Maths in Focus - Margaret Grove - ch4

4

Geometry 1 TERMINOLOGY Altitude: Height. Any line segment from a vertex to the opposite side of a polygon that is perpendicular to that side

Polygon: General term for a many sided plane figure. A closed plane (two dimensional) figure with straight sides

Congruent triangles: Identical triangles that are the same shape and size. Corresponding sides and angles are equal. The symbol is /

Quadrilateral: A four-sided closed figure such as a square, rectangle, trapezium etc.

Interval: Part of a line including the endpoints

Similar triangles: Triangles that are the same shape but different sizes. The symbol is zy

Median: A line segment that joins a vertex to the opposite side of a triangle that bisects that side

Vertex: The point where three planes meet. The corner of a figure

Perpendicular: A line that is at right angles to another line. The symbol is =

Vertically opposite angles: Angles that are formed opposite each other when two lines intersect

Chapter 4 Geometry 1

INTRODUCTION GEOMETRY IS USED IN many areas, including surveying, building and graphics.

These fields all require a knowledge of angles, parallel lines and so on, and how to measure them. In this chapter, you will study angles, parallel lines, triangles, types of quadrilaterals and general polygons. Many exercises in this chapter on geometry need you to prove something or give reasons for your answers. The solutions to geometry proofs only give one method, but other methods are also acceptable.

DID YOU KNOW? Geometry means measurement of the earth and comes from Greek. Geometry was used in ancient civilisations such as Babylonia. However, it was the Greeks who formalised the study of geometry, in the period between 500 BC and AD 300.

Notation In order to show reasons for exercises, you must know how to name figures correctly. •B The point is called B.

The interval (part of a line) is called AB or BA.

If AB and CD are parallel lines, we write AB < CD.

This angle is named +BAC or +CAB. It can sometimes be named +A. ^

Angles can also be written as BAC or

This triangle is named 3ABC.

BAC

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To name a quadrilateral, go around it: for example, BCDA is correct, but ACBD is not.

Producing a line is the same as extending it.

This quadrilateral is called ABCD.

Line AB is produced to C.

+ABD and +DBC are equal.

DB bisects +ABC.

AM is a median of D ABC.

AP is an altitude of D ABC.

Types of Angles Acute angle

0c1 xc1 90c

Chapter 4 Geometry 1

Right angle

A right angle is 90c. Complementary angles are angles whose sum is 90c.

Obtuse angle

90c1 xc1180c

Straight angle

A straight angle is 180c. Supplementary angles are angles whose sum is 180c.

Reflex angle

180c1 xc1 360c

Angle of revolution

An angle of revolution is 360c.

Vertically opposite angles

+AEC and +DEB are called vertically opposite angles. +AED and +CEB are also vertically opposite angles.

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Vertically opposite angles are equal. That is, +AEC = +DEB and +AED = +CEB.

Proof Let +AEC = xc Then +AED = 180c - xc (+CED straight angle, 180c) Now +DEB = 180c - (180c - xc) (+AEB straight angle, 180c) = xc Also +CEB = 180c - xc (+CED straight angle, 180c) ` +AEC = +DEB and +AED =+CEB

EXAMPLES Find the values of all pronumerals, giving reasons. 1.

Solution x + 154 = 180 (+ABC is a straight angle, 180c) x + 154 - 154 = 180 - 154 ` x = 26 2.

Solution 2x + 142 + 90 2x + 232 2x + 232 - 232 2x 2x 2 x

= 360 (angle of revolution, 360c ) = 360 = 360 - 232 = 128 128 = 2 = 64

Chapter 4 Geometry 1

3.

Solution y + 2y + 30 = 90 3y + 30 3y + 30 - 30 3y 3y 3 y

(right angle, 90c)

= 90 = 90 - 30 = 60 60 3 = 20 =

4.

Solution x + 50 = 165 x + 50 - 50 = 165 - 50 x = 115 y = 180 - 165 = 15 w = 15

(+WZX and +YZV vertically opposite)

(+XZY straight angle, 180c) (+WZY and +XZV vertically opposite)

5.

CONTINUED

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Solution a = 90 b + 53 + 90 = 180 b + 143 = 180 b + 143 - 143 = 180 - 143 b = 37 d = 37 c = 53

(vertically opposite angles) (straight angle, 180c)

(vertically opposite angles) (similarly)

6. Find the supplement of 57c 12l.

Solution Supplementary angles add up to 180c. So the supplement of 57c 12l is 180c - 57c 12l = 122c 48l. 7. Prove that AB and CD are straight lines. A

D

(x + 30)c C

(6x + 10)c

(2x 2 + 10)c E (5x + 30)c B

Solution 6x + 10 + x + 30 + 5x + 30 + 2x + 10 = 360 ^ angle of revolution h 14x + 80 - 80 = 360 - 80 14x = 280 14x 280 = 14 14 x = 20 +AEC = (20 + 30)c = 50c +DEB = (2 # 20 + 10)c = 50c These are equal vertically opposite angles. ` AB and CD are straight lines

Chapter 4 Geometry 1

4.1 Exercises 1.

Find values of all pronumerals, giving reasons. (a)

yc

(i)

133c

(b)

(j)

(c) 2.

Find the supplement of (a) 59c (b) 107c 31l (c) 45c 12l

3.

Find the complement of (a) 48c (b) 34c 23l (c) 16c 57l

4.

Find the (i) complement and (ii) supplement of (a) 43c (b) 81c (c) 27c (d) 55c (e) 38c (f) 74c 53l (g) 42c 24l (h) 17c 39l (i) 63c 49l (j) 51c 9l

5.

(a) Evaluate x. (b) Find the complement of x. (c) Find the supplement of x.

(d)

(e)

(f)

(g)

(h) (2x + 30)c 142c

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6.

Find the values of all pronumerals, giving reasons for each step of your working.

8.

(a) Prove that CD bisects +AFE. Prove that AC is a straight line.

9.

D C

(b) (3x + 70)c (110 - 3x)c B

(c) A

10. Show that +AED is a right angle. A

(d)

B

(50 - 8y)c

(e)

C

(5y - 20)c

E

(f)

7.

Prove that AC and DE are straight lines.

(3y + 60)c

D

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Parallel Lines When a transversal cuts two lines, it forms pairs of angles. When the two lines are parallel, these pairs of angles have special properties.

Alternate angles

Alternate angles form a Z shape. Can you find another set of alternate angles?

If the lines are parallel, then alternate angles are equal.

Corresponding angles

Corresponding angles form an F shape. There are 4 pairs of corresponding angles. Can you find them?

If the lines are parallel, then corresponding angles are equal.

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Cointerior angles Cointerior angles form a U shape. Can you find another pair?

If the lines are parallel, cointerior angles are supplementary (i.e. their sum is 180c).

Tests for parallel lines

If alternate angles are equal, then the lines are parallel.

If +AEF = +EFD, then AB < CD.

If corresponding angles are equal, then the lines are parallel.

If +BEF = +DFG, then AB < CD.

If cointerior angles are supplementary, then the lines are parallel.

If +BEF + +DFE = 180c, then AB < CD.

Chapter 4 Geometry 1

If 2 lines are both parallel to a third line, then the 3 lines are parallel to each other. That is, if AB < CD and EF < CD, then AB < EF.

EXAMPLES 1. Find the value of y, giving reasons for each step of your working.

Solution +AGF = 180c - 125c = 55c

(+FGH is a straight angle)

`

(+AGF, +CFE corresponding angles, AB < CD)

y = 55c

2. Prove EF < GH.

Solution +CBF = 180c - 120c (+ABC is a straight angle) = 60c ` +CBF = +HCD = 60c But +CBF and +HCD are corresponding angles ` EF < GH

Can you prove this in a different way?

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Think about the reasons for each step of your calculations.

4.2 Exercises 1.

Find values of all pronumerals. (a)

(h)

(i)

(b) (j)

(c)

2.

Prove AB < CD. (a)

(d)

(b) (e)

(c)

A

(f)

(g)

B

104c

C 76c

D

E

Chapter 4 Geometry 1

A

(d)

(e)

B 138c

B

52c

E C

C

E 128c

D

23c

F 115c

G

H

F

Types of Triangles Names of triangles A scalene triangle has no two sides or angles equal.

A right (or right-angled) triangle contains a right angle.

The side opposite the right angle (the longest side) is called the hypotenuse. An isosceles triangle has two equal sides. The angles (called the base angles) opposite the equal sides in an isosceles triangle are equal.

An equilateral triangle has three equal sides and angles.

A D

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All the angles are acute in an acute-angled triangle.

An obtuse-angled triangle contains an obtuse angle.

Angle sum of a triangle

The sum of the interior angles in any triangle is 180c, that is, a + b + c = 180

Proof

Let +YXZ = ac, +XYZ = bc and +YZX = cc Draw line AB < YZ Then +BXZ = cc (+BXZ, +XZY alternate angles, AB < YZ) +AXY = bc (similarly) +YXZ + +AXY + +BXZ = 180c (+AXB is a straight angle) ` a + b + c = 180

Chapter 4 Geometry 1

Class Investigation 1. 2. 3. 4. 5.

Could you prove the base angles in an isosceles triangle are equal? Can there be more than one obtuse angle in a triangle? Could you prove that each angle in an equilateral triangle is 60c? Can a right-angled triangle be an obtuse-angled triangle? Can you find an isosceles triangle with a right angle in it?

Exterior angle of a triangle

The exterior angle in any triangle is equal to the sum of the two opposite interior angles. That is, x+y=z

Proof

Let +ABC = xc , +BAC = yc and +ACD = zc Draw line CE < AB zc = +ACE + +ECD +ECD = xc +ACE = yc ` z=x+y

(+ECD,+ABC corresponding angles, AB < CE) (+ACE,+BAC alternate angles, AB < CE)

EXAMPLES Find the values of all pronumerals, giving reasons for each step. 1.

CONTINUED

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Solution x + 53 + 82 = 180 (angle sum of D 180c) x + 135 = 180 x + 135 - 135 = 180 - 135 x = 45 2.

Solution +A = +C = x x + x + 48 = 180 2x + 48 = 180 2x + 48 - 48 = 180 - 48 2x = 132 132 2x = 2 2 x = 66

(base angles of isosceles D) (angle sum in a D 180c)

3.

Solution y + 35 = 141 (exterior angle of D) y + 35 - 35 = 141 - 35 ` y = 106 This example can be done using the interior sum of angles. +BCA = 180c - 141c = 39c y + 39 + 35 = 180 y + 74 = 180 y + 74 - 74 = 180 - 74 ` y = 106

(+BCD is a straight angle 180c) (angle sum of D 180c)

Chapter 4 Geometry 1

Think of the reasons for each step of your calculations.

4.3 Exercises 1.

Find the values of all pronumerals. (a)

(h)

(b) (i)

(j) (c)

(d)

(k)

(e)

(f)

(g)

153

2.

Show that each angle in an equilateral triangle is 60c.

3.

Find +ACB in terms of x.

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4.

5.

6.

Prove AB < ED.

(d)

8.

Prove D IJL is equilateral and D JKL is isosceles.

9.

In triangle BCD below, BC = BD. Prove AB ED.

Show D ABC is isosceles.

Line CE bisects +BCD. Find the value of y, giving reasons.

A B C

46c E 88c

D

7.

Evaluate all pronumerals, giving reasons for your working. (a)

10. Prove that MN QP . 32c

M

(b) 75c

O

73c

Q

(c)

P

N

Chapter 4 Geometry 1

Congruent Triangles Two triangles are congruent if they are the same shape and size. All pairs of corresponding sides and angles are equal. For example:

We write D ABC / D XYZ.

Tests To prove that two triangles are congruent, we only need to prove that certain combinations of sides or angles are equal.

Two triangles are congruent if • SSS: all three pairs of corresponding sides are equal • SAS: two pairs of corresponding sides and their included angles are equal • AAS: two pairs of angles and one pair of corresponding sides are equal • RHS: both have a right angle, their hypotenuses are equal and one other pair of corresponding sides are equal

EXAMPLES 1. Prove that DOTS / DOQP where O is the centre of the circle.

CONTINUED

The included angle is the angle between the 2 sides.

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Solution S: A: S:

OS = OQ +TOS = +QOP OT = OP

`

by SAS, DOTS / DOQP

(equal radii) (vertically opposite angles) (equal radii)

2. Which two triangles are congruent?

Solution To find corresponding sides, look at each side in relation to the angles. For example, one set of corresponding sides is AB, DF, GH and JL. D ABC / D JKL (by SAS) 3. Show that triangles ABC and DEC are congruent. Hence prove that AB = ED.

Solution A: +BAC = +CDE A: +ABC = +CED S: AC = CD

(alternate angles, AB < ED) (similarly) (given)

` by AAS, D ABC / D DEC ` AB = ED

(corresponding sides in congruent D s)

Chapter 4 Geometry 1

4.4 Exercises 1.

Are these triangles congruent? If they are, prove that they are congruent. (a)

2.

Prove that these triangles are congruent. (a)

B

(b)

Y 4.7

m

110c

2.3

4.7

m

m

Z

110 c C

A 2

.3 m

(b)

X

(c)

(c)

(d)

(d) (e)

(e)

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3.

A

Prove that (a) Δ ABD is congruent to Δ ACD (b) AB bisects BC, given D ABC is isosceles with AB = AC.

D

B

4.

Prove that triangles ABD and CDB are congruent. Hence prove that AD = BC.

C

(a) Prove that TABC and TADC are congruent. (b) Show that +ABC = +ADC. The centre of a circle is O and AC is perpendicular to OB.

7.

A

5.

In the circle below, O is the centre of the circle. A

O

D

B O

C

B

C

(a) Prove that TOAB and TOCD are congruent. (b) Show that AB = CD. 6.

(a) Show that TOAB and TOBC are congruent. (b) Prove that +ABC = 90c. ABCF is a trapezium with AF = BC and FE = CD. AE and BD are perpendicular to FC.

8.

In the kite ABCD, AB = AD and BC = DC.

F

A

B

E

D

C

(a) Show that TAFE and TBCD are congruent. (b) Prove that +AFE = +BCD.

Chapter 4 Geometry 1

9.

The circle below has centre O and OB bisects chord AC.

10. ABCD is a rectangle as shown below. A

B

D

C

C O B

A

(a) Prove that TOAB is congruent to TOBC. (b) Prove that OB is perpendicular to AC.

(a) Prove that TADC is congruent to TBCD. (b) Show that diagonals AC and BD are equal.

Investigation The triangle is used in many structures, for example trestle tables, stepladders and roofs. Find out how many different ways the triangle is used in the building industry. Visit a building site, or interview a carpenter. Write a report on what you find.

Similar Triangles Triangles, for example ABC and XYZ, are similar if they are the same shape but different sizes. As in the example, all three pairs of corresponding angles are equal. All three pairs of corresponding sides are in proportion (in the same ratio).

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We write: D ABC