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Step 9: Proportional reasoning

Scale drawings and angle relationships

Step 9 · Proportional reasoning topics

Use a scale to go between a drawing and real lengths, and find unknown angles from complementary, supplementary and vertical angles and the angle sum of a triangle.

Scale drawings

A scale drawing shows a real object with every length multiplied by the same number, the scale factor. A scale such as 1 cm : 5 km says how far each centimeter on the drawing stands for.

Example: Using a scale

On a map, 1 cm stands for 5 km. Two towns are 7.5 cm apart on the map. How far apart are they?

  1. Each cm stands for 5 km.
  2. 7.5 × 5 = 37.5.

Answer37.5 km

Example: Finding the scale

A 12 m wall is drawn 4 cm long. What is the scale?

  1. 12 ÷ 4 = 3.
  2. Each cm stands for 3 m.

Answer1 cm : 3 m

A 3 by 2 room (dark) drawn at scale factor 2 becomes 6 by 4: every length doubles, and the area is 4 times as big.

Angle pairs

Some pairs of angles always add to the same total. Complementary angles add to 90°. Supplementary angles add to 180°, like two angles side by side on a straight line. Vertical angles sit opposite each other where two lines cross, and they are equal.

Two straight lines cross. Angles opposite each other (vertical angles) are equal; angles next to each other add to 180°.

Example: Supplementary

Two angles sit on a straight line. One is 128°. What is the other?

  1. They add to 180°.
  2. 180 − 128 = 52.

Answer52°

Example: Vertical angles

Two lines cross and one angle is 40°. What are the other three?

  1. The angle opposite is also 40°.
  2. Each angle next to it is 180 − 40 = 140.

Answer40°, 140° and 140°

Example: With an equation

Two complementary angles measure x and 2x. What is x?

  1. x + 2x = 90.
  2. 3x = 90, so x = 30.

Answer30°

Angles in a triangle

The three angles of any triangle add to 180°.

A triangle with angles of 50°, 70° and 60°: 50 + 70 + 60 = 180.

Example: A missing angle

A triangle has angles of 50° and 75°. What is the third?

  1. 50 + 75 = 125.
  2. 180 − 125 = 55.

Answer55°

Example: An isosceles triangle

An isosceles triangle has a top angle of 40°, and its two base angles are equal. What is each base angle?

  1. 180 − 40 = 140.
  2. 140 ÷ 2 = 70.

Answer70°

Try it yourself

Work each one on paper first, then open the answer to check.

  1. A model car is built at a scale of 1 : 24. The model is 18 cm long. How long is the real car?

    Show answer

    432 cm: 18 × 24 = 432.

  2. A plan uses 1 inch : 4 feet. A room is 14 feet long. How long is it on the plan?

    Show answer

    3.5 inches: 14 ÷ 4 = 3.5.

  3. What angle is complementary to 35°?

    Show answer

    55°

  4. What angle is supplementary to 35°?

    Show answer

    145°

  5. Two angles of a triangle are 90° and 27°. What is the third?

    Show answer

    63°

  6. A square is drawn at scale factor 2. How many times as big is its area?

    Show answer

    4 times

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