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8 Geometry
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10&10a

8.02 Further triangle proofs

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Introduction

One of the main advantages of being able to prove things is that we can then use what we have proved to prove something else. By starting from some basic information, we can often follow a chain of proofs to discover complex and interesting properties that we would never have seen before.

Specifically for triangles, if we know how to  prove triangles congruent or similar  then we can use the properties of congruent or similar triangles to discover other properties in a diagram.

Use congruent and similar triangles

After proving that two triangles are similar or congruent, we gain access to all the properties of similar or congruent triangles.

If we can prove that two triangles are congruent using any of the congruence tests, we have also proved that any angle or side in one triangle must be equal to the corresponding angle or side in the other.

The same applies if we can prove two triangles are similar, except instead of equal sides we get sides in a common ratio.

In particular, knowing which angles are equal can help us find relationships between the lines that the angles lie between, since there are many line properties relating to equal angles.

Examples

Example 1

Consider the diagram.

Two right angled triangles B A E and C A D. A B equals 2, B C equals 3, B E equals 1. Ask your teacher for more information.
a

Why is BE parallel to CD?

Worked Solution
Create a strategy

Identify the relationship of the two given angles.

Apply the idea

From the diagram, we are given that \angle ABE=\angle ACD.

\angle ABE and \angle ACD form a pair of equal corresponding angles on BE and CD, so BE || CD.

b

Which angle is equal to \angle BEA?

Worked Solution
Create a strategy

Find the corresponding angle on the parallel lines.

Apply the idea

\angle CDA corresponds to \angle BEA on the parallel lines BE || CD.

So \angle BEA = \angle CDA.

c

How do we know that \triangle ABE ||| \triangle ACD?

Worked Solution
Create a strategy

Show that the triangles have equal angles.

Apply the idea

  • \angle A is a common. (Given)

  • \angle ABE=\angle ACD (Given)

  • \angle BEA = \angle CDA (Corresponding angles, BE || CD)

\triangle ABE ||| \triangle ACD because all three pairs of corresponding angles are equal.

d

What is the scale factor relating \triangle ABE to \triangle ACD ?

Worked Solution
Create a strategy

Divide the longer side length by the corresponding shorter side length.

Apply the idea
\displaystyle \text{Scale factor}\displaystyle =\displaystyle \dfrac{AC}{AB}
\displaystyle =\displaystyle \dfrac{2+3}{2}Substitute the lengths
\displaystyle =\displaystyle 2.5Evaluate
e

Solve for the value of f.

Worked Solution
Create a strategy

Multiply the length of the corresponding side in the smaller triangle by the scale factor.

Apply the idea

The corresponding side to CD is BE.

\displaystyle f\displaystyle =\displaystyle \text{Scale factor} \times BE
\displaystyle =\displaystyle 2.5 \times 1Substitute values
\displaystyle =\displaystyle 2.5Evaluate

Example 2

Consider the diagram below:

Triangle C D E and triangle C A B. D E is parallel to A B. C D is equal to A D.

Prove that CE=EB.

Worked Solution
Create a strategy

Prove the triangles are similar and then use corresponding sides in similar triangles.

Apply the idea
To prove: CE=EB
StatementsReasons
1.\angle C is common(Given)
2.\angle CDE = \angle CAB (Corresponding angles, DE \parallel AB)
3. \angle CED = \angle CBA(Similarly)
4.\triangle CDE ||| \triangle CAB (Equiangular, AAA)
5.CD=DA(Given)
6.\dfrac{CD}{CA}=\dfrac{CE}{CB}(Corresponding sides in similar triangles)
7.\dfrac{1}{2}=\dfrac{CE}{CB} (CA=2CD)
8.CB=2CE \text{ }
9.CE+EB=CB(Diagram)
10.CE+EB=2CE (CB=2CE)
11.CE=EB \text{ }
Idea summary

After proving that two triangles are similar or congruent, we can prove other properties of similar or congruent triangles. Such as corresponding sides or angles are equal, or corresponding sides are in the same ratio.

Outcomes

ACMMG243

Formulate proofs involving congruent triangles and angle properties

ACMMG244

Apply logical reasoning, including the use of congruence and similarity, to proofs and numerical exercises involving plane shapes

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