Plane Geometry — Lines, Triangles, Quadrilaterals and Circles

Competitive Mathematics · Geometry

7 Part3 worked examples52 practice questions

In geometry we look at how a figure is built (which side equals or is parallel to which) and use that to find angles and lengths. A problem may look hard, but it is usually a mix of a handful of rules — the angles of a triangle add up to $18 0^{\circ}$ , angles on parallel lines, Pythagoras, similarity and the circle theorems. This chapter takes the rules one at a time and solves new questions on each.

त्रिभुज : ∠ A + ∠ B + ∠ C = 18 0^{\circ}; चतुर्भुज : 36 0^{\circ}; n - भुज : (n - 2) 18 0^{\circ}

Three angle-sum rules

One habit that helps in every question: write every angle on the figure as you find it. The next angle usually becomes visible by itself.

Lines and angles

Complementary, supplementary and parallel-line angles

The kinds of angle, angles on a line, vertically opposite angles and angles made by a transversal.

Kinds of angle and their relations

Pairs that make

9 0^{\circ}

and

18 0^{\circ}

Acute, right, obtuse, straight and reflex angles; complementary and supplementary angles; vertically opposite angles.

Name	Measure
Acute	$0^{\circ} < θ < 9 0^{\circ}$
Right	$θ = 9 0^{\circ}$
Obtuse	$9 0^{\circ} < θ < 18 0^{\circ}$
Straight	$θ = 18 0^{\circ}$
Reflex	$18 0^{\circ} < θ < 36 0^{\circ}$
Complementary pair	sum $9 0^{\circ}$
Supplementary pair	sum $18 0^{\circ}$

**Angles on a straight line add up to $18 0^{\circ}$ ;** angles all round a point add up to $36 0^{\circ}$ .
When two lines cross, the opposite (vertical) angles are equal; adjacent angles are supplementary.
An angle bisector cuts an angle into two equal parts. The bisectors of the two angles of a linear pair are always at $9 0^{\circ}$ to each other.
Ratio questions: divide the total (90, 180 or 360) by the sum of the parts, then multiply by the part you need.

Example 1 · Basic

An angle is 24° more than its complement. Find the angle and its complement.

Solution, step by step

1Let the angle be $θ$ ; its complement is $90 - θ$ . $θ - (90 - θ) = 24$ .
2 $2 θ = 114$ ⟹ $θ = 5 7^{\circ}$ ; complement $= 3 3^{\circ}$

Answer 57° and 33°

Example 2 · Standard

Three consecutive angles on a straight line are

(2 x - 10)^{\circ}

(x + 30)^{\circ}

and

(3 x + 10)^{\circ}

. Find the largest angle.

Solution, step by step

1 $6 x + 30 = 180$ ⟹ $x = 25$ .
2Angles: $4 0^{\circ}, 5 5^{\circ}, 8 5^{\circ}$ — sum $18 0^{\circ}$ ✓; the largest is $8 5^{\circ}$

Answer 85°

BasicTry it yourself first

What is the complement of 35°?

BasicTry it yourself first

What is the supplement of 65°?

StandardTry it yourself first

What is the complement of 72°?

StandardTry it yourself first

What is the supplement of 118°?

BasicTry it yourself first

An angle is

\frac{2}{3}

of its supplement. Find the angle.

StandardTry it yourself first

An angle is

\frac{1}{5}

of its supplement. Find the angle.

StandardTry it yourself first

An angle is

\frac{3}{2}

of its supplement. Find the angle.

AdvancedTry it yourself first

An angle is

\frac{4}{5}

of its supplement. Find the angle.

BasicTry it yourself first

An angle is

\frac{1}{2}

of its complement. Find the angle.

StandardTry it yourself first

An angle is

\frac{2}{3}

of its complement. Find the angle.

AdvancedTry it yourself first

An angle is

\frac{3}{2}

of its complement. Find the angle.

BasicTry it yourself first

Two complementary angles are in the ratio 4 : 5. Find the larger angle.

StandardTry it yourself first

Two complementary angles are in the ratio 2 : 7. Find the larger angle.

AdvancedTry it yourself first

Two complementary angles are in the ratio 7 : 11. Find the larger angle.

BasicTry it yourself first

Two supplementary angles are in the ratio 4 : 5. Find the smaller angle.

StandardTry it yourself first

Two supplementary angles are in the ratio 3 : 7. Find the smaller angle.

AdvancedTry it yourself first

Two supplementary angles are in the ratio 5 : 13. Find the smaller angle.

BasicTry it yourself first

Two supplementary angles differ by 20°. Find the larger angle.

StandardTry it yourself first

Two complementary angles differ by 40°. Find the larger angle.

StandardTry it yourself first

Two supplementary angles differ by 36°. Find the larger angle.

AdvancedTry it yourself first

Two complementary angles differ by 66°. Find the larger angle.

AdvancedTry it yourself first

Two supplementary angles differ by 24°. Find the larger angle.

StandardTry it yourself first

Three consecutive angles on one side of a straight line are

(x + 20)^{\circ}

(2 x)^{\circ}

and

(x - 20)^{\circ}

. Find

x

StandardTry it yourself first

Three consecutive angles on one side of a straight line are

(x + 10)^{\circ}

(3 x)^{\circ}

and

(2 x + 20)^{\circ}

. Find

x

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(x + 30)^{\circ}

(2 x + 10)^{\circ}

and

(x - 20)^{\circ}

. Find

x

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(2 x - 10)^{\circ}

(3 x + 5)^{\circ}

and

(x + 35)^{\circ}

. Find

x

AdvancedTry it yourself first

Three consecutive angles on one side of a straight line are

(5 x - 10)^{\circ}

(3 x + 20)^{\circ}

and

(2 x)^{\circ}

. Find

x

StandardTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(3 x + 10)^{\circ}

and

(5 x - 20)^{\circ}

. Find the measure of each of these angles.

StandardTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(2 x + 30)^{\circ}

and

(4 x - 10)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(5 x - 15)^{\circ}

and

(3 x + 25)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(6 x - 30)^{\circ}

and

(4 x + 10)^{\circ}

. Find the measure of each of these angles.

AdvancedTry it yourself first

Two straight lines intersect. A pair of vertically opposite angles are

(7 x + 5)^{\circ}

and

(9 x - 25)^{\circ}

. Find the measure of each of these angles.

StandardTry it yourself first

AOB is a straight line and ray OC is drawn so that

∠ A OC = 5 0^{\circ}

. OP bisects

∠ A OC

and OQ bisects

∠ COB

. Find

∠ POQ

AdvancedTry it yourself first

AOB is a straight line and ray OC is drawn so that

∠ A OC = 6 4^{\circ}

. OP bisects

∠ A OC

and OQ bisects

∠ COB

. Find

∠ POQ

Parallel lines and a transversal

One line, eight angles, only two values

Corresponding, alternate and co-interior angles, and exterior angles.

∥

CD, transversal EF

When a transversal cuts two parallel lines, all eight angles take just one of two values: some $α$ , or its supplement $18 0^{\circ} - α$ . All you need to decide is which angle gets which value.

Pair	Relation
Corresponding angles	equal
Alternate angles	equal
Vertically opposite	equal
Co-interior angles	sum $18 0^{\circ}$
Linear pair	sum $18 0^{\circ}$

In a “Z” shape alternate angles are equal; in an “F” shape corresponding angles are equal; in a “C/U” shape co-interior angles are supplementary.
Bisectors of two co-interior angles are perpendicular (their halves add up to $\frac{180}{2} = 9 0^{\circ}$ ).
For a “bent” line through a point between parallels, draw a parallel through the point and think in two parts.

Example 3 · Standard

A transversal cuts two parallel lines and the co-interior angles on one side are in the ratio 7 : 2. Find both angles.

Solution, step by step

1Sum $18 0^{\circ}$ , parts $7 + 2 = 9$ , one part $= 2 0^{\circ}$ .
2Angles: $7 \times 20 = 14 0^{\circ}$ and $2 \times 20 = 4 0^{\circ}$

Answer 140° and 40°

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 6 5^{\circ}

, find the alternate angle

∠ G H C

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 6 5^{\circ}

, find the co-interior angle

∠ G HD

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 7 2^{\circ}

, find the corresponding angle

∠ DH F

BasicTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 5 8^{\circ}

, find the vertically opposite angle

∠ A GE

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 11 0^{\circ}

, find the linear-pair angle

∠ EGB

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 12 3^{\circ}

, find the alternate angle

∠ G H C

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 13 4^{\circ}

, find the co-interior angle

∠ G HD

StandardTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 4 7^{\circ}

, find the corresponding angle

∠ DH F

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 9 8^{\circ}

, find the vertically opposite angle

∠ A GE

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 7 6^{\circ}

, find the linear-pair angle

∠ EGB

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 8 9^{\circ}

, find the co-interior angle

∠ G HD

AdvancedTry it yourself first

∥

CD and a transversal EF cuts AB at G and CD at H (E above, F below; B and D on the right). If

∠ BG H = 3 7^{\circ}

, find the alternate angle

∠ G H C

BasicTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 2 : 3. Find the larger angle.

StandardTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 4 : 5. Find the larger angle.

StandardTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 1 : 5. Find the larger angle.

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 7 : 11. Find the larger angle.

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 5 : 13. Find the larger angle.

AdvancedTry it yourself first

A transversal cuts two parallel lines. The two co-interior angles on one side are in the ratio 7 : 5. Find the larger angle.

Triangles

The most-asked part

Angles, sides, congruence, similarity, the four centres and the main theorems.

Angles and sides

Sum

18 0^{\circ}

, exterior angle, side–angle link

Kinds of triangle, the angle sum, the exterior angle, the side inequality and a use of Pythagoras.

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The main theorems

Proportion, mid-point, median and similarity

The angle bisector theorem, basic proportionality, the mid-point theorem, Apollonius and similar triangles.

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The four centres of a triangle

Centroid, incentre, circumcentre, orthocentre

Which lines make each centre, and the angle formulas attached to them.

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Quadrilaterals and polygons

Four sides and more

Properties of quadrilaterals, the cyclic quadrilateral, mid-point figures and polygon angles.

Properties of quadrilaterals

The diagonals tell you what the figure is

Parallelogram, rectangle, rhombus, square, trapezium and cyclic quadrilateral.

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Polygon angles

(n - 2) \times 18 0^{\circ}

and

36 0^{\circ}

The sum of interior and exterior angles; finding each angle and the number of sides in a regular polygon.

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The circle

Angles, chords and tangents

Central and inscribed angles, chord properties, tangents, secants and common tangents.

Angles and chords in a circle

The angle at the centre is double

The central angle, the angle in a semicircle, the alternate segment and a chord’s distance.

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Tangents and secants

From an outside point to the circle

Tangent length, the tangent–secant theorem, intersecting chords, common tangents and circumscribed figures.

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Traps and fast tricks

Slips · tricks

Where marks go in geometry questions.

The commonest mistakes

Seven traps

The slips that cost marks.

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Fast tricks

Recognise and apply

Habits for speed.

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The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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Plane Geometry — Lines, Triangles, Quadrilaterals and Circles

Lines and angles

Kinds of angle and their relations

Parallel lines and a transversal

Triangles

Angles and sides

The main theorems

The four centres of a triangle

Quadrilaterals and polygons

Properties of quadrilaterals

Polygon angles

The circle

Angles and chords in a circle

Tangents and secants

Traps and fast tricks

The commonest mistakes

Fast tricks

The practice ladder

Level 1 — Basic

Level 2 — Standard

Level 3 — Advanced

Revision — formula cards

Free All Access Passevery mind map · Hindi + English · 12 months

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