Lines, Angles, and Triangles

Welcome to geometry — this is where aptitude gets visual. Lines, angles, and triangles form the backbone of every geometry question we’ll ever see. The good news? Most of it is pattern recognition. Once we know the properties, we can spot the answer almost instantly. Let’s build up from the basics.

Types of Angles

• Acute angle: Less than 90° (think: a cute little angle)
• Right angle: Exactly 90° (the square corner)
• Obtuse angle: Between 90° and 180° (the fat angle)
• Straight angle: Exactly 180° (a straight line)
• Reflex angle: Between 180° and 360° (the “wrap around” angle)

Quick fact: Complementary angles add up to 90°. Supplementary angles add up to 180°. These definitions come up all the time.

Parallel Lines and a Transversal

When a line (transversal) cuts two parallel lines, it creates 8 angles — but we really only need to know 2 values, because angles come in matching sets.

Here are the angle relationships to memorize:

• Corresponding angles are equal: a = e, b = f, c = g, d = h (same position at each intersection)
• Alternate interior angles are equal: c = e, d = f (opposite sides, between the lines)
• Alternate exterior angles are equal: a = h, b = g (opposite sides, outside the lines)
• Co-interior (same-side interior) angles are supplementary: c + f = 180°, d + e = 180°

The shortcut: If we know just ONE angle, we know all eight. Angles are either equal to it or supplementary (add up to 180°).

Triangle Basics

Angle Sum Property

The angles inside any triangle always add up to exactly 180°. No exceptions. This is probably the single most useful fact in geometry.

Exterior Angle Theorem

An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

In simple language: if we extend one side of a triangle, the angle formed outside equals the sum of the two “far away” interior angles. This is a massive time-saver.

Triangle Inequality

For three sides to form a triangle, the sum of any two sides must be greater than the third side. In simple language: the two shorter sides together must be longer than the longest side. If not, no triangle.

Types of Triangles

By sides:

• Equilateral: All 3 sides equal, all angles = 60°
• Isosceles: 2 sides equal, base angles are equal
• Scalene: All sides different, all angles different

By angles:

• Acute: All angles < 90°
• Right-angled: One angle = 90°
• Obtuse: One angle > 90°

Pythagoras Theorem

The Pythagoras theorem only works for right-angled triangles. The hypotenuse (the side opposite the right angle) is always the longest side.

Pythagorean Triplets — Memorize These!

These sets of three numbers always form a right triangle. Knowing them saves us from doing any calculation:

Trick: Any multiple of a Pythagorean triplet is also a Pythagorean triplet. So if we see sides 15, 20, 25 — that’s just 3-4-5 multiplied by 5. Instant right triangle!

Identifying the triangle type from sides:

• If c² = a² + b² → Right triangle
• If c² < a² + b² → Acute triangle
• If c² > a² + b² → Obtuse triangle

Congruence of Triangles

Two triangles are congruent when they’re exactly the same shape AND size — one is a perfect copy of the other. Think of it like photocopying a triangle.

The conditions for congruence:

• SSS (Side-Side-Side): All three sides are equal
• SAS (Side-Angle-Side): Two sides and the included angle are equal
• ASA (Angle-Side-Angle): Two angles and the included side are equal
• AAS (Angle-Angle-Side): Two angles and a non-included side are equal
• RHS (Right-Hypotenuse-Side): For right triangles — hypotenuse and one side are equal

Watch out: SSA (Side-Side-Angle) is NOT a valid congruence rule (the ambiguous case). AAA is also not valid — it gives similarity, not congruence.

Similarity of Triangles

Two triangles are similar when they have the same shape but can be different sizes. Think of it like zooming in or out on a triangle.

The conditions for similarity:

• AA (Angle-Angle): Two angles are equal (third is automatically equal since angles sum to 180°)
• SAS (Side-Angle-Side): Two pairs of sides are proportional and the included angle is equal
• SSS (Side-Side-Side): All three pairs of sides are proportional

Key property: If two triangles are similar with sides in ratio k, then:

• Corresponding altitudes, medians, angle bisectors are also in ratio k
• Areas are in ratio k²
• This area ratio (k²) is the most tested property in exams!

Area of a Triangle

We have three main formulas — use whichever fits the given information:

1. Base-Height formula: Area = ½ × base × height (most common)

2. Heron’s formula (when all 3 sides are known):

• First find semi-perimeter: s = (a + b + c) / 2
• Then Area = √[s(s-a)(s-b)(s-c)]

3. Two sides and included angle: Area = ½ × a × b × sin(C)

Special Triangle Areas

• Equilateral triangle with side a: Area = (√3/4)a², Height = (√3/2)a
• Right triangle with legs a and b: Area = ½ × a × b (legs are base and height!)
• Isosceles triangle with equal sides a and base b: Height = √(a² - b²/4)

Worked Examples

Example 1: Angle Finding

In a triangle, two angles are 45° and 72°. Find the third angle.

Sum of angles = 180° Third angle = 180° - 45° - 72° = 63°

Example 2: Exterior Angle

One exterior angle of a triangle is 120°. If one of the non-adjacent interior angles is 50°, find the other non-adjacent interior angle.

Exterior angle = sum of non-adjacent interior angles 120° = 50° + other angle Other angle = 120° - 50° = 70°

And the adjacent interior angle = 180° - 120° = 60°. Quick check: 50° + 70° + 60° = 180° ✓

Example 3: Pythagoras

A ladder 13 m long is placed against a wall. The foot of the ladder is 5 m from the wall. How high up the wall does the ladder reach?

We recognize 5 and 13 — this is the 5-12-13 triplet! Height = 12 m

Without recognizing the triplet: h² = 13² - 5² = 169 - 25 = 144, so h = 12 m.

Example 4: Heron’s Formula

Find the area of a triangle with sides 7, 8, and 9.

Semi-perimeter s = (7 + 8 + 9) / 2 = 12 Area = √[12 × (12-7) × (12-8) × (12-9)] Area = √[12 × 5 × 4 × 3] Area = √720 Area = √(144 × 5) Area = 12√5 ≈ 26.83 sq units

Example 5: Similar Triangles

Two similar triangles have sides in the ratio 3:5. If the area of the smaller triangle is 36 cm², find the area of the larger triangle.

For similar triangles, area ratio = (side ratio)² Area ratio = (3/5)² = 9/25 36/Area₂ = 9/25 Area₂ = 36 × 25/9 = 100 cm²

Common Exam Variations

1. Finding missing angles using angle sum property + parallel line properties combined
2. Identifying right triangles from given sides (check if they form a Pythagorean triplet or its multiple)
3. Area of shaded regions — usually involves subtracting one triangle’s area from another
4. Similar triangles to find unknown sides when two triangles share an angle (very common with overlapping triangles)
5. Median/centroid problems — the centroid divides each median in ratio 2:1 from vertex

Practice Problems

Q1: In a triangle PQR, the exterior angle at R is 110°. If ∠P = 60°, find ∠Q.

Q2: The sides of a triangle are 6, 8, and 10. Find its area and determine what type of triangle it is.

Q3: Two similar triangles have areas 64 cm² and 100 cm². If the longest side of the smaller triangle is 16 cm, find the longest side of the larger triangle.

Answers

A1: Exterior angle at R = ∠P + ∠Q → 110° = 60° + ∠Q → ∠Q = 50°. Check: ∠R = 180° - 110° = 70°, and 60° + 50° + 70° = 180° ✓

A2: Check: 6² + 8² = 36 + 64 = 100 = 10². It’s a right triangle (the 3-4-5 triplet × 2). Area = ½ × 6 × 8 = 24 cm².

A3: Area ratio = 64/100 = (side ratio)². Side ratio = √(64/100) = 8/10 = 4/5. Longest side of larger = 16 × 5/4 = 20 cm.