Geometric Proofs

Prove geometric facts step by step with parallel line angles, the triangle angle sum, triangle congruence (SSS, SAS, ASA, AAS), CPCTC and isosceles triangles.

Advanced30 min18 problems
Practice Problems

Definition

A geometric proof shows that a fact about a figure must be true. It is a chain of statements, and every statement has a reason: the given information, a definition, a postulate or a theorem that is already proven. A drawing helps you find the idea, but a measurement with a protractor is never a reason.
The toolbox for this lesson:
1. Parallel lines and a transversal. A transversal is a line that crosses two other lines. If the two lines are parallel:
  • Alternate interior angles (between the parallel lines, on opposite sides of the transversal) are equal.
  • Corresponding angles (in the same position at both crossings) are equal.
  • Same-side interior angles add up to 180∘180^\circ.
2. Triangle angle sum. In every triangle ∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^\circ.
3. Congruent triangles. Two triangles are congruent (≅\cong) when one of these sets of matching parts is equal:
  • SSS: three sides
  • SAS: two sides and the angle between them
  • ASA: two angles and the side between them
  • AAS: two angles and a side that is not between them
4. CPCTC. Corresponding Parts of Congruent Triangles are Congruent: once you know △ABC≅△DEF\triangle ABC \cong \triangle DEF, every matching side and angle is equal.
5. Isosceles triangles. If two sides of a triangle are equal, the angles opposite them (the base angles) are equal.
6. Midpoints and bisectors. A midpoint splits a segment into two equal parts, an angle bisector splits an angle into two equal angles.
A complete proof in two columns. Given: segments ADAD and BCBC cross at MM, and MM is the midpoint of both. Prove: AB=DCAB = DC.
StatementReason
AM=DMAM = DMDefinition of midpoint
BM=CMBM = CMDefinition of midpoint
∠AMB=∠DMC\angle AMB = \angle DMCVertical angles are equal
△AMB≅△DMC\triangle AMB \cong \triangle DMCSAS
AB=DCAB = DCCPCTC
The usual plan: find two triangles that contain the part you want, prove them congruent, then finish with CPCTC.

Try it now

Lines ABAB and CDCD are parallel (AA and CC on the left, BB and DD on the right). A transversal crosses ABAB at EE and CDCD at FF. ∠AEF=72∘\angle AEF = 72^\circ and ∠EFD=72∘\angle EFD = 72^\circ. Which reason justifies ∠EFD=∠AEF\angle EFD = \angle AEF?

Worked Examples

Lines ABAB and CDCD are parallel. Points AA and CC are on the left, BB and DD on the right. A transversal crosses ABAB at EE and CDCD at FF, and ∠AEF=65∘\angle AEF = 65^\circ. Find ∠EFD\angle EFD and ∠CFE\angle CFE and justify each step.
1

Locate the angle pair

∠AEF\angle AEF and ∠EFD\angle EFD both lie between the parallel lines, on opposite sides of the transversal → They are alternate interior angles
2

Apply the parallel line theorem

Alternate interior angles at parallel lines are equal, so ∠EFD=∠AEF\angle EFD = \angle AEF → ∠EFD=65∘\angle EFD = 65^\circ
3

Use the straight angle at FF

∠CFE\angle CFE and ∠EFD\angle EFD together form the straight angle CFDCFD: ∠CFE=180∘−65∘\angle CFE = 180^\circ - 65^\circ → ∠CFE=115∘\angle CFE = 115^\circ (linear pair)

Common Mistakes

Using AAA (three equal angles) to prove that triangles are congruent

Why it's wrong: Equal angles only give the same shape: a small and a large equilateral triangle have the same angles but are not congruent. Two sides and an angle that is not between them (SSA) also do not work in general, because two different triangles can fit.

Correct: Use only SSS, SAS, ASA or AAS, and in SAS check that the angle lies between the two sides.

Writing CPCTC before the triangles are proven congruent

Why it's wrong: CPCTC is only allowed after a line that says the triangles are congruent. Otherwise the proof assumes what it wants to show.

Correct: First a line like △ABC≅△DEF\triangle ABC \cong \triangle DEF (SAS), then the matching sides or angles with the reason CPCTC.

Using angle facts for parallel lines when the lines are not parallel, or saying same-side interior angles are equal

Why it's wrong: Alternate interior and corresponding angles are equal only at parallel lines. Same-side interior angles are not equal, they add up to 180∘180^\circ.

Correct: Check that the lines are given as parallel (AB∥CDAB \parallel CD), then name the pair exactly: alternate interior (equal), corresponding (equal) or same-side interior (sum 180∘180^\circ).

Writing the vertices of a congruence statement in the wrong order

Why it's wrong: △ABC≅△DEF\triangle ABC \cong \triangle DEF means A↔DA \leftrightarrow D, B↔EB \leftrightarrow E, C↔FC \leftrightarrow F. A wrong order makes CPCTC pair the wrong sides and angles.

Correct: Match the vertices by the equal parts. In Example 5 the correct statement is △ABC≅△EDC\triangle ABC \cong \triangle EDC, not △ABC≅△DEC\triangle ABC \cong \triangle DEC.

Interactive Visual

Angle Explorer

12345678
∠1 = 115°∠2 = 65°∠3 = 65°∠4 = 115°∠5 = 115°∠6 = 65°∠7 = 65°∠8 = 115°

Alternate interior angles are between the parallel lines, on opposite sides of the transversal. They are equal.

Drag the blue point to tilt the transversal. Pick a pair type, or tap two angle numbers to see how they are related.

Triangle Explorer

By sides:Isosceles
By angles:Acute

Area:

0.3 sq units

Explore the relationship between angles in a triangle.

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
Lines ABAB and CDCD are parallel (AA and CC on the left, BB and DD on the right). A transversal crosses ABAB at EE and CDCD at FF. ∠AEF=72∘\angle AEF = 72^\circ and ∠EFD=72∘\angle EFD = 72^\circ. Which reason justifies ∠EFD=∠AEF\angle EFD = \angle AEF?

Why It Matters

Geometric proofs are where mathematical proof was born, and their ideas are still used every day:
  • History of mathematics: Euclid's Elements (around 300 BC) built all of geometry from a few postulates with exactly these proofs, and it was a school textbook for more than 2000 years
  • Surveying: Congruent triangles let surveyors measure distances they cannot walk, like the width of a river or a pond
  • Architecture and engineering: Roof trusses and bridges use isosceles and congruent triangles, because a triangle with fixed sides cannot change shape
  • Road and city planning: Parallel streets crossed by a diagonal avenue repeat the same angles at every corner
  • Reasoning skills: Proving that something must be true, step by justified step, is the same skill lawyers, programmers and scientists need
Once you know these few theorems well, a large part of school geometry turns into short, convincing arguments.

Real World Applications

Roof Trusses

Carpenters build roof trusses as isosceles triangles: two rafters of equal length meet at the ridge, so both roof slopes have the same angle.

Example:

A truss with a 100∘100^\circ angle at the ridge has base angles of (180∘−100∘)÷2=40∘(180^\circ - 100^\circ) \div 2 = 40^\circ.

1Try It Yourself

A carpenter builds a truss with two rafters of equal length. The angle at the ridge is 120∘120^\circ.

What angle does each rafter make with the horizontal beam?

Step 1: Write the mathematical expression

Subtract the ridge angle from 180∘180^\circ and split the rest equally:

Measuring Across a Pond

Surveyors measure distances they cannot walk with congruent triangles, using the same SAS and CPCTC proof as in this lesson.

Example:

To find the width ABAB of a pond, walk from AA to a point CC and the same distance further to DD. Then walk from BB through CC the same distance to EE. Now △ACB≅△DCE\triangle ACB \cong \triangle DCE (SAS), so DE=ABDE = AB by CPCTC.

2Try It Yourself

The points are placed as in the example: AC=CD=35AC = CD = 35 m, BC=CE=40BC = CE = 40 m, and on dry land you measure DE=48DE = 48 m.

How wide is the pond, that is, how long is ABAB?

Step 1: Write the mathematical expression

By CPCTC, ABAB equals the measured side:

Street Maps

Parallel streets crossed by a diagonal avenue form the angle pattern of parallel lines and a transversal, which city planners use to design corners and crossings.

Example:

If a diagonal avenue makes a 58∘58^\circ angle with one street, it makes the same 58∘58^\circ corresponding angle with every parallel street.

3Try It Yourself

Two parallel streets are crossed by a diagonal avenue. At the first street, the angle between the avenue and the street on the north-east side is 58∘58^\circ.

At the second street, how large are the corresponding angle (north-east) and the angle next to it (north-west)?

Step 1: Write the mathematical expression

The neighbouring angle forms a linear pair:

Key Takeaways

  • 1Every statement in a geometric proof needs a reason: given, definition, postulate or theorem
  • 2At parallel lines cut by a transversal, alternate interior and corresponding angles are equal, and same-side interior angles add up to 180∘180^\circ
  • 3The triangle angle sum (180∘180^\circ) is proved with a line through one vertex parallel to the opposite side
  • 4Triangles are congruent by SSS, SAS, ASA or AAS, never by AAA and in general not by SSA
  • 5CPCTC: after proving two triangles congruent, all their corresponding sides and angles are equal
  • 6In an isosceles triangle the base angles are equal, and midpoints and bisectors give equal segments and equal angles

Frequently Asked Questions

Corresponding Parts of Congruent Triangles are Congruent. It is the reason you write after proving two triangles congruent, when you conclude that a matching side or angle is equal.
Corresponding Parts of Congruent Triangles are Congruent. It is the reason you write after proving two triangles congruent, when you conclude that a matching side or angle is equal.
With two sides and an angle that is not between them, the third side can sometimes swing into two different positions, which gives two different triangles. So SSA does not guarantee congruence in general.
In ASA the known side lies between the two angles, in AAS it does not. Both work: two angles fix the third angle (angle sum 180∘180^\circ), so AAS can always be turned into ASA.
Look at the segment or angle you must prove equal and find two triangles that contain it. Mark everything you know: given parts, vertical angles, a shared side. If you reach SSS, SAS, ASA or AAS, finish with CPCTC.

Glossary

Transversal
A line that crosses two (or more) other lines
Alternate interior angles
Two angles between two lines, on opposite sides of a transversal; equal when the lines are parallel
Corresponding angles
Two angles in the same position at the two crossings of a transversal; equal when the lines are parallel
Congruent triangles
Triangles with all three sides and all three angles equal, written △ABC≅△DEF\triangle ABC \cong \triangle DEF
CPCTC
Corresponding Parts of Congruent Triangles are Congruent: the reason used after a congruence is proven
Isosceles triangle
A triangle with two equal sides; the angles opposite them (base angles) are equal

Formula Card

Triangle angle sum

∠A+∠B+∠C=180∘\angle A + \angle B + \angle C = 180^\circ

True in every triangle

Linear pair and same-side interior angles

α+β=180∘\alpha + \beta = 180^\circ

Angles on a straight line, and same-side interior angles at parallel lines

Isosceles triangle

AB=AC⇒∠B=∠CAB = AC \Rightarrow \angle B = \angle C

Equal sides are opposite equal base angles

Congruence and CPCTC

△ABC≅△DEF\triangle ABC \cong \triangle DEF

By SSS, SAS, ASA or AAS; then all matching parts are equal (CPCTC)

Today's challengeOne problem, three tries

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