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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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MINI QUEST 11.1Angle Sums & IntersectionsRead the lesson
MINI QUEST 11.2Parallel Lines & TransversalsRead the lesson
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.

NOTICE → MARK → REASON

Two angles on a straight line

Mark the known angle and the missing angle in different colors. This is a separate learning diagram; it does not supply the values for your practice question.

Highlight parts and add your information

Choose a part here, or select it in the diagram. Mark what you know, what you need, or a reason. These are your scratch notes; the website does not grade or save them.

One small move: Select a part above. Find its given measurement in the question, then label that part below.

Part of the diagramWhat does your mark mean?Length, angle or reasoning

Use the problem’s stated values. A drawing can look shorter after rotation even when the actual length stays the same. Changing the model or reloading clears these notes.

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 =

\displaystyle 360^{\circ}

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:

\displaystyle a + b = 180^{\circ}

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 $\displaystyle 68^{\circ}$ . Find the other.

Their sum is $\displaystyle 180^{\circ}$ .
Subtract 68 from 180.

$\displaystyle 112^{\circ}$

WORKED EXAMPLE 2

Angles of $\displaystyle 85^{\circ}$ , $\displaystyle 110^{\circ}$ and $\displaystyle x^{\circ}$ fill a full turn with no overlap. Find x.

A full turn measures $\displaystyle 360^{\circ}$ .
Subtract the known angles.

$\displaystyle 165^{\circ}$

Common pitfalls

Possible mix-up: Any adjacent angles sum to $\displaystyle 180^{\circ}$ .

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

Problem 1
Two complementary angles measure $\displaystyle 34^{\circ}$ and $\displaystyle x^{\circ}$ . Find x.
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Problem 2
Two supplementary angles measure $\displaystyle 127^{\circ}$ and $\displaystyle x^{\circ}$ . Find x.
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Problem 3
Two straight lines intersect. One angle is $\displaystyle 73^{\circ}$ . What is the measure of its vertical opposite angle?
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Problem 4
Three nonoverlapping angles of $\displaystyle 95^{\circ}$ , $\displaystyle 138^{\circ}$ and $\displaystyle x^{\circ}$ fill one full turn. Find x.
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Problem 5
An angle is twice its complement. What is the angle in degrees?
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Problem 6
Two adjacent angles forming a straight angle measure $\displaystyle \left(3\, x + 8\right)^{\circ}$ and $\displaystyle \left(2\, x + 17\right)^{\circ}$ . Find x.
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Problem 7New context
A robot turns $\displaystyle 90^{\circ}$ clockwise, then $\displaystyle 145^{\circ}$ clockwise. How many more clockwise degrees complete exactly one full turn?
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Problem 8New context
Four paths meet at a point. Consecutive gaps measure $\displaystyle 2\, x^{\circ}$ , $\displaystyle 3\, x^{\circ}$ , $\displaystyle 4\, x^{\circ}$ and $\displaystyle 3\, x^{\circ}$ . They fill one turn without overlap. Find the largest gap in degrees.
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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:

\displaystyle a + b = 180^{\circ}

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 $\displaystyle 112^{\circ}$ . Find the interior angle on the same side of the transversal at the other line.

Same-side interior angles sum to $\displaystyle 180^{\circ}$ .
$\displaystyle 180 - 112 = 68$ .

$\displaystyle 68^{\circ}$

WORKED EXAMPLE 2

Corresponding angles measure $\displaystyle \left(4\, x + 6\right)^{\circ}$ and $\displaystyle \left(7\, x - 15\right)^{\circ}$ on parallel lines. Find x.

Set corresponding angles equal.
$\displaystyle 4\, x + 6 = 7\, x - 15$ , so $\displaystyle 21 = 3\, x$ .

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

Problem 1
Parallel lines are crossed by a transversal. An angle is $\displaystyle 64^{\circ}$ . What is its corresponding angle at the other intersection?
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Problem 2
With parallel lines, one alternate interior angle is $\displaystyle 118^{\circ}$ . What is the other alternate interior angle?
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Problem 3
Same-side interior angles on parallel lines are $\displaystyle 71^{\circ}$ and $\displaystyle x^{\circ}$ . Find x.
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Problem 4
Corresponding angles on parallel lines are $\displaystyle \left(5\, x - 4\right)^{\circ}$ and $\displaystyle 86^{\circ}$ . Find x.
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Problem 5
Same-side interior angles on parallel lines measure $\displaystyle 4\, x^{\circ}$ and $\displaystyle \left(2\, x + 30\right)^{\circ}$ . Find x.
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Problem 6
Can unequal corresponding angles prove two lines parallel? Enter yes or no.
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Problem 7New context
Two parallel roads are crossed by a straight path. An obtuse angle at one crossing is $\displaystyle 123^{\circ}$ . What is an acute angle at the other crossing?
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Problem 8New context
A transversal crosses parallel rails. Two alternate interior angles are $\displaystyle \left(3\, x + 14\right)^{\circ}$ and $\displaystyle \left(5\, x - 22\right)^{\circ}$ . What is their common angle measure?
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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.

\displaystyle a + b + c = 180^{\circ}

Triangulate a convex polygon

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

interior sum =

\displaystyle \left(n - 2\right)\, 180^{\circ}

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 =

\displaystyle \frac{360^{\circ}}{n}

WORKED EXAMPLE 1

A triangle has angles $\displaystyle x^{\circ}$ , $\displaystyle 2\, x^{\circ}$ and $\displaystyle 3\, x^{\circ}$ . Find the largest.

Six equal parts total $\displaystyle 180^{\circ}$ .
$\displaystyle x = 30^{\circ}$ , so $\displaystyle 3\, x = 90^{\circ}$ .

$\displaystyle 90^{\circ}$

WORKED EXAMPLE 2

Find each interior angle of a regular decagon.

The exterior angle is $\displaystyle \frac{360}{10} = 36^{\circ}$ .
The interior supplements it.

$\displaystyle 144^{\circ}$

Common pitfalls

Possible mix-up: All polygons have interior sum $\displaystyle 360^{\circ}$ .

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

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

Browse another level →

Angles & Geometric Relationships

Angle Sums & Intersections

Two angles on a straight line

An angle measures a turn

Adjacent angles add across a known region

Vertical angles follow from two straight lines

Two adjacent angles form a straight line. One measures 68∘\displaystyle 68^{\circ}68∘. Find the other.

Angles of 85∘\displaystyle 85^{\circ}85∘, 110∘\displaystyle 110^{\circ}110∘ and x∘\displaystyle x^{\circ}x∘ fill a full turn with no overlap. Find x.

Common pitfalls

Explain it to yourself

Parallel Lines & Transversals

A transversal connects two lines

Alternate interiors are equal

Do not omit the hypothesis

Parallel lines are cut by a transversal. One interior angle is 112∘\displaystyle 112^{\circ}112∘. Find the interior angle on the same side of the transversal at the other line.

Corresponding angles measure (4 x+6)∘\displaystyle \left(4\, x + 6\right)^{\circ}(4x+6)∘ and (7 x−15)∘\displaystyle \left(7\, x - 15\right)^{\circ}(7x−15)∘ on parallel lines. Find x.

Common pitfalls

Explain it to yourself

Triangle & Polygon Angle Sums

A triangle has a fixed interior sum

Triangulate a convex polygon

A walk around a convex polygon turns once

A triangle has angles x∘\displaystyle x^{\circ}x∘, 2 x∘\displaystyle 2\, x^{\circ}2x∘ and 3 x∘\displaystyle 3\, x^{\circ}3x∘. Find the largest.

Find each interior angle of a regular decagon.

Common pitfalls

Explain it to yourself

Two adjacent angles form a straight line. One measures $\displaystyle 68^{\circ}$ . Find the other.

Angles of $\displaystyle 85^{\circ}$ , $\displaystyle 110^{\circ}$ and $\displaystyle x^{\circ}$ fill a full turn with no overlap. Find x.

Parallel lines are cut by a transversal. One interior angle is $\displaystyle 112^{\circ}$ . Find the interior angle on the same side of the transversal at the other line.

Corresponding angles measure $\displaystyle \left(4\, x + 6\right)^{\circ}$ and $\displaystyle \left(7\, x - 15\right)^{\circ}$ on parallel lines. Find x.

A triangle has angles $\displaystyle x^{\circ}$ , $\displaystyle 2\, x^{\circ}$ and $\displaystyle 3\, x^{\circ}$ . Find the largest.