5.2 Plane Geometry — Lines, Triangles & Circles

5.2 Plane Geometry — Lines, Triangles & Circles

Plane geometry examines the properties, measurements, and relationships of two-dimensional figures. In the HEC Aptitude Test (HAT), geometry questions evaluate a candidate's ability to apply fundamental theorems regarding intersecting lines, parallel transversals, triangle angle-side rules, special right triangles, and circular geometry.

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1. Angle Relationships & Parallel Lines

When two lines intersect, they create distinct angle pairs:

• Adjacent Angles / Linear Pair: Angles that lie on a straight line and share a common arm. Their sum is always $180^\circ$ (supplementary).
• Vertical Angles: Non-adjacent angles formed by two intersecting lines. Vertical angles are always equal in measure.
• Complementary Angles: Two angles whose measures sum to $90^\circ$.
• Supplementary Angles: Two angles whose measures sum to $180^\circ$.

Parallel Lines Cut by a Transversal

When a line (transversal $T$) intersects two parallel lines ($L_1 \parallel L_2$), eight angles are formed. These angles fall into exact equality or supplementary patterns:

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Worked Example 2.1: Transversal Angle Derivation

Problem: Lines $L_1$ and $L_2$ are parallel. A transversal cuts $L_1$ and $L_2$. If an interior angle on one side of the transversal is $(4x + 15)^\circ$ and its consecutive interior angle is $(x + 20)^\circ$, calculate $x$ and the measure of each angle.

Step-by-Step Derivation:

1. Consecutive interior angles on parallel lines are supplementary (their sum equals $180^\circ$): $(4x + 15) + (x + 20) = 180$
2. Combine like terms and solve for $x$: $5x + 35 = 180 \implies 5x = 145 \implies x = 29$
3. Calculate the two angle measures: $\text{Angle 1} = 4(29) + 15 = 116 + 15 = 131^\circ$ $\text{Angle 2} = 29 + 20 = 49^\circ$
4. Check sum: $131^\circ + 49^\circ = 180^\circ$.

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2. Triangle Properties & Classification

Triangles are classified by side lengths (scalene, isosceles, equilateral) or by interior angle measures (acute, right, obtuse).

Fundamental Triangle Theorems

1. Interior Angle Sum Theorem: The sum of interior angles in any triangle is always $180^\circ$: $\alpha + \beta + \gamma = 180^\circ$
2. Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote (non-adjacent) interior angles: $\theta_{\text{ext}} = \alpha + \beta$
3. Triangle Inequality Theorem: For any triangle with side lengths $a, b, c$, the sum of the lengths of any two sides must be strictly greater than the third side: $a + b > c, \quad a + c > b, \quad b + c > a$
4. Isosceles Triangle Rule: Sides opposite equal angles are equal in length, and vice versa.
5. Equilateral Triangle Formulas: All sides are equal ($s$), and all angles equal $60^\circ$. $\text{Height } h = \frac{\sqrt{3}}{2}s, \quad \text{Area } A = \frac{\sqrt{3}}{4}s^2$

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3. Pythagorean Theorem & Special Right Triangles

For any right-angled triangle with legs $a$ and $b$ and hypotenuse $c$:

$a^2 + b^2 = c^2$

Common Pythagorean Triples

Memorizing common integer triples accelerates problem-solving during the HAT test:

• $3 - 4 - 5$ (and multiples $6-8-10$, $9-12-15$, $12-16-20$)
• $5 - 12 - 13$
• $8 - 15 - 17$
• $7 - 24 - 25$

Special Right Triangles

Two right triangle configurations occur repeatedly on standardized aptitude tests:

1. $45^\circ - 45^\circ - 90^\circ$ Triangle (Isosceles Right Triangle):
   • Side ratio: $1 : 1 : \sqrt{2}$
   • If legs are $x$, hypotenuse $c = x\sqrt{2}$.
2. $30^\circ - 60^\circ - 90^\circ$ Triangle:
   • Side ratio: $1 : \sqrt{3} : 2$
   • Side opposite $30^\circ$ angle (shortest leg) $= x$
   • Side opposite $60^\circ$ angle (longer leg) $= x\sqrt{3}$
   • Side opposite $90^\circ$ angle (hypotenuse) $= 2x$

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Worked Example 2.2: Special Right Triangle Calculation

Problem: In a $30^\circ - 60^\circ - 90^\circ$ triangle, the hypotenuse has a length of $16$ cm. Find the perimeter and area of the triangle.

Step-by-Step Derivation:

1. Identify the base variable $x$ representing the side opposite $30^\circ$: $\text{Hypotenuse} = 2x = 16 \implies x = 8 \text{ cm}$
2. Find the longer leg opposite $60^\circ$: $\text{Longer leg } = x\sqrt{3} = 8\sqrt{3} \text{ cm}$
3. Calculate the perimeter $P$: $P = \text{Short leg} + \text{Long leg} + \text{Hypotenuse} = 8 + 8\sqrt{3} + 16 = 24 + 8\sqrt{3} \text{ cm}$
4. Calculate the area $A$: $A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 8 \times 8\sqrt{3} = 32\sqrt{3} \text{ cm}^2$

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4. Circles: Circumference, Area, Arcs & Sectors

A circle is the locus of points equidistant from a central point $O$.

Key Metrics

• Radius ($r$): Distance from center to outer edge.
• Diameter ($d$): Straight line across circle passing through center ($d = 2r$).
• Circumference ($C$): Total boundary length: $C = 2\pi r = \pi d$
• Area ($A$): Total enclosed surface area: $A = \pi r^2$

Arc Length & Sector Area

An arc is a fraction of the circle's boundary, and a sector is a pie-shaped slice enclosed by two radii and an arc. For central angle $\theta$ (in degrees):

$\text{Arc Length } L = \frac{\theta}{360^\circ} \times 2\pi r$

$\text{Sector Area } A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$

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Worked Example 2.3: Arc Length & Sector Calculation

Problem: A circle has a radius of $9$ cm. Calculate the arc length and sector area for a central angle of $120^\circ$.

Step-by-Step Derivation:

1. Compute the fraction of the circle: $\text{Fraction} = \frac{120^\circ}{360^\circ} = \frac{1}{3}$
2. Calculate Arc Length $L$: $L = \frac{1}{3} \times 2\pi (9) = \frac{18\pi}{3} = 6\pi \text{ cm}$
3. Calculate Sector Area $A_{\text{sector}}$: $A_{\text{sector}} = \frac{1}{3} \times \pi (9^2) = \frac{81\pi}{3} = 27\pi \text{ cm}^2$

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5. Circle Theorems — Tangents & Inscribed Angles

1. Tangent-Radius Theorem: A line tangent to a circle is strictly perpendicular ($90^\circ$) to the radius drawn to the point of tangency.
2. Inscribed Angle Theorem: An angle inscribed in a circle (vertex on the circle circumference) is equal to half the measure of the central angle that subtends the same arc: $\theta_{\text{inscribed}} = \frac{1}{2} \theta_{\text{central}}$
3. Thales's Theorem: Any angle inscribed in a semicircle (subtended by a diameter) is always a right angle ($90^\circ$).