Proof of Some Basic Theorem and Deductive Proofs – SS1 Mathematics Lesson Note (Second Term)

A deductive proof shows that a statement is true by reasoning step by step from given facts and facts already proved.

Behavioural objectives

By the end of the lesson, students should be able to:

1. Prove that the sum of the angles of a triangle is 180°.
2. Identify corresponding, alternate and vertically opposite angles.
3. State the properties of a parallelogram and of a rhombus.
4. State and prove the intercept theorem.

What is a deductive proof?

A deductive proof is a chain of logical steps, each backed by a reason, that leads from known facts to a conclusion.

A written geometry proof has four parts:

• Given: the facts we are told about the figure.
• To prove: the statement we must show to be true.
• Construction: extra lines drawn to help the reasoning.
• Proof: numbered steps, each with a reason in brackets.

Reasons allowed in a proof:

• Definitions: for example, a parallelogram has both pairs of opposite sides parallel.
• Earlier theorems: results already proved.
• Angle facts: alternate, corresponding and vertically opposite angles, and angles on a straight line.

What is the sum of the angles of a triangle?

The sum of the angles of a triangle is 180°.

Given: Any triangle ABC.

To prove: ∠A + ∠B + ∠C = 180°

Construction: Produce BC to a point X. Draw CP parallel to BA, on the same side of BC as A.

Lettering:

• a₁: angle BAC of the triangle.
• b₁: angle ABC of the triangle.
• c: angle ACB of the triangle.
• a₂: angle ACP.
• b₂: angle PCX.

Proof:

1. a₁ = a₂ (alternate angles, BA parallel to CP, transversal AC).
2. b₁ = b₂ (corresponding angles, BA parallel to CP, transversal BX).
3. c + a₂ + b₂ = 180° (angles on a straight line BCX).
4. c + a₁ + b₁ = 180° (replacing a₂ and b₂ using steps 1 and 2).
5. Therefore ∠A + ∠B + ∠C = 180°.

Worked example

1. Two angles of a triangle are 65° and 48°.
2. Third angle = 180° − (65° + 48°) = 180° − 113°.
3. Third angle = 67°.

What does the proof show about an exterior angle?

An exterior angle of a triangle equals the sum of the two interior opposite angles.

• From the proof: exterior angle ACX = a₂ + b₂.
• Substituting: a₂ + b₂ = a₁ + b₁, which are angles A and B.
• Example: if A = 68° and B = 30°, then exterior angle ACX = 98°.

What are the relationships between angles on lines?

Angles on lines follow fixed rules, and each rule can be used as a reason in a proof.

Angles on a straight line

The sum of angles on a straight line is 180°.

That is, A + B + C = 180°.

Angles on parallel lines cut by a transversal

A transversal is a line that cuts two or more other lines.

The figure below shows parallel lines cut by a transversal, with the angles numbered 1 to 8.

Corresponding angles

Corresponding angles are angles in the same position at the two points where the transversal cuts the parallel lines.

From the figure above, the following angles are corresponding and equal:

• ∠1 = ∠5 and ∠3 = ∠7
• ∠2 = ∠6 and ∠4 = ∠8

Alternate angles

Alternate angles are angles on opposite sides of the transversal, between the parallel lines, and they are equal.

The alternate angles are shown in the diagram above.

Vertically opposite angles

Vertically opposite angles are the angles facing each other where two straight lines cross, and they are equal.

From the diagram above, the vertically opposite angles are:

• ∠1 = ∠3
• ∠2 = ∠4

Summary table of angle rules

What is a parallelogram?

A parallelogram is a quadrilateral which has both pairs of opposite sides parallel.

Rhombuses, rectangles and squares are special examples of parallelograms.

Properties of a parallelogram

1. The opposite sides are parallel.
2. The opposite sides are equal.
3. The opposite angles are equal.
4. The diagonals bisect one another.

What is a rhombus?

A rhombus is a parallelogram with all four sides of equal length.

Properties of a rhombus

1. All four sides are equal.
2. The opposite sides are parallel.
3. The opposite angles are equal.
4. The diagonals bisect one another at right angles.
5. The diagonals bisect the angles.

How do a parallelogram and a rhombus compare?

What are the special parallelograms?

• Rectangle: has all the properties of a parallelogram, and all four angles are right angles.
• Rhombus: a parallelogram with four equal sides.
• Square: has all the properties of a rhombus, and all four angles are right angles.

What is the intercept theorem?

If three or more parallel lines cut off equal intercepts on a transversal, then they cut off equal intercepts on any other transversal.

Given: Three parallel lines cutting a fourth line at A, B and C so that /AB/ = /BC/, and cutting another line at X, Y and Z respectively.

To prove: /XY/ = /YZ/

Construction: Draw XP and YQ parallel to ABC to cut BY and CZ at P and Q respectively.

Proof:

1. AXPB is a parallelogram (opposite sides parallel).
2. /XP/ = /AB/ (opposite sides of a parallelogram are equal).
3. /YQ/ = /BC/ (opposite sides of parallelogram YQCB).
4. /XP/ = /YQ/ (since /AB/ = /BC/, given).
5. In triangles XPY and YQZ:
   • /XP/ = /YQ/ (proved in step 4).
   • ∠PXY = ∠QYZ (corresponding angles, XP parallel to YQ).
   • ∠XYP = ∠YZQ (corresponding angles, YP parallel to ZQ).
6. Therefore, triangle XPY ≅ triangle YQZ (AAS).
7. /XY/ = /YZ/ (corresponding sides of congruent triangles).

Evaluation

1. State the theorem about the sum of the angles of a triangle and name the construction used to prove it.
2. Write down two pairs of corresponding angles and two pairs of vertically opposite angles from the figures above.
3. List the properties of a rhombus.
4. State the intercept theorem.
5. Two angles of a triangle are 50° and 60°. Find the third angle and the exterior angle at that vertex.

Assignment

1. In triangle ABC, ∠BAC = 68° and ∠ABC = 30°. BC is produced to X. The bisectors of ∠ABC and ∠ACX meet at P. Calculate ∠BCP and ∠BPC.