Geometry I — Angles, Triangles & Symmetry

By the end of this chapter you will:

1. Use the angle-sum theorems for triangles, quadrilaterals, and straight lines
2. Spot vertical and parallel-line angle relationships
3. Chase angles through isosceles chains
4. Count lines of symmetry for regular shapes

5.0 The angle-chasing mindset

BC baseline: G6 covers angle measurement and classification (acute, right, obtuse, reflex) and polygon angles. What BC doesn't teach: chain reasoning ("$\angle A = 50°$ and the triangle is isosceles → therefore $\angle B = 65°$ → therefore $\angle C = …$"), parallel-line angle theorems (Z, F, C shapes), or counting lines of symmetry. Those three skills are the heart of Gauss geometry.

Most Gauss geometry questions in Parts A and B are angle chases: you're given a few angles, told about parallel lines or isosceles triangles, and asked to find $x$. The solution path is usually 2–3 short deductions.

🔑 Mark up the diagram. Annotate every angle you can deduce — equal marks for equal angles, parallel arrows on parallel lines. The answer often falls out the moment the diagram is fully labeled.

5.1 The angle-sum facts

2024 Q6 — straight angle

$\angle PQR$ is a straight angle. Find $x$ given the diagram shows $146°$ adjacent to $x°$.

Straight angle $= 180°$, so $146 + x = 180 \Rightarrow x = \mathbf{34}$. Answer (E). ✅

5.2 Vertical, parallel & corresponding angles

Vertical angles (formed by two intersecting lines) are equal.

Parallel-line angle pairs (where a transversal crosses two parallel lines):

🔑 If the problem mentions parallel lines, look for the $Z$, $F$, or $C$ shape formed by the transversal. The $Z$ gives alternate-interior (equal), the $F$ gives corresponding (equal), the $C$ gives co-interior (sum $180°$).

5.3 Isosceles & equilateral chains

Isosceles theorem: in a triangle with two equal sides, the angles opposite those sides are equal.

Equilateral: all three sides equal $\implies$ all three angles equal $60°$.

Example

In $\triangle ABC$, $AB = AC$ and $\angle A = 40°$. Find $\angle B$.

Since $AB = AC$, $\angle B = \angle C$.
$\angle A + \angle B + \angle C = 180° \Rightarrow 40 + 2 \angle B = 180 \Rightarrow \angle B = \mathbf{70°}$.

5.4 Lines of symmetry

A line of symmetry is a line such that folding along it makes the two halves coincide.

2024 Q3 — vertical line of symmetry

Which of the following shapes has a vertical line of symmetry?

The answer is the rectangle in choice (E) (the standard textbook rectangle, with its sides parallel to the page edges and a clean vertical mirror axis through its center).

⚠️ Trap: a parallelogram looks symmetric but typically isn't — it has rotational symmetry but no reflective lines of symmetry unless it's a rhombus or rectangle.

5.5 Trap Alerts ⚠️

1. Diagrams are NOT to scale. Don't measure angles with a protractor on the test booklet; you'll be misled.
2. Vertical angles ≠ supplementary angles. Vertical = opposite (equal). Supplementary = adjacent on a straight line (sum to 180°).
3. Two-sides-equal-two-angles-equal runs both directions: equal angles also imply equal sides.
4. Parallelogram is NOT symmetric in the reflection sense (unless it's a rectangle or rhombus).

5.6 Mnemonic

"Triangle = 180°, quad = 360°, equal sides give equal angles, look for $\frac{Z}{F}$/C with parallels."

Practice Set

1. (Part A) In a triangle, two angles are $50°$ and $60°$. Find the third.
2. (Part B) In $\triangle PQR$, $PQ = PR$ and $\angle Q = 75°$. Find $\angle P$.
3. (Part B) Two parallel lines are cut by a transversal. One pair of co-interior angles measures $x°$ and $(2x + 30)°$. Find $x$.
4. (Part C) In an isosceles triangle $ABC$ with $AB = AC$, the angle bisector from $A$ meets $BC$ at $M$. If $\angle BAC = 40°$, find $\angle AMB$.

Answers: 1) 70°; 2) Since $PQ = PR$, $\angle Q = \angle R = 75°$, so $\angle P = 30°$; 3) $x + 2x + 30 = 180 \Rightarrow x = 50$; 4) The bisector is perpendicular to the base in an isosceles triangle, so $\angle AMB = 90°$.

End of chapter. Next: Geometry II — Area, Perimeter & Scaling Laws.