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12 Geometry
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Stage 4

12.09 Using triangle congruence

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Lesson

Introduction

Once we have  shown two triangles are congruent  using some of the side and angle information, we know the other sides and angles must match up as well.

Corresponding sides and angles in congruent triangles

In two congruent triangles, any sides or angles that match up are referred to as corresponding.

If two triangles are congruent, then:

  • The sides in the same relative position are equal, and are called corresponding sides of congruent triangles.

  • The angles in the same relative position are equal, and are called corresponding angles of congruent triangles.

Examples

Example 1

It is known that \triangle STU \equiv \triangle ABC.

Two congruent triangles S T U and A B C. Angles T and B are equal. Angles U and C are right angles.

Which two of the following equalities do we know to be true?

A
TU=BC
B
US=AB
C
TU=CA
D
ST=AB
Worked Solution
Create a strategy

When two triangles are congruent, the corresponding sides will have the same position with respect to the matching angles.

Apply the idea
\displaystyle US\displaystyle =\displaystyle CASides opposite right angles
\displaystyle ST\displaystyle =\displaystyle ABSides opposite equal angles
\displaystyle TU\displaystyle =\displaystyle BCRemaining two sides of congruent triangles

So options A and D are the correct answers.

Idea summary

If two triangles are congruent, then:

  • The sides in the same relative position are equal, and are called corresponding sides of congruent triangles.

  • The angles in the same relative position are equal, and are called corresponding angles of congruent triangles.

Match information across congruent triangles

If two corresponding sides or angles must be equal in congruent triangles then knowing the value of one gives us the value of the other.

Examples

Example 2

Consider the two triangles below:

Two triangles E F G and H J K. Angles E, F, H, and J are 67 degrees and J K has length m.
a

Together with the given information, which other condition would make sure that these two triangles are congruent?

A
\angle FGE = \angle JKH
B
EG=HJ
C
EF=HJ
D
EF=HK
Worked Solution
Create a strategy

Take note of the given corresponding parts and determine which option would give us enough information to satisfy any of the congruence tests.

Apply the idea

We are given two pairs of equal angles which are all the same size.

So we need one pair of equal sides to allow us to prove congruence using AAS.

We need the equal sides to be in the same position in relation to the given angles. So EF=HJ is the additional information that would make sure that the two triangles are congruent by AAS.

So option C is the correct answer.

b

Given that EF=HJ, and EG=7, find the value of m.

Worked Solution
Create a strategy

Use the properties of congruent triangles and isosceles triangles.

Apply the idea

Since EF=HJ we know that the two triangles are congruent.

Since the two given angles in each triangle are equal, these triangles will be isosceles, with the sides opposite the equal angles being equal.

So EG=HK=FG=JK=7 since EG=7.

m=7

Idea summary

If two corresponding sides or angles must be equal in congruent triangles then knowing the value of one gives us the value of the other.

Outcomes

MA4-17MG

classifies, describes and uses the properties of triangles and quadrilaterals, and determines congruent triangles to find unknown side lengths and angles

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