Three facts fix a triangle (usually)
Tonight's idea
Two triangles are congruent if one fits exactly on top of the other: same shape, same size. A triangle has three sides and three angles. How few must you be told before only one triangle is possible?
Three sides (SSS). Three sticks joined at their ends make a rigid frame. Nothing can move.
Two sides and the angle between them (SAS). Two sticks on a hinge, locked at that angle. The third side is the gap between their free ends.
Two angles and the side between them (ASA). From each end of the side a line sets off at a known angle. Two lines cross at one point only.
Two sides and an angle not between them (SSA). Now the second side is free to swing, and it can reach the third side in two places. Two different triangles fit the same three facts.
Three angles (AAA) fix the shape but not the size.
Watch one: the two triangles in the figure
Angle \(A = 30^\circ\), \(AB = 8\) cm, \(BC = 5\) cm. Draw the angle and mark \(B\) 8 cm along one arm. \(C\) is on the other arm, 5 cm from \(B\): on a circle of radius 5 about \(B\). The circle cuts the arm twice.
Where? Drop the height from \(B\) to \(H\). Triangle \(ABH\) is half an equilateral triangle, so \(BH = 4\) and \(AH = 4\sqrt3\). Then \(BHC\) has hypotenuse 5 and one leg 4, so \(HC = 3\), either side of \(H\).
\(AC = 4\sqrt3 - 3 \approx 3.93\) cm, or \(4\sqrt3 + 3 \approx 9.93\) cm.
Where you met fitting one shape onto another: the grade 8 sheet Slide, flip, turn, scale.
Your turn
- Does each set of facts fix exactly one triangle? Name the rule, or say what can still change. (a) Sides 5, 6, 7. (b) Angles 40°, 60°, 80°. (c) Sides 4 and 7 with 50° between them. (d) A side of 6 with angles of 30° and 70° at its ends.
- A triangle has angles of 50° and 60°, and the side opposite the 50° angle is 9 cm. Find the third angle. Is the triangle fixed? Which rule tells you?
- A kite \(ABCD\) has \(AB = AD\) and \(CB = CD\). Draw the diagonal \(AC\). (a) Why are triangles \(ABC\) and \(ADC\) congruent? (b) What does that tell you about angles \(B\) and \(D\)? (c) What does \(AC\) do to the angle at \(A\)?
- Angle \(A = 30^\circ\) and \(AB = 10\). The third fact is the length \(BC\). How many triangles fit when \(BC\) is (a) 4, (b) 5, (c) 6, (d) 12? Start with the height from \(B\).
Stretch: the right angle rescues SSA
Suppose the known angle is 90°: you are told the hypotenuse and one other side. This is SSA, yet only one triangle fits. Give two reasons: one using Pythagoras, and one using a circle drawn as in the figure.
Answers
- (a) Yes, SSS. (b) No: AAA gives the shape, and the triangle can be any size. (c) Yes, SAS. (d) Yes, ASA.
- \(180 - 50 - 60 = 70^\circ\). The 9 cm side is opposite the 50° angle, so the angles at its two ends are 60° and 70°. Both are known, so ASA fixes the triangle. Two angles and any one side are always enough, because the third angle comes free.
- (a) \(AB = AD\), \(CB = CD\), and \(AC\) belongs to both triangles: SSS. (b) They are matching angles of congruent triangles, so they are equal. (c) Angles \(BAC\) and \(DAC\) match, so \(AC\) cuts the angle at \(A\) in half. It does the same at \(C\).
- The height from \(B\) to the other arm is 5 (half of 10, as in the example). (a) None: 4 is too short to reach. (b) One: \(BC\) is the height itself, and the triangle has a right angle at \(C\). (c) Two, as in the figure. (d) One: 12 is longer than \(AB\), so the circle cuts the line once on each side of \(A\), and only one of those points is on the arm.
Stretch. Pythagoras: the third side is \(\sqrt{\text{hypotenuse}^2 - \text{side}^2}\), so all three sides are known and SSS applies. Circle: the right angle is at \(A\), so \(A\) is the foot \(H\) of the height, and the two points where the circle cuts the line are \(C_1\) and \(C_2\) with \(H\) half way between. Triangles \(BHC_1\) and \(BHC_2\) are mirror images in \(BH\), so they are congruent: one triangle, drawn twice.