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E2.2 · Solve angle problems with lines and polygons

Learn to solve angle problems with lines and polygons through clear examples and targeted practice.

Ontario Grade 8 Mathematics

Spatial Sense

Use angle relationships to find missing measures

An angle measures the turn between two lines or line segments. We measure angles in degrees. In this lesson, you will use a few reliable angle facts to solve problems with intersecting lines and polygons. A polygon is a closed shape made from straight sides. Before finding a missing angle, first notice how the lines or sides are arranged.

What you will learn

  • Recognize straight angles, vertically opposite angles, and angles around a point.
  • Use angle sums in triangles and quadrilaterals to find a missing angle.
  • Choose a useful angle rule and explain why it applies.

Start with angles made by lines

Imagine a straight road with a smaller road crossing it. The two angles on one side of the straight road together make a half-turn. A half-turn measures 180°. So, if one angle is 65°, the next angle beside it is 115°, because 65° + 115° = 180°.
Angles beside each other that share a side and a vertex are adjacent angles. When their other sides form a straight line, they are a linear pair. Their measures add to 180°. The vertex is the point where the sides of an angle meet.
When two lines cross, the angles directly across from each other are vertically opposite angles. They have equal measures. For example, if one angle is 72°, the angle directly across from it is also 72°. The two remaining angles are each 108°, since they sit beside a 72° angle on a straight line.
Several angles meeting at one point make a full turn when they go all the way around. Their measures add to 360°. This is useful when three or more lines meet at one point.
180∘,360∘180^\circ,\quad 360^\circ
  • Angles on a straight line add to 180°.
  • Vertically opposite angles are equal.
  • Angles around a point add to 360°.

Use angle sums in polygons

A triangle has three interior angles. Interior angles are the angles inside a shape. The three angles in any triangle add to 180°. If two angles are 48° and 67°, the third is 65°, because 48° + 67° + 65° = 180°.
A quadrilateral is a polygon with four sides. Its four interior angles add to 360°. One way to see this is to draw a diagonal from one corner to the opposite corner. The diagonal divides the quadrilateral into two triangles. Each triangle has an angle sum of 180°, so together the angles add to 360°.
Use the shape to choose the rule. For a triangle, use 180°. For a quadrilateral, use 360°. For angles on a straight line or around a point, use the matching line rule. Then subtract the known angles from the total to find the missing measure.
Keep the unit, degrees, in your answer. Check that the result makes sense. A missing angle in a triangle must leave the three angles adding to 180°.
△:180∘,quadrilateral:360∘\triangle: 180^\circ,\quad \text{quadrilateral}: 360^\circ
  • The interior angles of a triangle add to 180°.
  • The interior angles of a quadrilateral add to 360°.
  • A diagonal can split a quadrilateral into two triangles.

A dependable solving plan

First, mark the angle you need to find. Next, name the fact that fits the picture: straight line, opposite angles, full turn, triangle, or quadrilateral. Write the total or equality for that fact.
Then combine the known angle measures and find what is missing. You can use subtraction when the total is known. Finally, check by putting the missing measure back into the angle relationship.
Do not choose a rule only because a number looks familiar. The angles must actually form the relationship named by the rule. For instance, opposite angles are equal only when two straight lines cross.
  • Identify the shape or line arrangement before calculating.
  • Use only the angle fact that matches the diagram.
  • Check the answer in the original angle relationship.

Practice with care

For each practice question, sketch or mark the stated shape and label the known angles. Say the rule in words before you calculate. This helps prevent using a triangle rule on a straight line, or a straight-line rule on a whole polygon.
Try these: two adjacent angles form a straight line and one is 124°; find the other. A triangle has angles 39° and 86°; find the third. A quadrilateral has angles 81°, 95°, and 102°; find the fourth.
Answers: the adjacent angle is 56°, the triangle angle is 55°, and the quadrilateral angle is 82°. For each, add the answer to the given angles to check the correct total.
  • Show the angle relationship before working out the missing measure.
  • Use addition to check a missing angle.

Worked example

Two lines cross

Two straight lines cross. One angle measures 74°. Find the vertically opposite angle and either angle beside the 74° angle.
Crossing lines
Crossing linesThe angles opposite each other are equal; adjacent angles on a straight line total 180°.xyABCDO

The angles opposite each other are equal; adjacent angles on a straight line total 180°.

  1. Find the opposite angle
    The two lines cross, so the angle directly opposite the 74° angle is vertically opposite to it. Vertically opposite angles are equal.
    74∘74^\circ
  2. Find an adjacent angle
    An angle beside the 74° angle and the 74° angle make a straight line. Their measures add to 180°. Subtract 74° from 180°.
    180∘−74∘=106∘180^\circ-74^\circ=106^\circ
Answer: The vertically opposite angle is 74°. Each adjacent angle is 106°.
Check: The adjacent angle checks because 74° + 106° = 180°. The crossing lines create two pairs of equal opposite angles.

Worked example

Missing angle in a triangle

A triangle has two interior angles measuring 52° and 71°. Find its third interior angle.
Triangle with three interior angles
Triangle with three interior anglesThe triangle's angles are labelled 52°, 71°, and 57°; their sum is 180°.xyABC

The triangle's angles are labelled 52°, 71°, and 57°; their sum is 180°.

  1. Choose the total
    The shape is a triangle, so its three interior angles add to 180°. Add the two known angles first.
    52∘+71∘=123∘52^\circ+71^\circ=123^\circ
  2. Find the remaining measure
    Subtract the known total from 180° to find the angle still needed to complete the triangle's angle sum.
    180∘−123∘=57∘180^\circ-123^\circ=57^\circ
Answer: The third interior angle measures 57°.
Check: The three angles add to 180°: 52° + 71° + 57° = 180°.

Worked example

Missing angle in a quadrilateral

Three interior angles of a quadrilateral are 88°, 104°, and 97°. Find the fourth angle.
Quadrilateral with four interior angles
Quadrilateral with four interior anglesThe fourth angle is 71°; together the four interior angles total 360°.xyABCD

The fourth angle is 71°; together the four interior angles total 360°.

  1. Add the known angles
    A quadrilateral has four interior angles that add to 360°. Add the three known measures.
    88∘+104∘+97∘=289∘88^\circ+104^\circ+97^\circ=289^\circ
  2. Subtract from the total
    The missing angle is the amount needed to bring the known total up to 360°.
    360∘−289∘=71∘360^\circ-289^\circ=71^\circ
Answer: The fourth interior angle measures 71°.
Check: Adding all four angles gives 360°: 88° + 104° + 97° + 71° = 360°.

Common mistakes and how to avoid them

Adding adjacent angles to 360° even though they form a straight line.
Correction: Angles along a straight line add to 180°. Use 360° only for a full turn around a point or for the interior angles of a quadrilateral.
Treating angles beside each other as equal.
Correction: Adjacent angles are not always equal. If they form a straight line, they add to 180°. Vertically opposite angles, not adjacent angles, are equal.
Using 180° as the total for a quadrilateral.
Correction: A quadrilateral has four interior angles with a total of 360°. The 180° total applies to a triangle.

Lesson summary

  • Angles on a straight line add to 180°, and angles around a point add to 360°.
  • Vertically opposite angles are equal.
  • The interior angles of a triangle add to 180°; those of a quadrilateral add to 360°.
  • Choose the rule that matches the diagram, calculate the missing angle, and check the total.

Check your understanding

Question 1

Two angles form a straight line. One is 113°. What is the other?
  1. 67°
  2. 113°
  3. 247°
  4. correctIndex imprecise
Show answer and explanation
67°
Angles on a straight line add to 180°, and 180° − 113° = 67°.

Question 2

Two straight lines cross. One angle is 46°. What is the angle directly opposite it?
  1. 134°
  2. 46°
  3. 90°
  4. correctIndex imprecise
Show answer and explanation
46°
Vertically opposite angles are equal, so the opposite angle is 46°.

Question 3

A quadrilateral has interior angles 90°, 110°, and 75°. What is its fourth interior angle?
  1. 85°
  2. 95°
  3. 275°
  4. correctIndex imprecise
Show answer and explanation
85°
The first three angles total 275°. A quadrilateral totals 360°, so the fourth angle is 360° − 275° = 85°.

Key terms

Adjacent angles
Two angles beside each other that share a vertex and one side.
Interior angle
An angle inside a polygon.
Linear pair
Two adjacent angles whose outside sides form a straight line.
Polygon
A closed shape made from straight sides.
Vertically opposite angles
The pair of angles directly across from each other when two straight lines cross.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation E2.2. It is a study resource, not an official curriculum publication.

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