Congruent triangles

Skills and methods in this topic, easy to hard

1. 1Congruent figures and transformationsTranslate, rotate, reflect: congruent; enlarge or reduce: not

   MethodSides 3 and 4 enlarged twice become 6 and 8: not congruent

   Another way: Tracing paperSlid, turned or flipped, it covers the other exactly: congruent

2. 2Corresponding parts of congruent trianglesMatch the letters in order: △ABC ≅ △PQR, AB = PQ, ∠C = ∠R

   Method△ABC ≅ △PQR, BC = 7 cm: QR = 7 cm (corr. sides, ≅△s)

   Another way: By the picture: equal angles face matching sides∠A = ∠P, so BC goes with QR

3. 3Conditions for congruent trianglesSSS, SAS, ASA, AAS, RHS; never AAA or SSA

   MethodAB = PQ, ∠B = ∠Q, BC = QR: ∠B lies between the sides, SAS

   Another way: Find the third angle firstTwo angles equal, so the third is too: AAS becomes ASA

4. 4Proving triangles congruentOne pair a line, with its reason; look for common sides, opposite angles, parallels

   MethodAB = AC (given), BD = CD (given), AD = AD (common side), △ABD ≅ △ACD (SSS)

   Another way: Work back from the endTo prove BD = CD: look at △ABD and △ACD

5. 5Properties of isosceles trianglesBase angles equal; the bisector, median and height from the top are one line

   MethodAD ⊥ BC, BC = 10 cm: BD = 10 ÷ 2 = 5 cm (prop. of isos. △)

   Another way: By congruent triangles△ABD ≅ △ACD (RHS), so BD = CD

6. 6Proving a triangle isoscelesEqual angles face equal sides: ∠B = ∠C gives AB = AC

   Method∠C = 180° − 40° − 70° = 70° = ∠B, so AB = AC (sides opp. equal ∠s)

   Another way: By congruent triangles△ABD ≅ △ACD (AAS), AB = AC

7. 7Bisecting an angle and a line segmentTwo arcs of one radius; the line through where they cross bisects

   MethodOP = OQ, PR = QR, OR = OR: △OPR ≅ △OQR (SSS)

   Another way: A rhombus's diagonalsAP = BP = AQ = BQ: PQ bisects AB at right angles

8. 8Angles compasses can drawStart from 60° and 90°: bisect, add, take away

   Method75° = 45° + 30°: half of 90° and half of 60°

   Another way: Take it from a straight line150° = 180° − 30°

Worked examples

From different skills in this topic. Have a go first, then open each one for the answer and the working.

1. Example 1Congruent figures and transformations →

   After which of these transformations is the image not congruent to the figure?

   • A. an enlargement, 3 times
   • B. a translation
   • C. a rotation
   • D. a reflection

   Choose the right answer

   ▶Show the answer and working

   Answer: A. an enlargement, 3 times

   Enlarged 3 times, every side becomes 3 times as long: the same shape, a different size, so not congruent.

2. Example 2Properties of isosceles triangles →

   In the picture, AB = AC, AD ⊥ BC and BC = 20 cm. BD =

   ▶Show the answer and working

   Answer: 10 cm

   In an isosceles triangle the height to the base is also its median (prop. of isos. △): BD = 20 ÷ 2 = 10 cm.

3. Example 3Angles compasses can draw →

   A 90° angle is drawn and bisected 1 times; next to it, a 90° angle is drawn and bisected 2 times. Together the two make °

   ▶Show the answer and working

   Answer: 67.5

   90° bisected 1 times is 45°, 90° bisected 2 times is 22.5°, and together 45° + 22.5° = 67.5°.