Parallel Lines & Transversals - Vocabulary

Core Concepts

• When two parallel lines are cut by a transversal, several angle relationships hold that let you transfer information from one angle to others in the diagram.
• Angles in certain positions are congruent; others are supplementary.
• If you know one angle measure, you can determine all others in the diagram by chaining these relationships.

Key Angle Relationships (theorems)

• Corresponding angles are congruent:
  • If an angle at one intersection is in a given relative position to the transversal, the angle in the same relative position at the other intersection has the same measure. Symbolically, if ∠A corresponds to ∠B, then m∠A = m∠B.
• Alternate interior angles are congruent:
  • The pair of interior angles on opposite sides of the transversal have equal measures: m∠(interior ext{ on left}) = m∠(interior ext{ on right}).
• Alternate exterior angles are congruent:
  • The pair of exterior angles on opposite sides of the transversal are equal: m∠(exterior ext{ left}) = m∠(exterior ext{ right}).
• Consecutive interior angles are supplementary (same-side interior):
  • The two interior angles on the same side of the transversal sum to 180°: m∠(interior ext{ A}) + m∠(interior ext{ B}) = 180°.
• Vertical angles are congruent:
  • Angles opposite each other at an intersection are equal: m∠A = m∠C.
• Linear pair (adjacent angles on a straight line) are supplementary:
  • If two angles form a linear pair, their measures sum to 180°: m∠A + m∠B = 180°.

Notation and practice principles

• Angles are typically labeled around the intersections (e.g., 1–8 in common diagrams) and we refer to them as ∠1, ∠2, …, ∠8.
• The measure of an angle is denoted as m∠X (or m∠X°).
• If a given angle measure is known, use the above relationships to deduce the rest:
  • Vertical angle to a given angle: same measure as the given angle.
  • Linear pair: supplementary to the given angle ⇒ value = 180° − m∠given.
  • Corresponding/alternate interior/exterior: the corresponding angle, or the alternate interior/exterior angle, has the same measure as the given angle.
  • At the opposite intersection, corresponding angles have the same measure as the angle at the original intersection.

Example 1 (Page 1): Given m∠? = 65°

• If one angle is 65°, then:
  • The vertical angle is also 65°.
  • The corresponding angle (at the other intersection, in the same relative position) is 65°.
  • The alternate interior angle(s) associated with the given angle are 65°.
  • The alternate exterior angle(s) associated with the given angle are 65°.
  • The linear pair (the angle adjacent on a straight line) is 180° - 65° = 115°.
  • Consequently, the obtuse angles in the diagram are 115°, and the acute ones are 65°.
• Therefore, if a problem states “Given m∠2 = 65°, find the missing angles,” you can fill in:
  • All corresponding/alternate interior/exterior angles: m∠ = 65°.
  • All linear-pair angles: m∠ = 115°.
  • Opposite angle at the same intersection: m∠ = 65°.
• Summary pattern: with a single acute angle of 65°, the diagram contains only 65° and 115° angles.

Example 2 (Page 1): Given m∠6 = 142°

• If one angle is 142°, then the complementary set of angle measures will be 142° for angles in the same congruent positions (corresponding/alternate interior/exterior/vertical) and 38° for the angles that are linear-pair or supplementary to those.
  • Linear pair to 142°: 180° - 142° = 38°.
  • Therefore, the diagram contains angles of 142° and 38° only.
• Practical steps:
  • Identify which angles are in the same position across the two intersections (congruent to 142°).
  • Identify which angles form linear pairs with 142° (or are consecutive interior with an interior angle equal to 142°). Those will be 38°.

Example 4 & 5 (Page 2): Two given angles to deduce the rest

• General approach when two angle measures are given (e.g., m∠12 = 121° and m∠6 = 75°):
  • Use vertical and linear-pair relationships to propagate known values to adjacent angles.
  • Use corresponding/alternate interior/exterior relationships to transfer known values to angles at the other intersection.
  • Where a pair of angles are known to be congruent, set their measures equal; where they are supplementary, set their sum to 180°.
  • Propagate step-by-step to fill all missing angles.
• Example outcome pattern (without diagram-specific labels):
  • From a given angle of 121°: the linear pair is 59°; any corresponding/alternate angle to 121° is 121°; the corresponding/alternate angles to 59° are 59°.
  • From a given angle of 75°: the linear pair is 105°; any corresponding/alternate angle to 75° is 75°; the corresponding/alternate angles to 105° are 105°.
• The exact numeric values for each missing angle depend on the labels of the angles (which one is corresponding to which, etc.). The key is to apply the same set of rules as above and propagate consistently.

Parallel Lines, Transversals, and Algebra (solving with variables)

• Directions from the transcript:
  • If lines are parallel, solve for the missing variable(s) in angle expressions.
• Common structure in these problems:
  • The angle measures are given as algebraic expressions (e.g.,
    (ax + b) ) or constants, and you must determine x using angle relationships.
  • Two primary types of equations arise:
  • Equality (congruence): corresponding angles or vertical angles yield m∠A = m∠B \,\Rightarrow\, ax + b = cx + d.
  • Supplementarity: same-side interior or linear-pair yields m∠A + m∠B = 180 \,\Rightarrow\, (ax + b) + (cx + d) = 180.
• Common templates you will encounter:
  • Corresponding/Vertical: ax + b = cx + d.
  • Consecutive interior: ax + b + cx + d = 180.
• After solving for x:
  • Substitute back to determine the numeric angle measures for the required angles.
• Example note on the transcript items (1–6):
  • Each item presents a pair of algebraic expressions or a combination of an algebraic expression and a numeric angle, and you determine x by applying the appropriate relation (equal or supplementary).
  • Then compute the referenced angle measures from the found x-value.

Notation recap and practical tips

• Always label angles by their position relative to the transversal (top intersection vs bottom; left vs right) to decide which relationship applies (corresponding, alternate interior/exterior, linear pair).
• If an angle is known to be acute (e.g., 65°) in a diagram with parallel lines, all acute angles will be congruent to that value, and all obtuse angles will be the supplementary 180° − value.
• If an angle is known to be obtuse (e.g., 142°), the complementary set of angles will be 142° and 38° accordingly.
• In algebraic problems, start by identifying whether the given pair of angles are congruent or supplementary, then set up the equation(s) and solve for x. Substitute back to obtain the requested angle measures.

Quick reference: relationship-to-equation mapping

• Congruent (corresponding, alternate interior, alternate exterior, vertical):
  • Equation form: \text{expression}1 = \text{expression}2
• Supplementary (same-side interior, linear pair):
  • Equation form: \text{expression}1 + \text{expression}2 = 180

Practical tips for exam prep

• Memorize the two core numerical results that arise from a single given angle on parallel lines:
  • The linear pair is 180° apart: sum to 180°.
  • Corresponding/alternate angles are equal to the given angle (and propagate to the other intersection).
• Practice with both numeric angles (e.g., 65°, 115°) and algebraic expressions to become fluent at switching between equalities and sums to 180°.
• When solving algebraic problems, keep track of units (degrees) and ensure your x-value yields valid angle measures (0° < m∠ ≤ 180°) for all angles involved.

Summary takeaway

• Parallel lines cut by a transversal create a rigid set of angle relationships that dramatically reduce the amount of work needed to find all angle measures.
• Vertical angles are equal; linear pairs sum to 180°; corresponding/alternate interior/exterior angles are equal across the two intersections.
• With a single angle measure, you can deduce the rest; with two algebraic expressions, use equality or summation-to-180° to form solvable equations for the unknowns.