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Geometry

Level · High-school Euclidean geometry Chapters · 8 Explorers · 3 interactive Feeds into · SAT · ACT · IGCSE
1

Foundations & Reasoning

Geometry begins with undefined terms — point, line, plane — and builds outward through definitions and logical argument. Most early results come from a handful of angle relationships.

● Definition 1.1 — Angle pairs

Two angles are complementary if they sum to \( 90^{\circ} \) and supplementary if they sum to \( 180^{\circ} \). A linear pair (adjacent angles on a straight line) is always supplementary, and vertical angles (opposite at an intersection) are always equal.

Midpoint
\( M=\Big(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\Big) \)
Complementary
\( \alpha+\beta=90^{\circ} \)
Supplementary
\( \alpha+\beta=180^{\circ} \)
Vertical angles
equal in measure
Worked example 1.A

Find the complement of \( 35^{\circ} \).

\[ 90^{\circ}-35^{\circ}=55^{\circ}. \]

Worked example 1.B

Angles \( \angle 1 \) and \( \angle 2 \) form a linear pair with \( \angle 1=125^{\circ} \). Find \( \angle 2 \).

\[ 180^{\circ}-125^{\circ}=55^{\circ}. \]

📝 Chapter Quiz
2

Lines & Angles

When a transversal cuts two parallel lines, eight angles appear — but only two distinct measures. Knowing which pairs are equal and which are supplementary unlocks most angle-chase problems.

▲ Reference — Parallel lines & a transversal

Corresponding angles are equal; alternate (interior or exterior) angles are equal; co-interior (same-side interior) angles are supplementary, summing to \( 180^{\circ} \).

Tilt the transversal below. Every acute angle stays equal to every other acute angle (corresponding and alternate), and each is supplementary to the obtuse one.

⊙ Explorer · Transversal anglesinteractive
acute angle—
obtuse angle—

The two marked angles are corresponding — always equal. Acute + obtuse \( =180^{\circ} \).

Corresponding
equal
Alternate
equal
Co-interior
sum to \( 180^{\circ} \)
Angles on a line
sum to \( 180^{\circ} \)
Worked example 2.A

Two parallel lines are cut by a transversal. One interior angle is \( 72^{\circ} \). Find its alternate interior angle.

Alternate interior angles are equal:

\[ 72^{\circ}. \]

Worked example 2.B

A co-interior angle measures \( 110^{\circ} \). Find the other co-interior angle.

\[ 180^{\circ}-110^{\circ}=70^{\circ}. \]

📝 Chapter Quiz
3

Triangles & Congruence

The angles of every triangle sum to \( 180^{\circ} \) — no matter its shape. Congruence criteria then let us prove two triangles identical from only three pieces of information.

▲ Reference — Triangle facts

Interior angles sum to \( 180^{\circ} \). An exterior angle equals the sum of the two remote interior angles. Triangles are congruent by SSS, SAS, ASA, AAS, or HL (right triangles).

Drag the apex. The three interior angles change, yet their sum holds at exactly \( 180^{\circ} \).

⊙ Explorer · The angle suminteractive
∠A—
∠B—
∠C—
sum—

\( \angle A+\angle B+\angle C=180^{\circ} \) for every triangle you can make.

Angle sum
\( A+B+C=180^{\circ} \)
Exterior angle
\( =\) sum of remote interior
Isosceles
base angles equal
Congruence
SSS · SAS · ASA · AAS · HL
Worked example 3.A

Two angles of a triangle are \( 40^{\circ} \) and \( 60^{\circ} \). Find the third.

\[ 180^{\circ}-40^{\circ}-60^{\circ}=80^{\circ}. \]

Worked example 3.B

The angles of a triangle are in the ratio \( 1:2:3 \). Find them.

Six parts total \( 180^{\circ} \), so one part is \( 30^{\circ} \):

\[ 30^{\circ},\ 60^{\circ},\ 90^{\circ}. \]

📝 Chapter Quiz
4

Similarity

Similar figures have the same shape but not necessarily the same size: equal angles and sides in a constant ratio. The ratio — the scale factor — governs lengths, areas and volumes differently.

● Definition 4.1 — Similar polygons

Two polygons are similar when corresponding angles are equal and corresponding sides are in proportion. Triangles need only two equal angles (AA) to be similar.

Side ratio
\( \dfrac{a'}{a}=k \)
Perimeter ratio
k
Area ratio
\( k^{2} \)
Triangle test
AA similarity
Worked example 4.A

A \( 6 \)-ft person casts a \( 4 \)-ft shadow while a tree casts a \( 20 \)-ft shadow. Find the tree's height.

Similar triangles give \( \dfrac{6}{4}=\dfrac{h}{20} \):

\[ h=\frac{6\times 20}{4}=30\ \text{ft}. \]

Worked example 4.B

Two similar figures have side ratio \( 1:3 \). What is the ratio of their areas?

\[ 1^{2}:3^{2}=1:9. \]

📝 Chapter Quiz
5

Right Triangles & Trigonometry

The right angle is geometry's workhorse. Pythagoras links the three sides; the special \( 45\text{–}45\text{–}90 \) and \( 30\text{–}60\text{–}90 \) triangles give exact ratios; and SOH-CAH-TOA handles the rest.

▲ Reference — Right-triangle toolkit

Pythagoras: \( a^{2}+b^{2}=c^{2} \). A \( 45\text{–}45\text{–}90 \) triangle has sides \( 1:1:\sqrt2 \); a \( 30\text{–}60\text{–}90 \) triangle has sides \( 1:\sqrt3:2 \). Ratios: \( \sin=\tfrac{\text{opp}}{\text{hyp}} \), \( \cos=\tfrac{\text{adj}}{\text{hyp}} \), \( \tan=\tfrac{\text{opp}}{\text{adj}} \).

Pythagoras is really a statement about areas: the square on the hypotenuse equals the sum of the squares on the legs. Resize the legs and watch the three areas balance.

⊙ Explorer · Squares on the sidesinteractive
a² + b²9 + 16
c² (hypotenuse)25

The amber squares (on the legs) always add up to the indigo square (on the hypotenuse).

Pythagoras
\( a^{2}+b^{2}=c^{2} \)
45–45–90
\( 1:1:\sqrt2 \)
30–60–90
\( 1:\sqrt3:2 \)
Ratios
SOH-CAH-TOA
Worked example 5.A

A right triangle has legs \( 5 \) and \( 12 \). Find the hypotenuse.

\[ \sqrt{5^{2}+12^{2}}=\sqrt{169}=13. \]

Worked example 5.B

In a \( 30\text{–}60\text{–}90 \) triangle the short leg is \( 1 \). Find the hypotenuse.

The hypotenuse is twice the short leg:

\[ 2. \]

📝 Chapter Quiz
6

Polygons & Quadrilaterals

Beyond the triangle lie quadrilaterals and regular polygons, each with predictable angle sums and — for the special quadrilaterals — distinctive side, angle and diagonal properties.

▲ Reference — Polygon angles

Interior angles of an \( n \)-gon sum to \( (n-2)\times 180^{\circ} \); exterior angles always sum to \( 360^{\circ} \). In a parallelogram, opposite sides and opposite angles are equal and the diagonals bisect each other.

Interior sum
\( (n-2)\,180^{\circ} \)
Regular interior
\( \dfrac{(n-2)180^{\circ}}{n} \)
Exterior angle
\( \dfrac{360^{\circ}}{n} \)
Parallelogram
opposite angles equal
Worked example 6.A

Find the sum of the interior angles of an octagon.

\[ (8-2)\times 180^{\circ}=1080^{\circ}. \]

Worked example 6.B

Find each interior angle of a regular hexagon.

\[ \frac{(6-2)\times 180^{\circ}}{6}=120^{\circ}. \]

📝 Chapter Quiz
7

Circles

Circles bring their own angle and length rules: the relationship between central and inscribed angles, the right angle in a semicircle, and arcs and sectors as fractions of the whole.

▲ Reference — Circle facts

Circumference \( C=2\pi r \); area \( A=\pi r^{2} \). The inscribed angle is half the central angle on the same arc, so the angle in a semicircle is \( 90^{\circ} \). An arc and sector scale by \( \dfrac{\theta}{360} \).

Circumference
\( C=2\pi r \)
Area
\( A=\pi r^{2} \)
Arc length
\( \dfrac{\theta}{360}\,2\pi r \)
Sector area
\( \dfrac{\theta}{360}\,\pi r^{2} \)
Worked example 7.A

A triangle is inscribed in a circle with one side as the diameter. Find the angle opposite that side.

The angle in a semicircle is a right angle:

\[ 90^{\circ}. \]

Worked example 7.B

Find the area of a \( 90^{\circ} \) sector of a circle with \( r=4 \). Take \( \pi=3.14 \).

\[ \frac{90}{360}\,\pi r^{2}=\tfrac14\times 3.14\times 16=12.56. \]

📝 Chapter Quiz
8

Area, Surface Area & Volume

The measurement strand: areas of plane figures and the surface areas and volumes of the common solids — prisms, cylinders, cones, pyramids and spheres.

▲ Reference — Key formulas

Rectangle area \( =bh \); triangle \( =\tfrac12 bh \). Prism volume \( =A_{\text{base}}\times h \); cylinder \( =\pi r^{2}h \); sphere \( =\tfrac43\pi r^{3} \); cone \( =\tfrac13\pi r^{2}h \).

Rectangle
\( A=bh \)
Triangle
\( A=\tfrac12 bh \)
Prism / cylinder
\( V=A_{\text{base}}h \)
Sphere / cone
\( \tfrac43\pi r^{3},\ \tfrac13\pi r^{2}h \)
Worked example 8.A

Find the volume of a cube of side \( 4 \).

\[ 4^{3}=64. \]

Worked example 8.B

Find the volume of a cylinder with \( r=2 \), \( h=5 \). Take \( \pi=3.14 \).

\[ \pi r^{2}h=3.14\times 4\times 5=62.8. \]

📝 Chapter Quiz
9

Transformations & Symmetry

Rigid motions — translations, reflections and rotations — preserve size and shape; dilations scale a figure.

▲ Reference — Transformations

Translations, reflections and rotations are rigid motions that preserve distance. A dilation by factor \( k \) scales lengths by \( k \) and areas by \( k^{2} \).

Translation
slide
Reflection
flip
Rotation
turn
Dilation
scale by \( k \)
Worked example 9.A

A dilation with scale factor \( 3 \) maps a side of length \( 4 \). Find the image length.

\[ 3\times 4=12. \]

📝 Chapter Quiz
10

Coordinate Geometry

Placing figures on the plane lets us use distance, midpoint and slope to prove geometric facts.

▲ Reference — On the plane

Distance \( \sqrt{(x_2-x_1)^{2}+(y_2-y_1)^{2}} \); midpoint \( \left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right) \). Parallel lines share a slope; perpendicular slopes multiply to \( -1 \).

Distance
\( \sqrt{\Delta x^{2}+\Delta y^{2}} \)
Midpoint
average the ends
Parallel
equal slopes
Perpendicular
\( m_1m_2=-1 \)
Worked example 10.A

Find the distance from \( (0,0) \) to \( (6,8) \).

\[ \sqrt{36+64}=10. \]

📝 Chapter Quiz
11

Probability

Probability measures the chance of an event, including area-based (geometric) models.

▲ Reference — Probability

\( P=\tfrac{\text{favourable}}{\text{total}} \). Geometric probability compares lengths or areas. Independent events multiply, and \( P(\text{not }A)=1-P(A) \).

Probability
\( \tfrac{\text{fav}}{\text{total}} \)
Geometric
ratio of areas
Independent
\( P(A)P(B) \)
Complement
\( 1-P \)
Worked example 11.A

A point lands in a \( 10\times10 \) square. Find the chance it lands in a \( 2\times2 \) corner.

\[ \frac{4}{100}=0.04. \]

📝 Chapter Quiz
§

Practice set

Fifty problems spanning all eight chapters — foundations through area and volume — tagged by difficulty. Click any problem to reveal a full worked solution.

1

\( M \) is the midpoint of \( \overline{AB} \) with \( AB=12 \). Find \( AM \).

Basic
\[ \tfrac12\times 12=6. \]
2

Find the complement of \( 35^{\circ} \).

Basic
\[ 90^{\circ}-35^{\circ}=55^{\circ}. \]
3

Find the supplement of \( 110^{\circ} \).

Basic
\[ 180^{\circ}-110^{\circ}=70^{\circ}. \]
4

Two angles are vertical and one measures \( 70^{\circ} \). Find the other.

Intermediate

Vertical angles are equal:

\[ 70^{\circ}. \]
5

An angle bisector splits an \( 80^{\circ} \) angle. Find each part.

Intermediate
\[ \tfrac12\times 80^{\circ}=40^{\circ}. \]
6

\( \angle 1 \) and \( \angle 2 \) form a linear pair with \( \angle 1=125^{\circ} \). Find \( \angle 2 \).

Intermediate
\[ 180^{\circ}-125^{\circ}=55^{\circ}. \]
7

Parallel lines are cut by a transversal. Find the corresponding angle to \( 65^{\circ} \).

Basic

Corresponding angles are equal:

\[ 65^{\circ}. \]
8

Find the alternate interior angle to \( 72^{\circ} \).

Intermediate
\[ 72^{\circ}. \]
9

One co-interior angle is \( 110^{\circ} \). Find the other.

Intermediate
\[ 180^{\circ}-110^{\circ}=70^{\circ}. \]
10

Two corresponding angles measure \( 3x \) and \( 75^{\circ} \). Find \( x \).

Intermediate

Equal, so \( 3x=75 \):

\[ x=25. \]
11

Alternate angles measure \( 2x \) and \( x+40 \). Find \( x \).

Intermediate

\( 2x=x+40 \):

\[ x=40. \]
12

Angles on a straight line are \( 120^{\circ} \) and \( x \). Find \( x \).

Basic
\[ 180^{\circ}-120^{\circ}=60^{\circ}. \]
13

Two angles of a triangle are \( 40^{\circ} \) and \( 60^{\circ} \). Find the third.

Basic
\[ 180^{\circ}-40^{\circ}-60^{\circ}=80^{\circ}. \]
14

An isosceles triangle has an apex angle of \( 40^{\circ} \). Find each base angle.

Intermediate

Base angles are equal: \( \tfrac{180^{\circ}-40^{\circ}}{2} \):

\[ 70^{\circ}. \]
15

State the measure of each angle in an equilateral triangle.

Intermediate
\[ \frac{180^{\circ}}{3}=60^{\circ}. \]
16

An exterior angle of a triangle has remote interior angles \( 50^{\circ} \) and \( 60^{\circ} \). Find it.

Intermediate

Exterior angle = sum of remote interior angles:

\[ 50^{\circ}+60^{\circ}=110^{\circ}. \]
17

A right triangle has one acute angle of \( 35^{\circ} \). Find the other acute angle.

Intermediate
\[ 90^{\circ}-35^{\circ}=55^{\circ}. \]
18

A triangle has angles \( x \), \( x \) and \( 80^{\circ} \). Find \( x \).

Intermediate

\( 2x+80=180 \):

\[ x=50. \]
19

The angles of a triangle are in the ratio \( 1:2:3 \). Find the largest.

Advanced

Six parts \( =180^{\circ} \), one part \( =30^{\circ} \); largest is 3 parts:

\[ 90^{\circ}. \]
20

State the sum of the exterior angles of any triangle.

Intermediate
\[ 360^{\circ}. \]
21

Two similar triangles have scale factor 2. A side of the small one is 5. Find the matching side of the large one.

Basic
\[ 5\times 2=10. \]
22

Similar triangles satisfy \( \dfrac{3}{6}=\dfrac{x}{8} \). Find \( x \).

Intermediate
\[ x=\frac{3\times 8}{6}=4. \]
23

Two similar figures have side ratio \( 1:3 \). Find the ratio of their areas.

Intermediate
\[ 1^{2}:3^{2}=1:9. \]
24

A \( 6 \)-ft person casts a \( 4 \)-ft shadow; a tree casts a \( 20 \)-ft shadow. Find the tree's height.

Intermediate
\[ \frac{6}{4}=\frac{h}{20}\Rightarrow h=30\ \text{ft}. \]
25

In similar triangles, sides 4 and 10 correspond. A second side of the small triangle is 6. Find its match.

Intermediate

Scale factor \( \tfrac{10}{4}=2.5 \):

\[ 6\times 2.5=15. \]
26

Two similar solids have scale factor 3. By what factor does the area scale?

Advanced
\[ 3^{2}=9. \]
27

A right triangle has legs 3 and 4. Find the hypotenuse.

Basic
\[ \sqrt{3^{2}+4^{2}}=5. \]
28

A right triangle has legs 5 and 12. Find the hypotenuse.

Intermediate
\[ \sqrt{5^{2}+12^{2}}=\sqrt{169}=13. \]
29

In a \( 45\text{–}45\text{–}90 \) triangle each leg is 1. Find the hypotenuse.

Intermediate
\[ \sqrt{1^{2}+1^{2}}=\sqrt2. \]
30

In a \( 30\text{–}60\text{–}90 \) triangle the short leg is 1. Find the hypotenuse.

Intermediate
\[ 2. \]
31

A right triangle has hypotenuse 10 and one leg 6. Find the other leg.

Intermediate
\[ \sqrt{10^{2}-6^{2}}=\sqrt{64}=8. \]
32

Evaluate \( \sin 30^{\circ} \).

Intermediate
\[ \tfrac12. \]
33

Evaluate \( \cos 60^{\circ} \).

Advanced
\[ \tfrac12. \]
34

Find the sum of the interior angles of a quadrilateral.

Basic
\[ (4-2)\times 180^{\circ}=360^{\circ}. \]
35

Find each interior angle of a regular hexagon.

Intermediate
\[ \frac{(6-2)\times 180^{\circ}}{6}=120^{\circ}. \]
36

Find the sum of the interior angles of an octagon.

Intermediate
\[ (8-2)\times 180^{\circ}=1080^{\circ}. \]
37

In a parallelogram one angle is \( 70^{\circ} \). Find its opposite angle.

Intermediate

Opposite angles of a parallelogram are equal:

\[ 70^{\circ}. \]
38

A rectangle has one diagonal of length 10. Find the length of the other diagonal.

Intermediate

A rectangle's diagonals are equal:

\[ 10. \]
39

Find the exterior angle of a regular pentagon.

Intermediate
\[ \frac{360^{\circ}}{5}=72^{\circ}. \]
40

A polygon has interior angles summing to \( 1440^{\circ} \). How many sides has it?

Advanced

\( (n-2)180=1440\Rightarrow n-2=8 \):

\[ n=10. \]
41

Find the circumference of a circle of radius 7. Take \( \pi=\tfrac{22}{7} \).

Basic
\[ 2\pi r=2\times\tfrac{22}{7}\times 7=44. \]
42

Find the area of a circle of radius 5. Take \( \pi=3.14 \).

Intermediate
\[ \pi r^{2}=3.14\times 25=78.5. \]
43

A triangle is inscribed in a circle with one side a diameter. Find the angle opposite that side.

Intermediate

Angle in a semicircle:

\[ 90^{\circ}. \]
44

A central angle is \( 100^{\circ} \). Find the inscribed angle on the same arc.

Intermediate
\[ \tfrac12\times 100^{\circ}=50^{\circ}. \]
45

Find the arc length of a quarter circle of radius 8. Take \( \pi=3.14 \).

Intermediate
\[ \tfrac14\times 2\pi r=\tfrac14\times 2\times 3.14\times 8=12.56. \]
46

Find the area of a \( 90^{\circ} \) sector of a circle with \( r=4 \). Take \( \pi=3.14 \).

Advanced
\[ \tfrac14\,\pi r^{2}=\tfrac14\times 3.14\times 16=12.56. \]
47

Find the area of a rectangle measuring \( 8\times 5 \).

Basic
\[ 8\times 5=40. \]
48

Find the volume of a cube of side 4.

Intermediate
\[ 4^{3}=64. \]
49

Find the volume of a cylinder with \( r=2 \), \( h=5 \). Take \( \pi=3.14 \).

Intermediate
\[ \pi r^{2}h=3.14\times 4\times 5=62.8. \]
50

A triangle has base 12 and height 5. Find its area.

Advanced
\[ \tfrac12\times 12\times 5=30. \]
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