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Prealgebra / LEVEL 11 · DIFFICULTY 4/5

Angles & Geometric Relationships

Use stated geometric relationships to justify angle calculations.

3 stages · 24 practice problems · two 6-question assessment forms

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  1. MINI QUEST 11.1Angle Sums & IntersectionsRead the lesson
  2. MINI QUEST 11.2Parallel Lines & TransversalsRead the lesson
  3. MINI QUEST 11.3Triangle & Polygon Angle SumsRead the lesson
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STAGE 11.1 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Angle Sums & Intersections

Goal: Recognize full turns, straight angles and vertical-angle equality.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

An angle measures a turn

A full turn is 360 degrees, a straight angle is 180 degrees and a right angle is 90 degrees. These are measures, not lengths.

full turn = 360°

Adjacent angles add across a known region

Nonoverlapping angles that fill a straight angle sum to 180 degrees. Complementary angles sum to 90; supplementary angles sum to 180.

straight-line pair: a + b = 180°

Vertical angles follow from two straight lines

Opposite angles formed by two intersecting straight lines are equal because each is supplementary to the same adjacent angle.

vertical angles have equal measures
WORKED EXAMPLE 1

Two adjacent angles form a straight line. One measures 68°. Find the other.

  1. Their sum is 180°.
  2. Subtract 68 from 180.

112°

WORKED EXAMPLE 2

Angles of 85°, 110° and x° fill a full turn with no overlap. Find x.

  1. A full turn measures 360°.
  2. Subtract the known angles.

165°

Common pitfalls

Possible mix-up: Any adjacent angles sum to 180°.

They must form a straight angle to use that sum.

Possible mix-up: Vertical angles share a side.

Vertical angles are opposite at an intersection; adjacent angles share a side.

Explain it to yourself

Prove vertical angles equal by using two supplementary-angle equations.

Preview the eight practice prompts
  1. Two complementary angles measure 34° and x°. Find x.
  2. Two supplementary angles measure 127° and x°. Find x.
  3. Two straight lines intersect. One angle is 73°. What is the measure of its vertical opposite angle?
  4. Three nonoverlapping angles of 95°, 138° and x° fill one full turn. Find x.
  5. An angle is twice its complement. What is the angle in degrees?
  6. Two adjacent angles forming a straight angle measure (3x + 8)° and (2x + 17)°. Find x.
  7. A robot turns 90° clockwise, then 145° clockwise. How many more clockwise degrees complete exactly one full turn? New context
  8. Four paths meet at a point. Consecutive gaps measure 2x°, 3x°, 4x° and 3x°. They fill one turn without overlap. Find the largest gap in degrees. New context
Open stage 11.1 in the student workspace →

STAGE 11.2 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Parallel Lines & Transversals

Useful preparation: Angle Sums & Intersections · Solving & Checking Linear Equations

Goal: Use parallel-line hypotheses to connect angles at different intersections.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A transversal connects two lines

Label angles by their positions before applying a rule. Corresponding positions align at the two intersections.

corresponding angles are equal when lines are parallel

Alternate interiors are equal

For parallel lines, angles inside the strip and on opposite sides of the transversal are equal. Same-side interior angles are supplementary.

same-side interior: a + b = 180°

Do not omit the hypothesis

These cross-intersection rules require parallel lines, or can sometimes be used in reverse to prove parallelism. Vertical-angle rules need only the intersection itself.

equal corresponding angles ⇒ parallel lines
WORKED EXAMPLE 1

Parallel lines are cut by a transversal. One interior angle is 112°. Find the interior angle on the same side of the transversal at the other line.

  1. Same-side interior angles sum to 180°.
  2. 180 − 112 = 68.

68°

WORKED EXAMPLE 2

Corresponding angles measure (4x + 6)° and (7x − 15)° on parallel lines. Find x.

  1. Set corresponding angles equal.
  2. 4x + 6 = 7x − 15, so 21 = 3x.

7

Common pitfalls

Possible mix-up: Alternate interior angles are always supplementary.

With parallel lines they are equal.

Possible mix-up: Any two lines cut by a transversal have equal corresponding angles.

That equality requires parallel lines.

Explain it to yourself

Which angle facts remain valid if the two lines are not parallel?

Preview the eight practice prompts
  1. Parallel lines are crossed by a transversal. An angle is 64°. What is its corresponding angle at the other intersection?
  2. With parallel lines, one alternate interior angle is 118°. What is the other alternate interior angle?
  3. Same-side interior angles on parallel lines are 71° and x°. Find x.
  4. Corresponding angles on parallel lines are (5x − 4)° and 86°. Find x.
  5. Same-side interior angles on parallel lines measure 4x° and (2x + 30)°. Find x.
  6. Can unequal corresponding angles prove two lines parallel? Enter yes or no.
  7. Two parallel roads are crossed by a straight path. An obtuse angle at one crossing is 123°. What is an acute angle at the other crossing? New context
  8. A transversal crosses parallel rails. Two alternate interior angles are (3x + 14)° and (5x − 22)°. What is their common angle measure? New context
Open stage 11.2 in the student workspace →

STAGE 11.3 · 8 PRACTICE PROBLEMS · 2 NEW-CONTEXT APPLICATIONS

Triangle & Polygon Angle Sums

Useful preparation: Parallel Lines & Transversals

Goal: Use triangulation and turning angles to find unknown measures.

Before you begin: Before starting, explain one example from each prerequisite. If an idea is unfamiliar, follow the prerequisite lesson links.

A triangle has a fixed interior sum

The three interior angles of a Euclidean triangle sum to 180 degrees. An exterior angle obtained by extending one side equals the sum of the two remote interiors.

a + b + c = 180°

Triangulate a convex polygon

Drawing diagonals from one vertex partitions a convex n-gon into n − 2 triangles, giving its interior sum.

interior sum = (n − 2)180°

A walk around a convex polygon turns once

Taking one exterior turning angle at each vertex totals 360 degrees. In a regular polygon those exterior angles are equal.

regular exterior angle = 360°/n
WORKED EXAMPLE 1

A triangle has angles x°, 2x° and 3x°. Find the largest.

  1. Six equal parts total 180°.
  2. x = 30°, so 3x = 90°.

90°

WORKED EXAMPLE 2

Find each interior angle of a regular decagon.

  1. The exterior angle is 360/10 = 36°.
  2. The interior supplements it.

144°

Common pitfalls

Possible mix-up: All polygons have interior sum 360°.

The sum depends on the number of sides.

Possible mix-up: Equilateral and equiangular mean the same for every polygon.

These are different properties; a regular polygon has both.

Explain it to yourself

Explain why the exterior turning-angle sum does not depend on the number of sides.

Preview the eight practice prompts
  1. A triangle has angles 48° and 67°. Find its third angle.
  2. An isosceles triangle has vertex angle 38°. Find either equal base angle.
  3. Find the interior-angle sum of a heptagon, in degrees.
  4. Find each exterior turning angle of a regular nonagon.
  5. A regular polygon has each exterior angle 24°. How many sides does it have?
  6. A triangle exterior angle is 132°. One remote interior angle is 57°. Find the other remote interior angle.
  7. A designer builds a regular 12-sided window. What is each interior angle? New context
  8. The interior angles of a convex pentagon are x°, x°, x°, 120° and 135°. Find x. New context
Open stage 11.3 in the student workspace →
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