unit six geometry topics covered - algonquin &...

UNIT SIX

GEOMETRY

Topics Covered: Angle properties Triangle Properties

Quadrilateral Properties Polygons

Triangle Geometry Quadrilateral Geometry

Grade 9 Academic Workbook Fletcher’s Meadow SS

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Angle Properties Find the measure of the angle.

1.

XD

ZF

m ZRQ = _______

m PQD = 73

2.

UG

LR

m VTG = ______

m UTV = 72

3.

SB

JK

m SPQ = _______

m PQJ = 109

4.

YD

HQ

m GNQ = ______

m HNG = 78

5.

NG

UR

m HTU = _______

m UTM = 124

6.

AB

HM

m HZJ = _______

m LJB = 57

Lines and Angles Complete.

1.

Name a pair of adjacent angles.

2.

Name a pair of vertically opposite angles.

3.

Name a pair of complementary angles.

Complete.

4.

Name a pair of supplementary angles.

5.

Name a pair of adjacent angles.

6.

Name a pair of vertically opposite angles.


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Find the measure of the angle.

7.

m RNF = _________

m FNH = 38

m RNH = 90o

8.

m JLS = _________

m SLG = 54

m JLG = 90o

9.

m DPC = _________

m TPY = 109


10.

m VXC = _________

m VXF = 47

11.

m LDQ = _________

m XDP = 52

12.

m ZCP = _________

m RCF = 30

Fill in the blank with an angle.

13.

QD

ES

DPK and _____ are alternate interior angles

14.

QM

ZL

PXM and _____ are vertically opposite angles

15.

NA

GX

AED and ______ are alternate interior angles


16.

KP

CV

m PLE = _________

m CEL = 118

17.

BD

PF

m DAM = _________

m AMF = 92

18.

PY

JT

m TED = _________

m YLE = 73


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19.

m RAV = _________

m RAK = 48

20.

m XFE = _________

m EFH = 30

m XFH = 90o

21.

m VQD = _________

m PQD = 142

Triangle Properties

The _________ of the interior angels of any triangle is always ______________.

Exterior Angle of a Triangle

The _________ of the exterior angles of any triangle is always ______________.

Triangle Geometry

1. Two angles of a triangle are given. Find the third angle.

a) °° 30,90 b) °° 60,40 c) °° 85,55 d) °° 40,70 e) °° 25,130 f) °° 60,60

2. Which of the triangles in #2 are: acute? right? obtuse? 3. ABC is a right triangle with °=∠ 90A . Find C∠ if

a) °=∠ 60B b) °=∠ 20B c) °=∠ 45B

4. Calculate the measure of each unknown angle. a) b)

A a

b B

c C

p

150° 95° 50°

105°

w


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Interior and Exterior Angles of a Triangle

An_________________________ of a triangle is equal to the _________ of the two opposite _____________ angles.

1. Calculate the measure of each unknown angle.

a) b)

2. Two exterior angles of a triangle are given. Find the third exterior angle.

a) °°110,80 b) °° 140,65 c) °°130,80 d) °° 120,120

3. Find the measure of the angle labeled x .

4. Describe (with justification) a triangle that has: a) three equal exterior angles b) two equal exterior angles c) no equal exterior angles 5. Find all the angles in each figure below.

d

e

f

d + e = f

110°

49°

z d

91°

45°


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6. Find the measure of each unknown angle.

7. Find all of the interior angles in each triangle below.

8. Find the angles in a triangle if the measure of the angles are a) x, 2x, and 3x b) x-10, x, and x + 10 9. Find the measure of each unknown angle in the triangle below. 10. In ., ACABABC =∆

a) If you knew the measure of B∠ , explain how you could find the measure of A∠ . b) If you know the measure of A∠ , explain how you could find the measure of B∠ .


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Quadrilateral Properties The ________of the interior angles of any quadrilateral is_______ . The ________of the exterior angles of any quadrilateral is_______ .

1. Find the measures of the indicated angles.

2. The measures of three of the interior angles of a quadrilateral are given. Find the measure of the

fourth interior angle. a) 42o, 91o, and 115o b) 103o, 92o, and 78o

3. The measures of three of the interior angles of a quadrilateral are given. Find the measure of the fourth interior angle.

a) 42o, 91o, and 115o b) 103o, 92o, and 78o

4. One angle of a parallelogram is four times the measure of the exterior angle adjacent to it. Find the measure of all interior angles, then draw the diagram.

5. Find the measure of each unknown angle.

C

D

A B b

c

d

a


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6. Find the measure of each unknown angle. 7. Draw an example of a quadrilateral with each set of interior angles, or explain why the quadrilateral is

not possible. a) four acute angles b) exactly two acute angles c) one right angle and two obtuse angles 8. Find all the angles in each figure below. a) b)

x

x

x - 30

x - 40

x

x + 2

x + 4 x + 6


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Angle Relationships in Polygons

The sum of the interior angles of any concave polygon is given by the formula ________________, where _____ is the number of __________ of the polygon. The sum of the exterior angles of any polygon is always _____________.

1. Find the sum of the interior angles of a polygon with

a) 12 sides b) 38 sides 2. Find the measure of each interior angle of a regular polygon with a) 9 sides b) 24 sides 3. How many sides does a polygon have if the sum of its interior angles is a) 1440o? b) 2340o? 4. How many sides does a polygon have if each of its interior angles measures a) 144o? b) 135o? 5. Complete the table below.

Polygon Number of

sides

Number of

Diagonals from

one vertex

Number of

Triangles in

polygon

Sum of

Interior

Angles

Pentagon 5 2 3 540o

Hexagon

Octagon

Dodecagon 12

6. Ron, who is a furniture maker, is designing an octagonal wooden container.

a) At what angle will the adjacent sides of the wooden container meet if its shape is a regular octagon?

b) Do you think the angles between the adjacent sides of the wooden container will all be equal if one pair of opposite sides are twice as long as the other sides? Draw a diagram to justify your answer.

7. Describe (if possible) a regular polygon whose sum of interior angles is 2100o. Explain your thinking.


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m∠ JGK = 150°IL = 1.2 cm

GK = 2.3 cm

m∠ GHI = 75°

LK

G

H

I

J

Midpoints in Triangles

In any ABC∆

Where D is the midpoint of AB E is the midpoint of AC Then two properties follow 1 - _______________________________________ 2- _______________________________________

1. Find the length of XY in each of the triangles below.

2. What is the length of the cross-brace (labeled PQ) in this bridge support?

Provide mathematical justification to support your answer. 3. Points D and E are the midpoints of sides AB and AC in ∆ABC. (a) Find the measure of ∠ADE

(b) Provide mathematical justification to support your answer. 4. Points D, E and F are midpoint of sides AB, BC and AC. Find the measures of: (a) ∠ABC (b) ∠ACB (c) ∠ BAC 5. K is a midpoint of GH and L is a midpoint of IH. GK = 2.3cm and IL = 1.2cm.

a) Find the length of HK and HI. b) Find the measure of ∠HKL and ∠HLK.

A

B C

D E

m∠ ABC = 93.00°

E

D

C

A

B

m∠ DEF = 80°

m∠ EDF = 24°F

E

D

A

B

C

Grade 9 Academic Workbook Fletchers Meadow SS

Properties of Quadrilaterals Investigation Plot each set of ordered pairs. Join points once plotted. (Everyone must graph all six shapes)

a) A(-4, -4) B(-4, 4) C(4, 4) D(4,-4) b) E(-6, 8) F(6, 8) G(6, 2) H(-6, 2) c) I(-1, 3) J(5, 1) K( 4, -3) L(-3, -5)

d) M(-2, 2) N(3, 2) O(0, -2) P(-5, -2) e) Q(-3, 3) R(5, 3) S(3, -3) T(-5, -3) f) U(0, 4) V(4, 2) W(0, -4) X(-4, 2)

You are each responsible for answering the following questions about your assigned shape on lined paper. Show

CALCULATIONS where necessary. Hint: find the slopes and lengths of all lines and compare them to each

other.

1. What is the shape formed by joining the ordered pairs? How do you know? Prove it!

2. Make a point at the midpoint of each side of your shape. How do you know where to draw the midpoints?

3. What shape is formed by joining the midpoints? How do you know? Prove it!

4. Draw the 2 diagonals of each shape. Are they related to each other in any way? If so, how?

****** I have been assigned shape _____. *******


Summary of Quadrilaterals **the inner figure is always a parallelogram but some are even more specific…

Shape Sides and Angles

Properties of Diagonals The Midpoints

Form a … Same Length Perpendicular Bisect Bisect Angles

Square

Regular Quad All angles 900 Sides congruent Opp sides parallel

Rectangle

All angles 900 Opp sides congruent Opp sides parallel

Rhombus

Sides congruent Opp angles congruent Opp sides parallel

Parallelogram Opp sides congruent Opp sides parallel

Kite

Two pairs of adjacent equal sides One pair of angles equal

Isosceles Trapezoid

Non parallel sides equal length Two pairs of congruent angles

Trapezoid One pair of opp sides parallel


A

C

B

D

A

C

B

D

EF

HG

EF

HG

Unit 6 Unit Review

1. Find the measure of x and y. a) AB|| CD y 105º x

b) AB|| CD y x

130º

c) 45º x

30º y

d) EF|| GH 2x + 30

4x - 50

e) 2x + 30

2y

3x

8x - 30

2. In triangle ABC, < A = 90°, AB = 40cm and AC = 30cm. D is the midpoint of AB and E is the midpoint of AC. Find the length of DE. (Hint: Draw a diagram first!) 3. Rectangle ABCD has a length of 8m and a width of 6m. Find the distance between EF.

H

G

F

E

A

B C

D

4. Find the value of x. 3x - 10

80°

2x + 50

x + 19 2x + 5

5. The exterior angle of a 7 sided figure is 2x + 6. What is the value of x?

Grade 9 Academic Workbook

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unit six geometry topics covered - algonquin &...

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