through with
Prove:

Quick Check: Applying the Angle Sum Theorem

In triangle :

Find .

Think through this before advancing.

Answer: Finding the Missing Angle

Apply the Angle Sum Theorem:

Exterior Angle Theorem as a Corollary

Corollary: An exterior angle equals the sum of the two remote interior angles.

Extend beyond . The exterior angle and form a straight pair (180°). Subtract from both sides of the angle sum equation.

Quick Check: Exterior Angle Theorem

In the figure below, is an exterior angle of where is on the extension of .

Find .

Name the theorem you'll use, then compute.

From Angle Sum to Isosceles Triangles

What if two sides of a triangle are equal?

Draw with and measure and .

The angles opposite the equal sides appear equal. Can we prove this is always true?

Stating the Isosceles Triangle Theorem

Theorem: If two sides of a triangle are congruent, the angles opposite those sides are congruent.

If , then .

Historically called pons asinorum — "bridge of donkeys."

Setting Up the SAS Congruence Proof

Given: . Prove: .

Construction: Draw angle bisector of to point on .

Three facts about and :

• (given)
• (bisector)
• (reflexive)

Two-Column Proof: Isosceles Triangle Theorem

Given: , . bisects , on .
Prove:

Isosceles Base Angles via Reflection

Reflect across the perpendicular bisector of .

• under reflection (symmetric about bisector)
• → lies on the bisector → fixed
• Rigid motion preserves angles:

Two Valid Approaches Reach Same Conclusion

Both proofs reach the same conclusion: .

Neither is more correct. Multiple paths build stronger conviction.

Converse: Equal Angles Imply Equal Sides

Converse: If two angles of a triangle are congruent, then the opposite sides are congruent.

If , then .

• Congruent angles → isosceles triangle
• Equilateral → all angles equal → all angles 60° (three sides equal + angle sum)

Quick Check: Applying the Isosceles Theorem

An isosceles triangle has vertex angle .

Find the measure of each base angle.

Use the theorems we've built today — name them before you compute.

Answer: Finding the Isosceles Base Angles

Isosceles: . Apply Angle Sum Theorem:

Each base angle measures 70°.

Equilateral Triangle: All Angles Are Sixty Degrees

If is equilateral, each angle measures 60°.

• → (Isosceles Theorem)
• → (Isosceles Theorem)
• All three equal → →

Quick Check: Choosing the Right Tool

Choosing the Right Theorem to Apply

Key Takeaways from Lesson One

✓ Angle Sum: — parallel line through vertex
✓ Isosceles Theorem: equal sides → equal base angles (SAS or reflection)
✓ Converse: equal angles → equal sides

180° is for triangles only
Proof requires the Parallel Postulate

Preview: What Comes Next in Deck Two

In Lesson 2 we prove two more triangle theorems:

• Midsegment Theorem: connecting two midpoints gives a segment parallel to the third side, half its length
• Centroid Theorem: three medians meet at one point, 2:1 from vertex

Both use similarity and coordinates.