Teach Me Maths

Geometric Proof

Build clear geometric proofs using angle facts, parallel lines, congruent triangles and circle theorems, with a reason for each deduction.

Build a chain of justified statements

A proof explains why a result must hold for every configuration meeting the conditions. Mark the given facts first. Write a statement and its reason together, and finish by stating exactly what you were asked to prove.

Useful reasons include angles on a straight line, vertically opposite angles, corresponding or alternate angles between parallel lines, base angles of an isosceles triangle, triangle angle sum, and a named circle theorem. A diagram is evidence of the configuration, not evidence that two lengths or angles are equal.

Worked example 1

In triangle ABC, AB=AC. Points D and E lie inside sides AB and AC respectively, and AD=AE. Prove that DE is parallel to BC.

  2. Let \angle BAC=x^\circ. Since AB=AC, the base angles ABC and BCA are equal.

  3. Your turn. Give \angle ABC in terms of x (without a degree symbol).

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    \angle ABC=(180-x)/2 degrees, using the triangle angle sum.

  4. Also AD=AE, so \angle ADE=(180-x)/2 degrees. Therefore \angle ADE=\angle ABC. These are corresponding angles on transversal AB
  5. Answer: DE\parallel BC, because equal corresponding angles imply parallel lines.

When triangles are congruent

Congruent triangles have the same shape and size. Establish a valid test: SSS, SAS, ASA (or AAS), or RHS for right-angled triangles. In SAS, the equal angle must be between the equal sides. AAA proves similarity, not congruence; two sides and a non-included angle are not generally enough.

List corresponding vertices in matching order. After proving congruence, you can conclude that corresponding sides and angles are equal.

Worked example 2

ABCD is a parallelogram. Using diagonal AC and congruent triangles, prove that AB=CD and BC=DA.

  2. \angle BAC=\angle DCA because AB\parallel CD (alternate angles). Also \angle BCA=\angle DAC because BC\parallel AD (alternate angles).

  3. Your turn. If the two matching angles are x^\circ and y^\circ, write the third angle in either triangle, without a degree symbol.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    The third angle is (180-x-y)^\circ in each triangle.

  4. AC is common to both triangles. Therefore \triangle ABC\cong\triangle CDA by ASA, using the two equal angles at the ends of AC
  5. Answer: AB=CD and BC=DA, as corresponding sides of congruent triangles.

Worked example 3

ABCD is a rhombus with \angle ABC=64^\circ. Prove that diagonal BD bisects \angle ABC, and hence find \angle ABD.

  2. In triangles ABD and CBD, AB=CB and AD=CD because all sides of a rhombus are equal. Also BD is common.
  3. The triangles are congruent by SSS. Thus \angle ABD=\angle CBD (corresponding angles), so BD bisects \angle ABC

  4. Your turn. Calculate \angle ABD in degrees.

    64\div2=32

  5. Answer: BD bisects the angle by SSS congruence, and \angle ABD=32^\circ

Worked example 4

PA and PB are tangents to a circle with centre O, where OP=13 cm and the radius is 5 cm. Prove that PA=PB using congruent triangles, then find their lengths.

  2. \angle OAP=\angle OBP=90^\circ (radius perpendicular to tangent). The hypotenuse OP is common, and OA=OB (radii). Thus \triangle OAP\cong\triangle OBP by RHS.
  3. Therefore PA=PB as corresponding sides. To find the length, use PA^2=OP^2-OA^2

  4. Your turn. Calculate 13^2-5^2

    PA^2=144

  5. Answer: the tangents are equal by RHS congruence, and PA=PB=12 cm.

Worked example 5

AB is a diameter. Points C and D lie on opposite sides of AB on the circle, and AC\parallel BD. Prove that BC\parallel AD.

  2. Your turn. What is \angle ACB in degrees?

    An angle in a semicircle is 90^\circ

  3. Also \angle ADB=90^\circ (angle in a semicircle). Therefore BC\perp AC and AD\perp BD
  4. Since AC\parallel BD, a line perpendicular to BD is also perpendicular to AC. So both AD and BC are perpendicular to AC
  5. Answer: BC\parallel AD, because two lines perpendicular to the same line in a plane are parallel.

Common mistakes

  • Assuming the result to be proved. Start from given facts and established theorems.
  • Trusting the picture’s appearance. Use given or proved facts; do not measure the diagram.
  • Giving an angle equality without a reason. Name the geometric rule that makes the angles equal.
  • Using an invalid congruence test. AAA shows similarity; SAS needs the included angle. Use a valid congruence test.
  • Stopping before the required conclusion. Link your proved facts to the exact statement requested.

Now try it: Geometric Proof practice questions

More on this topic: Geometric Proof worksheet with full solutions