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Year 8AC9M8SP01

Congruence and Similarity

Identify the conditions for congruence and similarity of triangles and explain the conditions for other sets of common shapes to be congruent or similar, including those formed by transformations.

Congruent means identical. Similar means the same shape at a different size. Each has a short list of tests, and you only need one of them.

Method

Congruent: same shape, same size
≡\equiv
All matching sides and angles are equal.
Similar: same shape, different size
∼\sim
Angles equal, sides in the same ratio.
The congruence tests
SSS, SAS, ASA, RHSSSS,\ SAS,\ ASA,\ RHS
One test is enough. You never need all four.
The similarity tests
AAA, SSS ratio, SAS ratioAAA,\ SSS \text{ ratio},\ SAS \text{ ratio}
Equal angles alone prove similarity.
SSA is not a test
Two sides and a non-included angle can give two different triangles.

Worked example

e.g. Two triangles have sides 5,7,9 and 9,5,7. Are they congruent?\text{Two triangles have sides } 5, 7, 9 \text{ and } 9, 5, 7. \text{ Are they congruent?}
  1. List the sides of each in order.5,7,9and5,7,95, 7, 9 \quad \text{and} \quad 5, 7, 9
  2. Check whether all three match.all three are equal\text{all three are equal}
  3. Name the test that applies.SSSSSS
  4. State the conclusion with the reason.congruent [SSS]\text{congruent } [SSS]
  5. Note that the order they were listed in does not matter.only the set of lengths matters\text{only the set of lengths matters}

Practice

Name the test. A conclusion without a reason scores half.

1
Triangles with angles 40,60,80 and 40,60,80. Congruent or similar?\text{Triangles with angles } 40,60,80 \text{ and } 40,60,80. \text{ Congruent or similar?}
AnswerSimilar only — size is unknown [AAA]\text{Similar only — size is unknown } [AAA]
2
Sides 3,4,5 and 6,8,10. What are they?\text{Sides } 3,4,5 \text{ and } 6,8,10. \text{ What are they?}
AnswerSimilar, ratio 1:2\text{Similar, ratio } 1:2
3
Two right triangles, equal hypotenuse and one equal side. Test?\text{Two right triangles, equal hypotenuse and one equal side. Test?}
AnswerRHSRHS
4
Why is AAA not enough for congruence?\text{Why is AAA not enough for congruence?}
AnswerThe triangles could be different sizes.\text{The triangles could be different sizes.}
5
A shape is translated. Is the image congruent?\text{A shape is translated. Is the image congruent?}
AnswerYes. Translation does not change size or shape.\text{Yes. Translation does not change size or shape.}
6
A shape is enlarged by scale factor 3. Congruent or similar?\text{A shape is enlarged by scale factor 3. Congruent or similar?}
AnswerSimilar.\text{Similar.}
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