Angles 2/26/2026

How are the Angles Related?

Warm-Up Exercises

  • Task 1: Calculate $-10 - (-8)$

  • Task 2: Calculate $(-3) + (5/4) - (7 - 4)^{2}$

  • Task 3: Calculate $5(2.6 - 2.25)$

  • Task 4: Calculate $35 + (-23) - (-6)$

Agenda

  • Warm-up

  • Core Problems 1-7

  • Closure

  • Homework: None

Learning Target

  • Objective: To learn how to use angle relationships, such as:

    • Triangle angle sum

    • Exterior angle of triangles

    • Angle relationships created by parallel lines cut by a transversal

  • Goal: Solve equations to find unknown angle measures in order to construct viable arguments and critique the reasoning of others.

Parallel Line Angle Pair Relationships

  • Source: Course Mathematics (Grade 8) Chapter 9

Introduction to Topic

  • Previous Learning: Students have previously learned vocabulary and considered relationships created by two intersecting lines.

  • New Focus: Examination of vocabulary and relationships created when a line intersects two parallel lines.

Learning Intentions

  • Goal: To understand and apply angle relationships formed when parallel lines are cut by a transversal.

Success Criteria

  • Students will demonstrate understanding of the angles formed under the conditions described (parallel lines cut by a transversal).

Notation and Symbols

  • Arrowheads at line ends indicate lines extend indefinitely.

  • Marks such as > and >> indicate that lines or segments are parallel.

  • A small box at the intersection point signifies that the lines are perpendicular, forming right angles.

Transversals

  • Definition: Lines that intersect two or more lines at different points.

    • Example:

    • Figure 1: Line $s$ parallel to line $t$, intersected by transversal line $r$.

    • Figure 2: Lines $x$ and $y$ are not parallel; line $w$ intersects them.

  • Observations: Transversals can intersect several lines, each at different points.

Angle Observation Activities

Task Instructions

  1. Place a sheet of paper over angle $ heta_a$ from page 391, copying the angle using a ruler.

  2. Slide the copy of angle $ hetaa$ to angle $ hetac$ and compare sizes.

  3. Repeat for angle pairs:

    • Trace $ hetab$ over $ hetad$.

    • Trace $ hetae$ over $ hetag$.

    • Trace $ hetaf$ over $ hetah$.

  4. Summarize the findings: Describe angle relationships involving congruency and the implication for parallel lines.

    • Include observations about when angles are congruent and what must be true if the lines are parallel.

Resource Page Description

  • Contains figures depicting pairs of parallel lines (p and q) intersected by transversal line (m).

  • Task: Calculate angle measures of angles $b$, $d$, $f$, $k$, $r$, and $s$ based on straight angle properties.

Angle Relationships

Corresponding Angles

  • Definition: Angles on the same side of two lines and same side of the transversal that intersects them.

    • Examples found on the resource page include:

    • Angles $ hetaa$ and $ hetad$;

    • Angles $ hetag$ and $ hetak$;

    • Angles $ hetar$ and $ hetat$.

  • Conjecture: If two parallel lines are cut by a transversal, then pairs of corresponding angles are congruent (equal in measure).

Alternate Interior Angles

  • Definition: Angles between a pair of lines and on opposite sides of a transversal.

    • Examples:

    • Angles $ hetac$ and $ hetad$;

    • Angles $ hetah$ and $ hetaj$;

    • Angles $ hetaw$ and $ hetas$.

  • Conjecture: If parallel lines are cut by a transversal, then alternate interior angles are congruent.

Same Side Interior Angles

  • Definition: Angles located on the same side of a transversal and between the two parallel lines.

  • Task: Examine angle pairs $b$ and $d$, $g$ and $j$, $r$ and $s$ for their sum and make conjectures regarding these relationships.

  • Conjecture: The sum of the measures of two interior angles on the same side of the transversal is supplementary to $180^{ ext{o}}$.

Solving Angle Measures

  • Example Problems:

  1. If $m∠2=67^{ ext{o}}$, determine $m∠5$.

  2. For angles defined as $m∠6 = 4x + 23^{ ext{o}}$ and $m∠4 = 3x + 17^{ ext{o}}$, find $m∠6$ by solving for $x$.

Classification Task

  • Classification of angle pairs as:

    • Corresponding

    • Alternate interior

    • Same side interior

    • Straight

    • None of the above

  • Identify conditions necessary for equal corresponding or alternate interior angles.

Practical Application of Conjectures

  • Use angle relationships learned to find angle measures in various figures. Present step-by-step procedures and justify calculations referencing angle conjectures (corresponding, alternate interior, vertical, straight).

Summary of Angle Relationships

  • Review the angle relationships covered throughout the lessons and utilize geometry vocabulary in explaining, supported by diagrams.