1. congruent and similar triangles, proofs, metric...

1. Congruent and Similar Triangles, Proofs, Metric Relations and Minimising a Distance (MASMTS408).notebook

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April 04, 2018

Triangle Geometry1) Naming angles

Often we can use one letter (capitalised) to name an angle.

When more than two lines meet at a vertex, then we must use three letters to name an angle.

A

B

C

PQ

RS

TW

X


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Complementary and Supplementary Angles

Complementary Angles add up to 90°.

Supplementary Angles add up to 180°.

A

B C

D

M NP

Q


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C

D

E

F

H Vertically Opposite angles are congruent.

Angle Pairs


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Parallel Lines and a Transversal

Transversal

When a transversal cuts across two parallel lines, several pairs of congruent angles are created


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Corresponding Angles are congruent.

Alternate Interior Angles are congruent.


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Alternate Exterior Angles are congruent.


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Example: Find the value of x.


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Example: Find the value of x.

M N

R S

Statement Justification

F

G

V

E

Q

T

P


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If two lines are parallel and they are intersected by a transversal, then the corresponding (or alt. interior or alt. exterior) angles are congruent....

Then...

We can also say that if the corresponding (or alt.interior or alt. exterior) angles created by a transversal intersecting two lines are congruent, then the lines must be parallel .

50°

50°

A B

C D

E

F

P

Q

Since the two alternate exterior angles are congruent, the two lines AB and CD must be parallel.


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1

2

34

56

910

11

8

7

12

A

B

Determine the measures of each angle. Justify your answers.

Measure Justification Measure Justification


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Proofs

Axiom: A statement that is not proven but is considered to be obvious, so it is accepted as true.

Conjecture: A statement that has yet to be proven.

Theorem: A statement that has been proven.

A proof involves taking given information that is accepted to be true (called a hypothesis) and, using knowledge and previously proved theorems, working logically to a conclusion.

This process is known as deductive reasoning.

All it takes to disprove a conjecture is one counter-example.


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Example: Given triangle ABC,where .

Prove that .

BC

A1

23

4


Hypothesis

Conclusion


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Congruent TrianglesTwo triangles are congruent when all three corresponding sides and all three corresponding angles have the same measurements.

A

B C

D E

F

To prove that two triangles are congruent, it is not necessary to show all six conditions.

(Note: Isometric and congruent mean the same thing.)


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There are minimum conditions for showing that triangles are congruent. These are known as Theorems of Congruence.

P

QR

V

U

T

1) If the three sides of one triangle are congruent to the three corresponding sides of another triangle, then the triangles are congruent. This is called the Side Side Side theorem (SSS).


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Prove that triangle MNP is congruent to triangle XYZ.

M

N

P

X

Y Z

= _

=_



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2) If two sides and the contained angle of one triangle are congruent to two corresponding sides and contained angle of another triangle, then the triangles are congruent. This is called the Side Angle Side theorem (SAS).

=

=


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Example: Prove that triangle HBC is congruent to triangle TSN.

H

B

N

S

T

C


_ =

_=


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3) If two angles and the contained side of one triangle are congruent to two corresponding angles and contained side of another triangle, then the triangles are congruent. This is called the Angle Side Angle theorem (ASA).

_

_


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1

2

3

4

5

6

7

Example: Prove that .

_

_

A

BT

SR

C


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Property of Congruent Triangles (PCT)

When 2 triangles are proven to be congruent, theircorresponding elements(sides and angles) are congruent.


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Prove that .


F

G

H

RS

T

_

_

=

=


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Similar Triangles

Two figures are similar when ...• All corresponding angles are congruent.• All corresponding sides are proportional.Therefore, similar figures have the same shape, but are not necessarily the same size.

_

_


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There are also minimum conditions to prove that two triangles are similar.

1) Side Side Side (SSS) If the corresponding sides of two triangles are proportional in length, then the triangles are similar.

2) Angle Angle (AA) If two corresponding angles of two triangles are congruent, then the triangles are similar.

3) Side Angle Side (SAS) If two triangles have one congruent angle contained between corresponding sides of proportional length, then the triangles are similar.

_

_


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Example: Prove that is similar to .



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Similar Triangles

Knowing that triangles are similar allows us to solve some geometric problems.

Example: Given that the triangles below are similar, solve triangle DEF. (To solve a triangle is to find all its measures.)

A

C B

D

F E


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Example:Determine the value of x.


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Other Theorems

1) Parallel Line to a Triangleʹs SideAny line parallel to one of a triangleʹs sides creates similar triangles.

A

B

DE

C

Work Book: Page 212, Questions 1 & 2


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1 a)

b) 1. 2.

2. a) b) c) d)


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2) Segment Joining the Mid‑Points of Two Sides of a Triangle

Any line segment that joins the midpoints of two sides in a triangle is parallel to the third side and is half the length of this third side.

=

_

_

=

Work Book: Page 212, Question 3


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3) Thalesʹ TheoremWhen two transversal lines are intersected by parallel lines, they are separated into segments of proportional lengths.

A

C

B

E

F

G

HDO

...


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Example: Determine the missing values.

Work Book: Page 213, Question 4


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Example: Prove that is similar to if AD is the altitude of the right triangle ABC.

A

BC D



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Metric Relations in Right Triangles

A

C

B

A special right triangle where the altitude is included...

creates three similar right triangles.


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A

C

Bc

ab

n m

h

n h

b

a

h m


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Proportionsmeans

extremes

When we refer to a "proportional mean", the means in the proportion are equal.


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The altitude to the hypotenuse of a right triangle is the proportional mean between the segments into which it divides the hypotenuse.

The altitude (h)divides the hypotenuse into 2 parts: m and n.So...

Altitude to the Hypotenuse Theorem

part 1 of hypotenuse = part 2 of hypotenusealtitudealtitude

or

(part 1 of hypotenuse) (part 2 of hypotenuse)or

in symbols:


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Example:Determine the value of x.

A

C

D B


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Each leg of a right triangle is the proportional mean between the hypotenuse and the projection of the leg on the hypotenuse.

Proportional Mean Theorem

legleg

projection of aprojection of b

or

or


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D BA

C

Examples: Determine the value of x.


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Determine the value of x.


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The product of the sides of the right angle is equal to the product of the hypotenuse and the altitude .

Product of the Sides Theorem


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1712

Example: Determine the value of x.

A

C

D B

x

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