IP Maths Notes (Lower Sec, Year 1-2): 06) Geometry, Congruency & Similarity

Geometry proofs reveal structure: angles encode relationships, congruency secures shape equality, similarity scales figures. This note consolidates core rules.

Learning targets

• Quote and apply angle properties of parallel lines, polygons, and circles.
• Use triangle congruency tests (SSS, SAS, ASA, RHS) logically.
• Apply similarity ratios to find missing lengths and areas.
• Structure geometry proofs with clear statements and reasons.

1. Angle properties refresher

• Straight line: adjacent angles sum to \( 180^\circ \).
• Vertically opposite angles are equal.
• Alternate angles and corresponding angles are equal when lines are parallel.
• Interior angles on the same side of a transversal sum to \( 180^\circ \).

Polygons

Sum of interior angles in an \( n \)-gon: \( (n - 2) \times 180^\circ \).

Regular polygon exterior angle: \( \frac{360^\circ}{n} \).

2. Triangle congruency

Use the appropriate congruency test:

• SSS: Three sides equal.
• SAS: Two sides and included angle equal.
• ASA: Two angles and included side equal.
• RHS: Right angle, hypotenuse, and one side equal.

Example: Triangles \( \triangle ABC \) and \( \triangle DEF \) have \( AB = DE \), \( AC = DF \), and \( \angle BAC = \angle EDF \). Conclude congruency by SAS.

3. Similarity and scale factors

Triangles are similar if they have:

• Three pairs of equal angles (AAA).
• Corresponding sides proportional (SSS similarity).
• Two sides in proportion and included angle equal (SAS similarity).

Scale factor \( k = \frac{\text{length in image}}{\text{length in object}} \).

• Perimeter scales by \( k \).
• Area scales by \( k^2 \).
• Volume scales by \( k^3 \).

Worked example — Similar triangles

Given \( \triangle ABC \) similar to \( \triangle PQR \) with \( AB = 6 \text{ cm} \), \( AC = 9 \text{ cm} \), \( PQ = 4 \text{ cm} \). Find \( PR \).

Similarity ratio \( k = \frac{PQ}{AB} = \frac{4}{6} = \frac{2}{3} \).

Therefore \( PR = k \times AC = \frac{2}{3} \times 9 = 6 \text{ cm} \).

4. Circle properties (core set)

• Tangent is perpendicular to radius at point of contact.
• Angles subtended by the same chord at the circumference are equal.
• Angle in a semicircle is \( 90^\circ \).

Worked example: Given circle centre \( O \) with chord \( AB \) and point \( C \) on the major arc, show \( \angle ACB \) equals \( \angle AOB/2 \).

Provide reasoning: angle at centre is twice angle at circumference from same chord.

5. Structuring proofs

Write proofs in two-column or paragraph form, explicitly stating the reason for every statement.

Sample short proof

Given \( \triangle ABC \) and \( \triangle ADC \) share side \( AC \). If \( \angle BAC = \angle DAC \) and \( BC = DC \), prove \( \triangle ABC \) congruent to \( \triangle ADC \).

1. \( AC \) common side (reflexive property).
2. \( \angle BAC = \angle DAC \) (given).
3. \( BC = DC \) (given).
4. Therefore triangles congruent by SAS.

Try it yourself

1. In parallelogram \( ABCD \), show opposite angles are equal.
2. Prove that the base angles of an isosceles triangle are equal.
3. Two triangles have sidelengths in ratio 3:5:4 and 6:10:8. Are they similar or congruent? Justify.
4. A circle has chord \( AB \) of length 12 cm. Point \( C \) on the circumference forms angle \( 50^\circ \) with chord \( AB \). Find the angle at the centre subtended by \( AB \).

Proceed to mensuration and circles at https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-07-Mensuration-and-Circles.