Exercises: Congruent triangles and the tests SSS, SAS, ASA

State what it means for two triangles to be congruent.

Triangle $ABC$ has $AB = 5$ cm, $BC = 7$ cm and $CA = 9$ cm. Triangle $DEF$ has $DE = 5$ cm, $EF = 7$ cm and $FD = 9$ cm. Are the triangles congruent? Name the test.

In triangles $PQR$ and $XYZ$: $PQ = XY = 6$ cm, $\angle Q = \angle Y = 50^\circ$ and $QR = YZ = 8$ cm. Name the test showing they are congruent and list the pairs of corresponding vertices.

Triangle $ABC$ has $\angle A = 40^\circ$, $AB = 7$ cm and $\angle B = 65^\circ$. Triangle $DEF$ has $\angle D = 40^\circ$, $DE = 7$ cm and $\angle F = 75^\circ$.

1. a) Find $\angle C$ and $\angle E$.

1. b) Are the triangles congruent? Name the test.

Two triangles have sides $4$ cm, $6$ cm and an angle of $30^\circ$. In the first triangle the $30^\circ$ angle is between the $4$ cm and $6$ cm sides. In the second it is opposite the $4$ cm side. Explain whether you can conclude the triangles are congruent.

In the diagram description, $M$ is the midpoint of $AB$ and also of $CD$, where $AB$ and $CD$ are two different line segments that cross at $M$.

1. a) Name the two triangles that are congruent.

1. b) Identify the equal pairs of sides and angles.

1. c) State the test used.

In triangle $ABC$, $AB = AC$. $D$ is the midpoint of $BC$. Prove that triangles $ABD$ and $ACD$ are congruent, and hence prove that $\angle B = \angle C$.

$ABCD$ is a parallelogram with diagonal $AC$. Using the fact that opposite sides of a parallelogram are equal, prove that triangles $ABC$ and $CDA$ are congruent. Name the test.

Triangles $ABC$ and $DEF$ have $AB = DE$, $BC = EF$ and $\angle A = \angle D$, where these angles are not the angles between the given sides.

1. a) Give an example, with lengths and angle sizes, showing that the triangles need not be congruent.

1. b) Explain why this does not contradict the SAS test.