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15.3 Accurate Drawings 1

GCSE Maths Foundation · Edexcel · Chapter 15: Constructions, Loci and Bearings

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: congruent triangles and constructing triangles with a ruler, protractor and compasses. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Know the four conditions for congruent triangles: SSS, SAS, ASA and RHS
  • Use congruence to find missing sides and angles
  • Construct a triangle given three sides, using compasses
  • Construct a triangle given two sides and an angle, or two angles and a side

Key words

  • Congruent: Exactly the same shape and size. One shape may be turned over or rotated.
  • Matching parts: Corresponding sides or angles: in the same position in each shape, e.g. matching sides or angles.
  • Included angle: The angle between two given sides.
  • Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
  • Construct: Draw accurately using a ruler, a protractor or compasses. Leave all arcs showing.

Conditions for congruent triangles

Two triangles are congruent if one fits exactly on top of the other. You do not need to know all six measurements: any one of these four sets of facts is enough.

Key fact: Three equal angles (AAA) is not enough. The triangles have the same shape but can be different sizes.

Worked example: which condition?

Triangle 1 has sides 5 cm and 8 cm with an angle of 62° between them.
Triangle 2 has sides 8 cm and 5 cm with an angle of 62° between them.
Two sides and the included angle are equal.

Answer: congruent, by SAS

Using congruent triangles

When two triangles are congruent, every pair of corresponding sides is equal and every pair of corresponding angles is equal. Match the vertices carefully: if triangle ABC is congruent to triangle PQR, then A matches P, B matches Q and C matches R.

Worked example: finding a missing angle

Triangle ABC is congruent to triangle PQR. Angle A = 47° and angle B = 71°. Find angle R.
R matches C. Angle C = 180° − 47° − 71° = 62°.

Answer: angle R = 62°

Constructing a triangle from three sides (SSS)

Worked example: sides 7 cm, 5 cm and 6 cm

1. Draw the longest side AB = 7 cm with a ruler.
2. Open the compasses to 5 cm. Put the point on A and draw an arc above the line.
3. Open the compasses to 6 cm. Put the point on B and draw an arc that crosses the first arc.
4. The arcs cross at C. Join AC and BC.

Answer: triangle ABC, with both construction arcs left showing

Key fact: Three lengths make a triangle only if the two shorter sides add up to more than the longest side. 3 cm, 4 cm and 8 cm do not make a triangle because 3 + 4 < 8.

Constructing triangles with a protractor (SAS and ASA)

Worked example: SAS

Draw AB = 6 cm. At A, measure 50° with a protractor and draw a long faint line.
Measure 4.5 cm along this line from A and mark C. Join C to B.

Answer: BC measures about 4.6 cm

Worked example: ASA

Draw AB = 6.5 cm. Measure 40° at A and 65° at B. Extend both lines until they meet at C.
Check with an angle fact: angle C should be 180° − 40° − 65° = 75°.

Answer: triangle ABC with angle C = 75°

Exam tip: Accuracy in the exam: lengths to within 2 mm and angles to within 2°.

Common mistakes

  • Rubbing out construction arcs. Examiners need to see them for the marks.
  • Using AAA as a reason for congruence. It only shows the triangles are similar.
  • Matching the wrong vertices: in ABC ≡ PQR, side BC matches QR, not PR.
  • Reading the wrong scale on the protractor. Check whether your angle should be acute or obtuse.

Revision questions

Try these, then check your answers below each one.

Question 1. Triangle P has sides 6 cm, 8 cm and 9 cm. Triangle Q has sides 9 cm, 6 cm and 8 cm. Which condition shows they are congruent? (Choose from: SSS, SAS, ASA, RHS.)

Answer: SSS

Working: All three sides match, so the triangles are congruent by SSS.

Question 2. Two triangles both have angles of 30°, 60° and 90°. What can you say? (Choose from: They are not necessarily congruent, They are congruent by ASA, They are congruent by RHS, They are congruent by AAA.)

Answer: They are not necessarily congruent

Working: Equal angles only give the same shape. The sizes can differ, so they are not necessarily congruent.

Question 3. Triangle ABC is congruent to triangle DEF, with A matching D, B matching E and C matching F. Angle B = 48° and angle C = 75°. Work out angle D in degrees.

Answer: 57 °

Working: Angle A = 180 - 48 - 75 = 57°, and angle D = angle A = 57°.

Related lessons

Continue in the full course

This lesson is section 15.3 (EGCMF1503) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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