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Define parallel lines and alternate interior angles, and solve for x in given geometry problems.

Mathematics

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Answers

2 parts
a

x=65x = 65

b

x=10x = 10

Step-by-step solution

6 steps

Here are the solutions to the geometry problems:

1. Define parallel lines (1 mark)
Parallel lines are lines in a plane that are always the same distance apart and never intersect or touch each other, no matter how far they are extended.

2. State the relationship between alternate interior angles when two parallel lines are cut by a transversal line (2 marks)
When two parallel lines are cut by a transversal line, the alternate interior angles are equal in measure.

3. Find the value of xx (6 marks)

Part a

The angles x∘x^\circ and 65∘65^\circ are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are equal.

x∘=65∘x^\circ = 65^\circ x=65\boxed{x = 65}

Part b

x=10x = 10

The angles 3x∘3x^\circ and (2x+10)∘(2x+10)^\circ are alternate interior angles. When two parallel lines are cut by a transversal, alternate interior angles are equal.

  1. Set the angles equal to each other.

    3x=2x+103x = 2x + 10

  2. Subtract $2x$ from both sides.

    3x−2x=103x - 2x = 10 x=10x = 10

  3. Verify the angles.

    3x=3(10)=30∘3x = 3(10) = 30^\circ 2x+10=2(10)+10=20+10=30∘2x+10 = 2(10)+10 = 20+10 = 30^\circ x=10\boxed{x = 10}

Part c· 3 steps

The angles (2x+15)∘(2x+15)^\circ and (3x−25)∘(3x-25)^\circ are consecutive interior angles (also known as same-side interior angles). When two parallel lines are cut by a transversal, consecutive interior angles are supplementary, meaning their sum is 180∘180^\circ.

  1. Set the sum of the angles equal to 180∘180^\circ.

    (2x+15)+(3x−25)=180(2x+15) + (3x-25) = 180

  2. Combine like terms.

    5x−10=1805x - 10 = 180

  3. Step 3

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