Parallel Line Theorems
When two parallel lines are cut by a transversal, which angles are congruent?
Corresponding, alternate interior, and alternate exterior angles.
Solve for x: (3x + 47)° = (x + 67)°.
Answer: x = 10.
What is the slope of a line parallel to y = 2x + 3?
2
Are the lines y = 3x + 1 and y = 3x – 2 parallel, perpendicular, or neither?
Parallel
Define a two-column proof.
A proof with statements and reasons in two columns.
When two parallel lines are cut by a transversal, which angles are supplementary?
Same-side (consecutive) interior angles.
Two angles form a linear pair and one measures 110°. Find the other.
70°
What is the slope of a line perpendicular to y = –½x + 4?
2/1 or 2
Are the lines y = –2x + 4 and y = ½x – 6 parallel, perpendicular, or neither?
Perpendicular
Given ℓ ∥ m and ∠1 ≅ ∠7, prove a ∥ b.
Corresponding Angles Postulate.
Identify the theorem that justifies why alternate interior angles are congruent.
Alternate Interior Angles Theorem.
What do vertical angles always have in common?
They are congruent.
Write the equation of a line through (3, 6) parallel to 2x – 6y = 12.
y = ⅓x + 5.
Explain how to determine if lines in standard form are parallel.
Convert to slope-intercept form and compare slopes.
Given a ∥ b and ∠5 is supplementary to ∠2, prove ℓ ∥ m.
Consecutive Interior Angles Converse.
If two lines are cut by a transversal and corresponding angles are congruent, then the lines are ___.
Parallel (Converse of the Corresponding Angles Theorem).
(3x + 23)° and 5x° are supplementary. Find x.
x ≈ 15.4°.
Write the equation of a line through (4, 2) perpendicular to y = 2x – 8.
y = –½x + 4.
3x + 5y = 10 and 6x + 10y = 24 → parallel, perpendicular, or neither?
Parallel
What justifies that perpendicular lines form right angles?
Definition of Perpendicular Lines.
Explain how to prove two lines are parallel using angle pairs.
Show corresponding, alternate interior, or alternate exterior angles are congruent.
Define corresponding, alternate interior, and same-side interior angles.
Describe or illustrate their positions relative to the transversal.
How can you tell if two lines are parallel or perpendicular just from slopes?
Parallel → same slope; Perpendicular → negative reciprocal slopes.
2x – 3y = 12 and 3x + 2y = 9 → parallel, perpendicular, or neither?
Perpendicular
What theorem can be used to prove perpendicular lines form congruent adjacent angles?
Perpendicular Lines Theorem.