Year 10AC9M10SP01

Congruence and Similarity Proofs

Apply deductive reasoning to proofs involving shapes in the plane and use theorems to solve spatial problems.

A proof is a chain of statements, each with a reason. Naming the test is what earns the mark.

Builds onYr 10 · Deductive Reasoning and Proof

Method

State, then justify

\text{statement} \;+\; [\text{reason}]

Every line needs both halves.

Congruence tests

SSS,\ SAS,\ ASA,\ AAS,\ RHS

One is enough.

Similarity tests

AAA,\ SSS \text{ ratio},\ SAS \text{ ratio}

Congruent triangles give equal parts

Once proved, every matching side and angle follows.

Similar triangles give equal ratios

Use them to find unknown lengths.

e.g.

\text{Prove the diagonals of a parallelogram bisect each other.}

Name the triangles formed by the diagonals. $\triangle AOB \text{ and } \triangle COD$
Use the parallel sides to find equal angles. $\angle OAB = \angle OCD \quad [\text{alternate angles}]$
Find a second pair. $\angle OBA = \angle ODC \quad [\text{alternate angles}]$
Use the parallelogram property for a side. $AB = CD \quad [\text{opposite sides of a parallelogram}]$
Name the test. $\triangle AOB \equiv \triangle COD \quad [AAS]$
Conclude from the congruence. $\therefore AO = OC \text{ and } BO = OD$

Name a test on every proof. A conclusion without one scores half.

\text{Two right triangles, equal hypotenuse and one leg. Test?}

Answer

RHS

\text{Triangles with all angles equal. Congruent or similar?}

Answer

\text{Similar only.}

\text{Sides } 4,6,8 \text{ and } 6,9,12. \text{ What are they?}

Answer

\text{Similar, ratio } 2:3

\text{Similar triangles, ratio } 1:3. \text{ A side of 5 matches what?}

Answer

15

\text{Why is SSA not a valid test?}

Answer

\text{Two different triangles can satisfy it.}

\text{What must every proof line contain?}

Answer

\text{A statement and a reason.}