DWG NO. 05.5 — Lesson 5 of 5

Indirect Proof

Unit 5: Relationships Within Triangles · ~15–30 min

Objective

Write an indirect proof by assuming the opposite of what you want to prove and showing that assumption leads to a contradiction.

Proving something by assuming it's false

Every proof you've written so far has argued forward: start from what's given, and build step by step toward the conclusion. An indirect proof — also called a proof by contradiction — works the opposite way. You temporarily assume the conclusion is false, then show that assumption breaks something you already know is true. Since the assumption leads to nonsense, it must have been wrong, which means the original conclusion has to be true.

Step 1: Assume the opposite (the negation) of what you're trying to prove.
Step 2: Reason forward from that assumption, using definitions, postulates, and theorems you already know.
Step 3: Show that this reasoning leads to a contradiction — something that conflicts with a given fact or a known true statement.
Step 4: Conclude that the assumption must be false, so the original statement must be true.

Worked Example 1 · A triangle cannot have two right angles

ProblemAssume the opposite: suppose triangle ABC has two right angles, angle A = 90° and angle B = 90°.

1.By the Triangle Angle Sum Theorem, angle A + angle B + angle C = 180°.

2.Substituting: 90 + 90 + angle C = 180, so angle C = 0°.

3.An angle of 0° is impossible in a triangle — this contradicts the definition of a triangle, which requires three positive angle measures.

The assumption is false, so a triangle cannot have two right angles.

Worked Example 2 · A scalene triangle has no congruent angles

ProblemGiven: triangle DEF is scalene (no two sides are congruent). Prove: no two angles are congruent.

1.Assume the opposite: suppose two angles of triangle DEF are congruent, say angle D ≅ angle E.

2.By the Converse of the Isosceles Triangle Theorem, the sides opposite those angles must be congruent.

3.But this contradicts the given fact that triangle DEF is scalene, with no two sides congruent.

The assumption is false, so no two angles of triangle DEF are congruent.

Guided practice

1.What is the first step of any indirect proof?
Show answer
Assume the opposite (negation) of the statement you are trying to prove.
2.State the assumption you would start with to indirectly prove: "Triangle GHI is not equilateral."
Show answer
Assume the opposite: triangle GHI is equilateral.
3.To indirectly prove that a triangle cannot have two obtuse angles, what would you assume, and what contradiction would you reach?
Show answer
Assume two angles are each greater than 90°. Their sum alone would exceed 180°, leaving nothing (or a negative amount) for the third angle — contradicting the Triangle Angle Sum Theorem, which requires all three angles to sum to exactly 180°.
4.Why does reaching a contradiction prove the original statement, rather than just proving the assumption is false?
Show answer
Because the assumption was the exact opposite of the original statement. If the assumption must be false, the only remaining possibility is that the original statement is true.
5.State the assumption you'd use to indirectly prove: "In triangle JKL, if JK ≠ KL, then angle L ≠ angle J."
Show answer
Assume the opposite of the conclusion: angle L = angle J (while keeping the given, JK ≠ KL).

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