Congruent triangles and parallelograms
A groundsman at a GAA club cuts two triangular corner flags from the same pattern. How can he be sure, without laying one on top of the other, that the two flags are exactly the same shape and size? In this page you will learn the minimum information you need, and then use it to prove facts about isosceles triangles and parallelograms. At the end we come back to the flags.
Congruent triangles
Two triangles are congruent if one fits exactly on top of the other. They may be turned or flipped, but the sizes of all sides and angles are the same. We write $\triangle ABC \equiv \triangle XYZ$.
The order of the letters matters: it tells you which vertices correspond. If $\triangle ABC \equiv \triangle XYZ$, then $A$ matches $X$, $B$ matches $Y$ and $C$ matches $Z$. So $|AB| = |XY|$ and $|\angle ABC| = |\angle XYZ|$.
You do not need to check all six measurements. Three well-chosen ones are enough. These are the conditions (Axiom 4):
State whether each pair of triangles is congruent. If so, name the condition (SSS, SAS or ASA); if not, say why not. a) Triangle 1 has sides $5 \text{ cm}$,…
State whether each pair of triangles is congruent. If so, name the condition (SSS, SAS or ASA); if not, say why not.
- a) Triangle 1 has sides $5 \text{ cm}$, $7 \text{ cm}$, $9 \text{ cm}$; triangle 2 has sides $9 \text{ cm}$, $5 \text{ cm}$, $7 \text{ cm}$.
- 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 $60^\circ$.
- 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 triangles $PQR$ and $XYZ$: $PQ = XY$, $PR = XZ$ and angle $QPR$ = angle $YXZ$. a) Name the axiom that proves the triangles are congruent. b) List the three pairs of…
In triangles $PQR$ and $XYZ$: $PQ = XY$, $PR = XZ$ and angle $QPR$ = angle $YXZ$.
- a) Name the axiom that proves the triangles are congruent.
- b) List the three pairs of corresponding vertices.
- c) Hence state which side of $XYZ$ equals $QR$ and which angle equals angle $PQR$.
Isosceles triangles
An isosceles triangle has two equal sides. Theorem 2: in an isosceles triangle the angles opposite the equal sides are equal. The converse is also true: if two angles of a triangle are equal, then the sides opposite them are equal.
The converse swaps "if" and "then". A statement being true does not mean its converse is true, so each must be checked. Here both are true.
In triangle $ABC$, $AB = AC$ and angle $B = 65^\circ$. a) Name the type of triangle and state the theorem that tells you angle $C$. b) Find angle $C$ and angle $A$.
In triangle $ABC$, $AB = AC$ and angle $B = 65^\circ$.
- a) Name the type of triangle and state the theorem that tells you angle $C$.
- b) Find angle $C$ and angle $A$.
In triangle $ABC$, angle $B$ = angle $C$ = $52^\circ$ and $BC = 10 \text{ cm}$. a) Find angle $A$. b) State the converse of the isosceles triangle theorem and use it to…
In triangle $ABC$, angle $B$ = angle $C$ = $52^\circ$ and $BC = 10 \text{ cm}$.
- a) Find angle $A$.
- b) State the converse of the isosceles triangle theorem and use it to say which two sides are equal.
- c) If $AB = 8.1 \text{ cm}$, state the length of $AC$.
Writing a proof
A proof is a chain of reasons, each one justified by an axiom or a theorem already known. The word implies links a statement to what follows from it: "$|AB| = |AC|$ implies $|\angle B| = |\angle C|$".
In triangles $PQR$ and $XYZ$: $PQ = XY$, $PR = XZ$ and angle $QPR$ = angle $YXZ$. a) Name the axiom that proves the triangles are congruent. b) List the three pairs of…
In triangles $PQR$ and $XYZ$: $PQ = XY$, $PR = XZ$ and angle $QPR$ = angle $YXZ$.
- a) Name the axiom that proves the triangles are congruent.
- b) List the three pairs of corresponding vertices.
- c) Hence state which side of $XYZ$ equals $QR$ and which angle equals angle $PQR$.
In triangle $ABC$, $AB = AC$. $D$ is the midpoint of $BC$. a) Prove that triangles $ABD$ and $ACD$ are congruent. b) Hence prove that angle $ABC$ = angle $ACB$. c) Prove…
In triangle $ABC$, $AB = AC$. $D$ is the midpoint of $BC$.
- a) Prove that triangles $ABD$ and $ACD$ are congruent.
- b) Hence prove that angle $ABC$ = angle $ACB$.
- c) Prove that $AD$ is perpendicular to $BC$.
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
Also, angles next to each other (for example $A$ and $B$) add to $180^\circ$, because they are co-interior angles between parallel lines.
Why are opposite sides equal? Draw diagonal $[AC]$. Alternate angles give $|\angle BAC| = |\angle DCA|$ and $|\angle BCA| = |\angle DAC|$, and $[AC]$ is common. So $\triangle ABC \equiv \triangle CDA$ [ASA], and the corresponding sides are equal.
$ABCD$ is a parallelogram with $AB = 8 \text{ cm}$, $BC = 5 \text{ cm}$ and angle $A = 70^\circ$. a) Write down the lengths of $CD$ and $AD$. b) Write down the sizes of…
$ABCD$ is a parallelogram with $AB = 8 \text{ cm}$, $BC = 5 \text{ cm}$ and angle $A = 70^\circ$.
- a) Write down the lengths of $CD$ and $AD$.
- b) Write down the sizes of angles $B$, $C$ and $D$.
- c) State the property of a parallelogram you used in each case.
$ABCD$ is a parallelogram. The diagonals $AC$ and $BD$ meet at $E$. Given $AE = 4.5 \text{ cm}$ and $BE = 3 \text{ cm}$: a) Write down the lengths of $AC$ and $BD$. b)…
$ABCD$ is a parallelogram. The diagonals $AC$ and $BD$ meet at $E$. Given $AE = 4.5 \text{ cm}$ and $BE = 3 \text{ cm}$:
- a) Write down the lengths of $AC$ and $BD$.
- b) Name a pair of congruent triangles in the diagram and state the condition that proves it.
- c) If $AB = 5 \text{ cm}$, write down the length of $CD$.
$ABCD$ is a quadrilateral with $AB \parallel DC$ and $AD \parallel BC$. The diagonal $AC$ is drawn. a) Name two pairs of equal alternate angles. b) Prove that triangles…
$ABCD$ is a quadrilateral with $AB \parallel DC$ and $AD \parallel BC$. The diagonal $AC$ is drawn.
- a) Name two pairs of equal alternate angles.
- b) Prove that triangles $ABC$ and $CDA$ are congruent.
- c) Hence prove that $AB = CD$ and $AD = BC$.
Converses and proving a parallelogram
Congruence also works the other way round. If we know the sides of a quadrilateral, we can show it is a parallelogram.
We have proved: "opposite sides equal implies parallelogram". Theorem 9 says the reverse: "parallelogram implies opposite sides equal". Each is the converse of the other, and here both are true.
Kate claims: "In quadrilateral $PQRS$, if $PQ = RS$ and $QR = PS$ then it is a parallelogram." She draws the diagonal $PR$. a) Prove that triangles $PQR$ and $RSP$ are…
Kate claims: "In quadrilateral $PQRS$, if $PQ = RS$ and $QR = PS$ then it is a parallelogram." She draws the diagonal $PR$.
- a) Prove that triangles $PQR$ and $RSP$ are congruent.
- b) Use the congruence to prove that $PQ \parallel SR$ and $QR \parallel PS$.
- c) Write the statement you have proved using the word "implies", and state its converse. Is the converse true?
Back to the flags: if the two flags have the same three side lengths (SSS), or two sides and the angle between them (SAS), they are congruent, so measuring three well-chosen things is enough.
Summary
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Congruent triangles have all corresponding sides and angles equal. The order of letters shows which vertices match.
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Valid conditions: SSS, SAS (included angle) and ASA. SSA and AAA are not enough.
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Isosceles triangle: angles opposite equal sides are equal (Theorem 2), and the converse is also true.
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Parallelogram: opposite sides and opposite angles are equal (Theorem 9) and the diagonals bisect each other (Theorem 10).
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In a proof, give a reason for every step. Use congruence to show corresponding parts are equal.
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The converse of "A implies B" is "B implies A", and it may be false.