Exercises: Congruent triangles and parallelograms

State whether each pair of triangles is congruent. If so, name the condition (SSS, SAS or ASA); if not, say why not.

1. a) Triangle 1 has sides $5 \text{ cm}$, $7 \text{ cm}$, $9 \text{ cm}$; triangle 2 has sides $9 \text{ cm}$, $5 \text{ cm}$, $7 \text{ cm}$.
2. b) Triangle 1 has sides $4 \text{ cm}$ and $6 \text{ cm}$ with an included angle of $50^\circ$; triangle 2 has sides $4 \text{ cm}$ and $6 \text{ cm}$ with an included angle of $60^\circ$.
3. c) Triangle 1 has angles $40^\circ$ and $70^\circ$ with the side $8 \text{ cm}$ between them; triangle 2 has angles $70^\circ$ and $40^\circ$ with the side $8 \text{ cm}$ between them.

In triangle $ABC$, $AB = AC$ and angle $B = 65^\circ$.

1. a) Name the type of triangle and state the theorem that tells you angle $C$.
2. b) Find angle $C$ and angle $A$.

$ABCD$ is a parallelogram with $AB = 8 \text{ cm}$, $BC = 5 \text{ cm}$ and angle $A = 70^\circ$.

1. a) Write down the lengths of $CD$ and $AD$.
2. b) Write down the sizes of angles $B$, $C$ and $D$.
3. c) State the property of a parallelogram you used in each case.

In triangles $PQR$ and $XYZ$: $PQ = XY$, $PR = XZ$ and angle $QPR$ = angle $YXZ$.

1. a) Name the axiom that proves the triangles are congruent.
2. b) List the three pairs of corresponding vertices.
3. c) Hence state which side of $XYZ$ equals $QR$ and which angle equals angle $PQR$.

In triangle $ABC$, angle $B$ = angle $C$ = $52^\circ$ and $BC = 10 \text{ cm}$.

1. a) Find angle $A$.
2. b) State the converse of the isosceles triangle theorem and use it to say which two sides are equal.
3. c) If $AB = 8.1 \text{ cm}$, state the length of $AC$.

$ABCD$ is a parallelogram. The diagonals $AC$ and $BD$ meet at $E$. Given $AE = 4.5 \text{ cm}$ and $BE = 3 \text{ cm}$:

1. a) Write down the lengths of $AC$ and $BD$.
2. b) Name a pair of congruent triangles in the diagram and state the condition that proves it.
3. c) If $AB = 5 \text{ cm}$, write down the length of $CD$.

In triangle $ABC$, $AB = AC$. $D$ is the midpoint of $BC$.

1. a) Prove that triangles $ABD$ and $ACD$ are congruent.
2. b) Hence prove that angle $ABC$ = angle $ACB$.
3. c) Prove that $AD$ is perpendicular to $BC$.

$ABCD$ is a quadrilateral with $AB \parallel DC$ and $AD \parallel BC$. The diagonal $AC$ is drawn.

1. a) Name two pairs of equal alternate angles.
2. b) Prove that triangles $ABC$ and $CDA$ are congruent.
3. c) Hence prove that $AB = CD$ and $AD = BC$.

Kate claims: "In quadrilateral $PQRS$, if $PQ = RS$ and $QR = PS$ then it is a parallelogram." She draws the diagonal $PR$.

1. a) Prove that triangles $PQR$ and $RSP$ are congruent.
2. b) Use the congruence to prove that $PQ \parallel SR$ and $QR \parallel PS$.
3. c) Write the statement you have proved using the word "implies", and state its converse. Is the converse true?