High School Geometry Worksheets | Printable Geometry Math Worksheets - Free Printable

I'll solve each problem using the Law of Sines. Let me work through them systematically.

The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C)

Section A - Find the missing sides:

Problem 1: Triangle ABC with angles A=29°, B=82°, side b (opposite angle B)
- First, find angle C: C = 180° - 29° - 82° = 69°
- Using Law of Sines: b/sin(82°) = ?/sin(29°)
- Wait, I need to identify what's given. Looking at the diagram:
- Angle A = 29°, Angle B = 82°
- Side b is opposite angle B
- We need to find side b, but we don't have any side length given!

Let me reconsider. In triangle notation, side a is opposite angle A, side b is opposite angle B, side c is opposite angle C. But I need at least one side length to use the Law of Sines. Let me check if there's information I'm missing from the diagram... Actually, looking more carefully, it seems no side lengths are provided for problem 1 in Section A. This might be an error in my interpretation or the problem setup.

Let me move to problems where I can see all the information clearly.

Problem 2: Triangle ABC with side AB = 9 cm, angle A = 74°, angle C = 33°, find side a (BC)
- Angle B = 180° - 74° - 33° = 73°
- Side AB = c = 9 cm (opposite angle C)
- Using Law of Sines: a/sin(74°) = 9/sin(33°)
- a = 9 × sin(74°)/sin(33°)
- a = 9 × 0.9613/0.5446 ≈ 15.9 cm

Problem 3: Triangle ABC with side BC = 15.7 cm, angle A = 68°, angle C = 40°, find side c (AB)
- Angle B = 180° - 68° - 40° = 72°
- Side BC = a = 15.7 cm (opposite angle A)
- Using Law of Sines: c/sin(40°) = 15.7/sin(68°)
- c = 15.7 × sin(40°)/sin(68°)
- c = 15.7 × 0.6428/0.9272 ≈ 10.9 cm

Section B - Find the missing angles:

Problem 1: Triangle XYZ with XZ = 11 cm, XY = 8 cm, angle Z = 30°, find angle x
- Using Law of Sines: 11/sin(x) = 8/sin(30°)
- sin(x) = 11 × sin(30°)/8 = 11 × 0.5/8 = 0.6875
- x = arcsin(0.6875) ≈ 43.4°

Problem 2: Triangle XYZ with XY = 23 cm, YZ = 14 cm, angle X = 27°, find angle z
- Using Law of Sines: 23/sin(z) = 14/sin(27°)
- sin(z) = 23 × sin(27°)/14 = 23 × 0.4540/14 ≈ 0.7461
- z = arcsin(0.7461) ≈ 48.2°

Problem 3: Triangle XYZ with XZ = 108 mm, ZY = 110 mm, angle X = 62°, find angle y
- Using Law of Sines: 108/sin(y) = 110/sin(62°)
- sin(y) = 108 × sin(62°)/110 = 108 × 0.8829/110 ≈ 0.8669
- y = arcsin(0.8669) ≈ 60.1°

Section C - Find the missing side:

Problem 1: Triangle PQR with PR = 18 cm, angle Q = 79°, angle R = 69°, find side x (QR)
- Angle P = 180° - 79° - 69° = 32°
- Side PR = q = 18 cm (opposite angle Q)
- Side QR = p = x (opposite angle P)
- Using Law of Sines: x/sin(32°) = 18/sin(79°)
- x = 18 × sin(32°)/sin(79°)
- x = 18 × 0.5299/0.9816 ≈ 9.72 cm

Problem 2: Triangle ABC with AB = 12 cm, angle A = 75°, angle B = 48°, find side b (AC)
- Angle C = 180° - 75° - 48° = 57°
- Side AB = c = 12 cm (opposite angle C)
- Side AC = b (opposite angle B)
- Using Law of Sines: b/sin(48°) = 12/sin(57°)
- b = 12 × sin(48°)/sin(57°)
- b = 12 × 0.7431/0.8387 ≈ 10.6 cm

Problem 3: Triangle XYZ with YZ = 8 cm, angle X = 29°, angle Z = 112°, find side z (XY)
- Angle Y = 180° - 29° - 112° = 39°
- Side YZ = x = 8 cm (opposite angle X)
- Side XY = z (opposite angle Z)
- Using Law of Sines: z/sin(112°) = 8/sin(29°)
- z = 8 × sin(112°)/sin(29°)
- z = 8 × 0.9272/0.4848 ≈ 15.3 cm

Now let me verify my calculations using Python: