- Mr Ausome

# 9 Tips to Mastering Proofs of Plane Geometry (O-Level A-Math)

Proofs of Plane Geometry (PG) is a ghastly topic that haunts many O-level A-Math students. It is not uncommon to see students spent countless hours staring at a single plane geometry question and eventually collapse into eternal coma. Before we discuss the cures to this evil syndrome, let us first take a look at the 4 stages of the PG (Plane Geometry) syndrome that experts have identified:

**4 Stages of Plane Geometry Syndrome**

**Stage 1: Ignorant Optimism**

Being told that the only new things in this chapter are two simple theorems (Alternate segment theorem and mid-point theorem), students think that they can just memorize and apply formulas to solve all plane geometry problems.

**Stage 2: Realization**

After trying a few questions, students realize that it involves all the geometrical proofs and angle properties from sec 1 to sec 4 E-math. When they look at the answers, it feels that the proofs are not too difficult and questions seem doable.

**Stage 3: Fear**

After attempting several more questions, students realize that they couldn’t do a single one by themselves. They had to refer to the answers for every single question. They start to feel hopeless and fearful.

**Stage 4: Maturity**

Not every student reaches this stage. But those that have attempted hundreds of questions from worksheets, assessment books, past year papers, tuition practice eventually grasp the concepts and gains some confidence. Yet, they still remain wary and apprehensive when facing plane geometry questions.

**9 Tips On Handling Proofs of Plane Geometry**

The following are some tips for students (O-level or IP) who are having difficulties with geometric proving questions:

**Tip 1:** Do not assume that all details are indicated in diagram. Always **read the wordy part of the question **and fill in any extra information onto diagram.

**Tip 2:** Use highlighter to **highlight triangles on the diagram** that you want to prove as “similar” or “congruent”.

**Tip 3:** When desperate, apply every single theorem to find every single angle in the diagram. You might get lucky and hit a few of the working marks. In the process, you might also accidently “See the light” as well.

**Tip 4:** Don’t forget about what you learnt in sec 1! **Ext angle of triangle = sum of int angles**. Although taught in sec 1, this is surprisingly often used in sec 4 A-math Plane Geometry.

**Tip 5**: When trying to prove two angles are the same, **let one of the unknown angles be x**. This helps to simplify your working.

**Tip 6: **Common questions and know what they mean:

**Tip 7:** Look out for figures with more than 1 radius given. **Radii of a circle are same length **and they form isosceles triangles.

**Tip 8:** When stuck, go through each of the rules in Plane Geometry, one by one.

Since plane geometry questions involve alot of staring, you can use the mnemonics "**M SPECTATORS**" to run through all the points one by one:

**Tip 9: **If unable to derive clear directions after 3 mins of staring, it is advisable to** skip the plane geometry question** and attempt it at the end.

The following are **5 carefully selected ****challenging questions** to test you on your current competency in geometric proving. Do your best to prove them without looking at the solutions! Refer to the tips above and the hints given if you are stuck. Only look at the answers as a last resort. Good luck!

**Example 1: **

**Example 2:**

**Example 3:**

**Example 4:**

**Example 5:**

**Solutions to above Examples 1-5**

To get professional help (Math tuition) on secondary school mathematics in Singapore, click here.

**Mr Ausome**