“Geometry: Angles and Triangles”

🟢 Lite — Quick Review (1h–1d)

Rapid summary of angles and triangles for NABTEB mathematics.

Types of Angles:

Angle Type	Size	Example
Acute	$0° < x < 90°$	45°
Right angle	$x = 90°$	—
Obtuse	$90° < x < 180°$	120°
Straight	$x = 180°$	—
Reflex	$180° < x < 360°$	270°
Complementary	Two angles adding to 90°	30° + 60°
Supplementary	Two angles adding to 180°	110° + 70°

Angle Properties:

• Angles around a point add to $360°$
• Angles on a straight line add to $180°$
• Vertically opposite angles are equal
• Angles in a triangle add to $180°$
• Angles in a quadrilateral add to $360°$

Types of Triangles:

Type	Properties
Equilateral	All sides equal, all angles = 60°
Isosceles	Two sides equal, two base angles equal
Scalene	All sides different, all angles different
Right-angled	One angle = 90°

Triangle Angle Rules:

1. Sum of interior angles = $180°$
2. Exterior angle = sum of two opposite interior angles
3. Base angles of an isosceles triangle are equal

Pythagoras’ Theorem: $$a^2 + b^2 = c^2$$ Where $c$ is the hypotenuse (longest side, opposite the right angle).

⚡ NABTEB Exam Tip: In an isosceles triangle, the altitude from the apex also bisects the base and the apex angle. This is useful in construction and proof questions.

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🟡 Standard — Regular Study (2d–2mo)

For NABTEB students who want thorough understanding.

Congruent Triangles:

Two triangles are congruent if all corresponding sides and angles are equal.

Tests for Congruence:

Test	Abbreviation	Description
Side-Side-Side	SSS	All three sides equal
Side-Angle-Side	SAS	Two sides and the included angle equal
Angle-Side-Angle	ASA	Two angles and the included side equal
Angle-Angle-Side	AAS	Two angles and a corresponding side equal
Right angle-Hypotenuse-Side	RHS	Right angle, hypotenuse, and one other side equal

Example RHS: Two right-angled triangles with the same hypotenuse and one other equal side are congruent.

Similar Triangles:

Two triangles are similar if their corresponding angles are equal (and sides are in proportion).

Tests for Similarity:

Test	Description
AAA	All three angles equal
SSS	All three sides in proportion
SAS	Two sides in proportion and included angle equal

Area of a Triangle: $$A = \frac{1}{2} \times \text{base} \times \text{height}$$ $$A = \frac{1}{2}ab\sin C \text{ (using two sides and included angle)}$$ $$A = \sqrt{s(s-a)(s-b)(s-c)} \text{ (Heron’s formula, where } s = \frac{a+b+c}{2}\text{)}$$

Circle Theorems:

Theorem	Statement
Angle in a semicircle	Angle subtended by a diameter is $90°$
Angles in the same segment	Angles subtended by the same chord are equal
Angle at the centre	Angle at the centre is twice the angle at the circumference
Cyclic quadrilateral	Opposite angles sum to $180°$
Tangent-radius	Tangent is perpendicular to the radius

Angles in Circles:

• Chord: a straight line joining two points on the circumference
• Arc: part of the circumference
• Segment: region between a chord and an arc

⚡ NABTEB Exam Tip: When solving problems with circles, always identify the chord, arc, or segment involved and apply the correct circle theorem.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage of geometry for thorough NABTEB preparation.

Polygon Interior Angles:

For an $n$-sided polygon:

• Sum of interior angles = $(n - 2) \times 180°$
• Each interior angle (regular polygon) = $\frac{(n-2) \times 180°}{n}$
• Each exterior angle (regular polygon) = $\frac{360°}{n}$

Regular Polygons:

Sides	Interior Angle	Exterior Angle
3	60°	120°
4	90°	90°
5	108°	72°
6	120°	60°
8	135°	45°

Parallel Lines:

When two parallel lines are intersected by a transversal:

• Corresponding angles are equal (e.g., both top-left)
• Alternate interior angles are equal (Z-shape)
• Co-interior (allied) angles sum to $180°$ (U-shape)

Coordinate Geometry:

Distance between two points $(x_1, y_1)$ and $(x_2, y_2)$: $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Midpoint of a line joining $(x_1, y_1)$ and $(x_2, y_2)$: $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Gradient (slope) of a line: $$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Equation of a straight line: $$y - y_1 = m(x - x_1)$$ $$y = mx + c \text{ (where } c \text{ is the y-intercept)}$$

Parallel lines have the same gradient: $m_1 = m_2$ Perpendicular lines have gradients such that: $m_1 \times m_2 = -1$

The Section Formula:

Dividing a line in ratio $m:n$ (internally): $$P = \left(\frac{nx_1 + mx_2}{m+n}, \frac{ny_1 + my_2}{m+n}\right)$$

Area of a Triangle from Coordinates:

For vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$: $$A = \frac{1}{2}|x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$$

Circle Equations:

Centre $(a, b)$, radius $r$: $$(x - a)^2 + (y - b)^2 = r^2$$

Locus:

The path traced by a point satisfying given conditions:

• Circle: Points equidistant from a fixed point (centre)
• Perpendicular bisector: Points equidistant from two fixed points
• Angle bisector: Points equidistant from two intersecting lines

Trigonometric Ratios in All Quadrants:

Quadrant	Sine	Cosine	Tangent
I (0°–90°)	+	+	+
II (90°–180°)	+	−	−
III (180°–270°)	−	−	+
IV (270°–360°)	−	+	−

Compound Angle Formulas:

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$ $$\cos(A + B) = \cos A \cos B - \sin A \sin B$$ $$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$

2D Vector Geometry:

For $\vec{u} = (u_1, u_2)$ and $\vec{v} = (v_1, v_2)$:

• Addition: $\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2)$
• Magnitude: $|\vec{u}| = \sqrt{u_1^2 + u_2^2}$
• Dot product: $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 = |\vec{u}||\vec{v}|\cos\theta$
• Angle between vectors: $\cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}||\vec{v}|}$

Transformations:

Transformation	Effect
Translation	Slides every point the same distance and direction
Reflection	Flips across a line (mirror image)
Rotation	Turns around a fixed point
Enlargement	Scales by a factor $k$ from a centre of enlargement

If $k > 1$: enlargement; $0 < k < 1$: reduction; $k < 0$: rotated enlargement.

⚡ NABTEB Quick Reference:

• Angles in triangle: $180°$
• Angles in quadrilateral: $360°$
• Interior angles of $n$-gon: $(n-2) \times 180°$
• Exterior angles (regular): $360°/n$
• Pythagoras: $a^2 + b^2 = c^2$
• Congruence tests: SSS, SAS, ASA, AAS, RHS
• Similarity tests: AAA, SSS, SAS
• Area (triangle): $\frac{1}{2}bh$ or $\frac{1}{2}ab\sin C$ or $\sqrt{s(s-a)(s-b)(s-c)}$
• Distance: $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
• Midpoint: $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$
• Gradient: $\frac{y_2-y_1}{x_2-x_1}$
• $y = mx + c$: $m$ = gradient, $c$ = y-intercept
• Perpendicular: $m_1 \times m_2 = -1$
• Circle: $(x-a)^2 + (y-b)^2 = r^2$
• Vector dot product: $u_1v_1 + u_2v_2 = |u||v|\cos\theta$