Two Triangular Stage Braces with a Non-Included Angle
A community theater is rebuilding two triangular side braces for a revolving set. Brace ABC has AB = 7.0 ft, BC = 5.0 ft, and angle A = 38°. Brace DEF has DE = 7.0 ft, EF = 5.0 ft, and angle D = 38°. The shop foreman says the braces are congruent by SAS because two sides and an angle match, so one template will fit both. A volunteer notes that angle A is not between AB and BC, and angle D is not between DE and EF. The theater wants to avoid rebuilding a brace after the set is assembled. They can order one additional measurement for each brace, but the measurement must be the same type on both. The available tools can measure either the remaining side or the included angle at B or E. The foreman also claims that even if the triangles are not guaranteed congruent, they must have the same area because the same three measurements are given.
1. Does the given information guarantee that triangles ABC and DEF are congruent? Support your answer by finding two possible values for AC in a triangle with AB = 7.0 ft, BC = 5.0 ft, and angle A = 38°.
Read the worked answer
No. The given angle is not included between the two given sides, so this is the ambiguous SSA case, not SAS. Use the Law of Sines in triangle ABC. Side BC = 5 is opposite angle A = 38°, and side AB = 7 is opposite angle C:
sin C / 7 = sin 38° / 5 sin C = 7 sin 38° / 5 ≈ 0.8619
Thus C ≈ 59.5° or C ≈ 180° − 59.5° = 120.5°. Both are valid because both are greater than 38° and leave a positive angle B.
Case 1: C ≈ 59.5°, so B ≈ 180° − 38° − 59.5° = 82.5°. AC = 5 sin 82.5° / sin 38° ≈ 8.05 ft.
Case 2: C ≈ 120.5°, so B ≈ 180° − 38° − 120.5° = 21.5°. AC = 5 sin 21.5° / sin 38° ≈ 2.98 ft.
Two noncongruent triangles satisfy the same two sides and non-included angle, so the braces are not guaranteed congruent.
2. What single additional measurement would let the shop guarantee congruence? Explain why it works, and explain why the original angle A is not enough.
Read the worked answer
Measuring the included angle at B and at E would work: then AB = DE, BC = EF, and the angles between those pairs of sides are equal, so the triangles are congruent by SAS. Equivalently, measuring the remaining sides AC and DF would work by SSS.
The original angle A is not between AB and BC; it is opposite BC. Two sides and a non-included angle do not determine a unique triangle, as the two AC values 8.05 ft and 2.98 ft show. The shop must control the angle between the two measured sides, or control all three sides.
3. The foreman says the braces must have equal area even if they are not congruent. Is that true? Use the two possible triangles to decide.
Read the worked answer
No. Using the included angle A between AB and AC, the area is
Area = (1/2)(AB)(AC) sin A = (1/2)(7)(AC)(sin 38°) ≈ 2.155(AC).
For AC ≈ 8.05 ft, area ≈ 17.35 ft². For AC ≈ 2.98 ft, area ≈ 6.42 ft².
The areas differ by more than a factor of 2.5. Equal pairs of sides and one non-included angle do not fix the shape or the area.