Grade: Grade 10 Subject: Mathematics Unit: Geometry Theorems Lesson: 3 of 6 SAT: Geometry+Trigonometry ACT: Math

Guided Practice

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This lesson provides step-by-step practice with triangle and circle theorems. Work through each problem carefully, following the solution strategies demonstrated.

Problem-Solving Strategy for Geometry Theorems

  1. Identify what you know: List given information and mark it on the figure
  2. Identify what you need to find: What is the question asking?
  3. Choose the right theorem: Which theorem applies to this situation?
  4. Set up and solve: Write equations and solve for unknowns
  5. Check your answer: Does it make sense? Do angles add up correctly?

Key Theorems Review

  • Triangle Angle Sum: The sum of angles in a triangle is 180 degrees
  • Exterior Angle Theorem: An exterior angle equals the sum of the two non-adjacent interior angles
  • Isosceles Triangle Theorem: Base angles of an isosceles triangle are congruent
  • Inscribed Angle Theorem: An inscribed angle is half the central angle that subtends the same arc
  • Central Angle Theorem: A central angle equals the measure of its intercepted arc

Worked Examples

Example 1: Triangle Angle Sum

Problem: In triangle ABC, angle A = 45 degrees and angle B = 72 degrees. Find angle C.

Solution:

  1. Apply the Triangle Angle Sum Theorem: A + B + C = 180 degrees
  2. Substitute known values: 45 + 72 + C = 180
  3. Simplify: 117 + C = 180
  4. Solve: C = 180 - 117 = 63 degrees

Answer: Angle C = 63 degrees

Example 2: Exterior Angle Theorem

Problem: In triangle PQR, the exterior angle at R measures 130 degrees. If angle P = 55 degrees, find angle Q.

Solution:

  1. Apply the Exterior Angle Theorem: Exterior angle = sum of non-adjacent interior angles
  2. So: 130 = angle P + angle Q
  3. Substitute: 130 = 55 + angle Q
  4. Solve: angle Q = 130 - 55 = 75 degrees

Answer: Angle Q = 75 degrees

Example 3: Inscribed Angle Theorem

Problem: In a circle, central angle AOB = 84 degrees. Find the inscribed angle ACB that subtends the same arc AB.

Solution:

  1. Apply the Inscribed Angle Theorem: Inscribed angle = (1/2) x Central angle
  2. Calculate: angle ACB = (1/2) x 84 = 42 degrees

Answer: Inscribed angle ACB = 42 degrees

Practice Problems

Try these problems on your own. Click to reveal the answer and solution.

Problem 1: In triangle XYZ, angle X = 38 degrees and angle Y = 54 degrees. Find angle Z.

Solution: Using Triangle Angle Sum: 38 + 54 + Z = 180

92 + Z = 180

Answer: Z = 88 degrees

Problem 2: An isosceles triangle has a vertex angle of 50 degrees. Find each base angle.

Solution: Let each base angle = x. Then: 50 + x + x = 180

50 + 2x = 180

2x = 130

Answer: Each base angle = 65 degrees

Problem 3: The exterior angle of a triangle is 115 degrees. One of the non-adjacent interior angles is 48 degrees. Find the other non-adjacent interior angle.

Solution: By Exterior Angle Theorem: 115 = 48 + x

Answer: x = 67 degrees

Problem 4: In a circle, an inscribed angle measures 35 degrees. What is the measure of the central angle subtending the same arc?

Solution: Central angle = 2 x Inscribed angle = 2 x 35

Answer: Central angle = 70 degrees

Problem 5: A central angle in a circle measures 144 degrees. Find the inscribed angle subtending the same arc.

Solution: Inscribed angle = (1/2) x Central angle = (1/2) x 144

Answer: Inscribed angle = 72 degrees

Problem 6: Triangle DEF has angles in the ratio 2:3:4. Find each angle.

Solution: Let angles be 2x, 3x, and 4x.

2x + 3x + 4x = 180

9x = 180, so x = 20

Answer: Angles are 40, 60, and 80 degrees

Problem 7: In an isosceles triangle, one base angle is (3x + 10) degrees and the vertex angle is (2x + 20) degrees. Find x and all three angles.

Solution: Base angles are equal: (3x + 10) + (3x + 10) + (2x + 20) = 180

8x + 40 = 180

8x = 140, x = 17.5

Base angles: 3(17.5) + 10 = 62.5 degrees each

Vertex angle: 2(17.5) + 20 = 55 degrees

Answer: x = 17.5; angles are 62.5, 62.5, and 55 degrees

Problem 8: Two inscribed angles subtend the same arc. One measures 47 degrees. What does the other measure?

Solution: Inscribed angles subtending the same arc are congruent.

Answer: The other inscribed angle also measures 47 degrees

Problem 9: An inscribed angle subtends a semicircle. What is its measure?

Solution: A semicircle has a central angle of 180 degrees.

Inscribed angle = (1/2) x 180 = 90 degrees

Answer: 90 degrees (this is the Thales' Theorem)

Problem 10: In triangle ABC, the exterior angle at C is (5x - 10) degrees. Angle A = (2x + 5) degrees and angle B = (x + 15) degrees. Find x and all angles.

Solution: By Exterior Angle Theorem: 5x - 10 = (2x + 5) + (x + 15)

5x - 10 = 3x + 20

2x = 30, x = 15

Angle A: 2(15) + 5 = 35 degrees

Angle B: 15 + 15 = 30 degrees

Exterior angle at C: 5(15) - 10 = 65 degrees

Interior angle C: 180 - 65 = 115 degrees

Answer: x = 15; angles are 35, 30, and 115 degrees

Check Your Understanding

Answer these questions to verify your mastery of the guided practice concepts.

  1. What is the first step in solving a geometry theorem problem?
  2. How does the Exterior Angle Theorem relate exterior and interior angles?
  3. What is the relationship between inscribed and central angles subtending the same arc?
  4. Why are base angles of an isosceles triangle always congruent?

Next Steps

  • If you scored 8-10 correct, move on to Word Problems
  • If you scored 5-7 correct, review Examples and try Practice again
  • If you scored below 5, revisit Triangle Theorems and Circle Theorems lessons