Geometric Twins · Lesson 2 of 10
Measuring the Sidelengths
“Construct a triangle from three side lengths and see why its reflected copy is congruent.”
• Construct a triangle by using a base and two compass arcs. • Explain why two intersections on opposite sides of the base give congruent triangles. • State and apply the SSS congruence condition. • Match three side lengths rather than relying on drawing orientation.
Measuring the Sidelengths
Meera and Rabia need a cardboard copy of a triangular frame too large to trace. Its sides measure 40 cm, 60 cm, and 80 cm. Must they also measure all three angles? To explore the question on paper, use 4 cm, 6 cm, and 8 cm instead. This smaller triangle is only a model; the actual cardboard copy must use the original full lengths.
Three lengths offer more control than the two arms of a hinged symbol. After a base is drawn, the third vertex must be the right distance from each endpoint. A compass describes all locations at one distance from an endpoint. The third vertex must lie on both of the resulting circles, so it must be at an intersection.
Problem
Construct triangle ABE with AB = 6 cm, AE = 4 cm, and BE = 8 cm.
- 1.Draw base AB of length 6 cm.
- 2.With A as centre and radius 4 cm, draw an arc in the region where the third vertex could lie.
- 3.With B as centre and radius 8 cm, draw another arc to cross the first. Name one intersection E.
- 4.Join AE and BE. The compass radii guarantee the two required lengths, and AB already has the third.
Does the second intersection change the shape?
The circles also meet at a point F on the other side of AB. Both triangles use the same base, and both have one side of length 4 cm and another of length 8 cm. This is not the troublesome two-shape situation we will meet later. The construction above and below AB is symmetric: E and F lie at reflected positions. A cutout of one triangle fits the other after a flip across AB.
This explains the Side–Side–Side condition, abbreviated SSS. If all three sides of one triangle match three corresponding sides of another, the triangles are congruent. The triangles may occupy different places or face different directions. The three lengths still fix one shape and size up to these allowed movements.
Two triangles are congruent if their three corresponding side lengths are equal.
Problem
Triangle RED has RE = 3.5 cm, ED = 5 cm, and RD = 6 cm. Triangle JAM has JA = 3.5 cm, AM = 5 cm, and JM = 6 cm. Are they congruent?
- 1.Match RE with JA, ED with AM, and RD with JM.
- 2.All three corresponding lengths agree, so SSS establishes congruence.
- 3.The matches give R ↔ J, E ↔ A, D ↔ M, hence triangle RED is congruent to triangle JAM.
What SSS lets us conclude
SSS does not tell us that any three positive lengths make a triangle. It says that triangles that exist with the same three side lengths are congruent. Construction is a useful check: if the arcs do not meet in a valid third vertex, the requested triangle has not been formed. Once SSS has established congruence, the corresponding angles also match even though we did not measure them separately.
Problem
One triangle has sides 4, 6, 8 cm; another has sides 4, 6, 7 cm. Does SSS establish congruence?
- 1.Two pairs agree, but the third side length is different.
- 2.SSS requires all three corresponding sides to be equal, not only two.
- 3.The differing third side prevents an exact fit, so these triangles are not congruent.
The two intersections in an SSS construction are reflected copies. Do not reject congruence merely because one lies above the base and one below it.
Quiz
What information is sufficient in SSS?
What fixes AE = 4 cm in the construction?
The SSS arcs meet above and below the base. What is the relationship between the resulting triangles?
Which pair satisfies SSS?
After SSS proves triangles congruent, what follows for corresponding angles?
Practice Problems
- Construct a triangle with sides 4 cm, 6 cm, and 8 cm and retain the compass arcs.
- Locate both intersections and explain how to superimpose the resulting triangles.
- Triangles have sides 5, 7, 9 cm and 9, 5, 7 cm. Explain the congruence condition.
- Why is a 4–6–8 cm paper model not itself congruent to the 40–60–80 cm frame?
- A learner says matching two sides is enough for SSS. Show what remains undetermined.
- Use paper cutouts to compare two triangles independently constructed with the same three lengths.
Key Takeaways
• A third vertex in an SSS construction lies at the intersection of two distance arcs. • The two opposite-side intersections are mirror copies and give congruent triangles. • SSS requires three matching corresponding side lengths. • SSS does not require separate angle measurements. • A scale model illustrates a construction but is not an equal-size copy of the original.