Construct triangles and test what given measures determine

Learning goal: Draw and construct triangles from specified side or angle measures using freehand sketches, ruler/protractor tools and a digital geometry tool. Determine when conditions give one shape up to rigid motion, more than one triangle or no triangle.

A freehand sketch helps plan a construction, but it does not verify exact conditions. Use a ruler for specified lengths, a protractor for angles and compass arcs or an equivalent digital distance tool for fixed-distance intersections. After constructing, measure the result and compare it with the intended conditions. Small tool errors do not change the exact mathematical question.

Three side lengths can determine a unique triangle up to translation, rotation or reflection when they satisfy the triangle inequalities. Construct a three-four-five example by drawing a five-unit base. Draw an arc of radius three from one endpoint and an arc of radius four from the other. Their intersection supplies the third vertex. The reflected intersection below the base gives a congruent mirror image, not a different set of side lengths and angles.

A triangle requires the sum of the two shorter sides to exceed the longest side. Lengths one, two and four fail because one plus two is less than four. The two short segments cannot reach across the base to meet. Lengths one, two and three also fail to form a nondegenerate triangle: the segments lie in a straight line at their limiting reach, enclosing no area. Equality is not sufficient.

Three angle measures must sum to 180 degrees in a plane triangle. Angles 50, 60 and 80 cannot form a triangle because they sum to 190. Angles 30, 60 and 90 do sum to 180, but they do not specify a size. A small and a large triangle can share those angles. Thus three angles can determine shape without determining one unique triangle size.

Two side lengths and the included angle fix the relationship between those sides. Draw a six-unit segment, construct a 60-degree ray at one endpoint, mark four units along the ray and connect to the other endpoint. The included angle lies between the two specified sides. The resulting triangle is determined up to rigid motion by those measures. Stating which angle is included is essential.

Two angles and a specified corresponding side can also determine a triangle when the conditions are consistent. Draw a six-unit base and construct angles of 50 and 60 degrees at its endpoints on the same side of the base. The rays meet, and the third angle is 70 degrees. Because the base length fixes size and the rays fix direction, this construction has one resulting shape up to rigid motion.

A side-side-angle condition in which the angle is not between the given sides may be ambiguous. Consider angle A = 30 degrees, side AC = ten units and side BC = seven units. Draw ray AB horizontally, then draw the 30-degree ray from A and mark C ten units along it. A circle centered at C with radius seven intersects the horizontal ray at two different points in this example. Either intersection can be B.

Both resulting triangles have the same specified 30-degree angle at A, the same ten-unit AC and the same seven-unit BC. Their AB lengths differ, so they are not congruent. The diagram shows both intersections with the actual circle and ray. This example demonstrates why a list of three measures must be interpreted by their positions, not merely counted.

The exact decimal lengths of AB are not needed to establish ambiguity here. A careful compass or digital construction reveals two intersections. Record the measurements you observe with an appropriate approximation label rather than inventing exact values from a sketch. The reasoning rests on both candidates satisfying the stated constraints and having different remaining lengths.

A digital tool should enforce the intended relationships. Place points with a fixed distance or angle construction rather than dragging until a picture looks close. Move the whole construction to check which relationships remain fixed. Compare the digital result with a ruler/protractor version and explain any differences as tool precision or construction choices, not as different geometry.

Worked impossible case: Sides two, three and six fail because two plus three is only five. No arrangement can make the two shorter sides connect the endpoints of a six-unit base. A drawing that appears to close must have violated a supplied length.

Worked multiple sizes: Angles 40, 60 and 80 sum to 180, so the angle condition is consistent. Without a side length, enlarging a valid triangle preserves all three angles and produces a different size. The result is more than one triangle, even though all such triangles share a shape.

Worked included-angle check: If four and six are the sides meeting at a specified 60-degree angle, constructing those two rays fixes the third vertex positions relative to their shared endpoint. Moving the 60-degree requirement to an opposite angle changes the constraints and may change the number of solutions. Name the angle's vertex and adjacent sides explicitly.

Required construction sequence: First sketch, then construct a three-four-five triangle with ruler and compass or fixed-distance circles. Construct a triangle from a six-unit side with 50- and 60-degree endpoint angles using a protractor. Use a digital geometry tool with distance and angle constraints to reproduce one case and the ambiguous ten/seven/30-degree case. Record which intersections are valid. If a digital tool is not yet available, preserve that task as unfinished rather than claiming the paper drawing completes it.

Independent performance: Present five condition sets: one valid three-side case, one impossible side case, one angle-only family, one included-angle construction and the ambiguous example. For each, state whether there is one congruence class, multiple possibilities or no triangle, with a construction or geometric reason. Distinguish a reflected copy from a genuinely different triangle and label approximate measurements honestly.