E2.3 · Solve with complementary, supplementary, opposite, and interior angles

Ontario Grade 6 Mathematics

Spatial Sense

Complementary, supplementary, opposite, and interior angles

What you will learn

• Identify complementary and supplementary angles and use their totals to find a missing angle.
• Use opposite angles to find a missing angle where two straight lines cross.
• Use the interior-angle totals of triangles and quadrilaterals to solve for missing angles.
• Check a solution by using the angle relationship again.

Use right-angle and straight-angle totals

• Complementary angles add to 90°.
• Supplementary angles add to 180°.
• To find one part, subtract the known measure from the correct total.

Use angles where straight lines cross

• Opposite angles at a crossing are equal.
• A pair of angles beside each other on a straight line adds to 180°.
• The position of an angle tells you which relationship to use.

Use interior angles in shapes

• The three interior angles of a triangle add to 180°.
• The four interior angles of a quadrilateral add to 360°.
• Add known angles first, then subtract from the shape's total.

A routine for solving and checking

• Choose the relationship before calculating.
• Use the matching total or equality.
• Check the result in the same relationship.

Worked example

Complete a right angle

1. Name the relationship
   The two angles fill a right angle, so they are complementary. Complementary angles add to 90°.
   $$a+b=90^\circ$$
2. Subtract the known angle
   The unknown angle is the part left after taking away 34° from the right-angle total.
   $$90^\circ-34^\circ=56^\circ$$
3. Check the total
   The two parts add back to 90°, so the missing measure fits the right angle.
   $$34^\circ+56^\circ=90^\circ$$

Worked example

Find opposite and beside angles

1. Find the opposite angle
   The angle directly across from 128° is an opposite angle. Opposite angles are equal, so it also measures 128°.
   $$128^\circ$$
2. Find an angle beside it
   The 128° angle and an angle beside it make a straight angle. Subtract 128° from 180° to find the beside angle.
   $$180^\circ-128^\circ=52^\circ$$
3. Check the relationships
   The opposite angle matches 128°, and the beside angle completes a straight angle with it.
   $$128^\circ+52^\circ=180^\circ$$

Worked example

Find a quadrilateral's fourth angle

1. Use the quadrilateral total
   A quadrilateral has four interior angles. Their measures add to 360°.
   $$a+b+c+d=360^\circ$$
2. Add the known angles
   Add the three given measures to find how much of the total is already accounted for.
   $$85^\circ+102^\circ+76^\circ=263^\circ$$
3. Subtract from the total
   The missing angle is the part left after subtracting 263° from 360°.
   $$360^\circ-263^\circ=97^\circ$$
4. Check all four angles
   Adding the missing angle back gives the quadrilateral's interior-angle total.
   $$85^\circ+102^\circ+76^\circ+97^\circ=360^\circ$$

Common mistakes and how to avoid them

Lesson summary

• Complementary angles add to 90°; supplementary angles add to 180°.
• Opposite angles where straight lines cross are equal.
• Triangle interior angles add to 180°; quadrilateral interior angles add to 360°.
• Name the relationship, find the missing measure, and check it.

Check your understanding

Key terms

Angle

The opening between two rays that meet at a point.

Vertex

The point where the two rays of an angle meet.

Complementary angles

Two angles whose measures add to 90°.

Supplementary angles

Two angles whose measures add to 180°.

Opposite angles

Angles directly across from each other where two straight lines cross.

Interior angle

An angle inside a shape, at one of its corners.

Quadrilateral

A shape with four straight sides.

Continue through Grade 6 Math

View the complete Grade 6 Math Ontario Grade 6 Mathematics curriculum and lessons

• E2.2 · Measure and construct angles to 360 degrees
• E2.4 · Find areas of trapezoids, rhombuses, kites, and composite polygons
• E1.1 · Classify quadrilaterals using sides, diagonals, and symmetry
• E1.2 · Build three-dimensional objects from top, front, and side views
• E1.3 · Plot four-quadrant coordinates and describe translations
• E1.4 · Combine translations, reflections, and rotations on a grid

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation E2.3. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

Official curriculum reference

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