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Triangles – Congruence Theorems

The apex bisector does three jobs

One line can halve an angle and a side. Can it also meet the side at 90°?

The isosceles triangle, its two equal sides ticked.
Step 1 of 4
  1. The isosceles triangle, its two equal sides ticked.
  2. The bisector splits the apex into two equal halves.
  3. It meets the base at its middle — the two base pieces tick equal.
  4. It meets the base at 90°.

The rule

Yes, in an isosceles triangle the apex bisector does all three.

In triangle ABC, AB = AC. The bisector AD of angle A meets the base BC at D. Triangles ABD and ACD match by SAS. Matching parts of matching triangles are equal. That gives three jobs.

  • It halves the apex. Angle BAD = angle CAD.
  • It halves the base. D is the midpoint: BD = DC.
  • It meets the base at 90°. Angle ADB = angle ADC.

Angles ADB and ADC fill a straight line. They total 180°. Equal angles that total 180° are 90° each.

Example: two rafters (roof beams) of 6 m meet at 50°. Each base angle is (180 − 50) ÷ 2 = 65°. The bisector cuts the base into two equal pieces. A scale drawing shows each piece is about 2.5 m. The bisector meets the base at 90°.

Slide the apex angle. Watch the bisector.

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This track

  1. 0 Start here
  2. 1 SAS: two sides and the angle between
  3. 2 SSA can give two triangles
  4. 3 RHS: a right angle, the hypotenuse, a leg
  5. 4 AAS: two angles and any side
  6. 5 Equal sides give equal base angles
  7. 6 Checkpoint: SAS, SSA, RHS and AAS
  8. 7 Equal angles give equal sides
  9. 8 One counter-example ends a claim
  10. 9 Matching parts decide congruence
  11. 10 The apex bisector does three jobs
  12. 11 Checkpoint: converse, counter-example and bisector
  13. 12 The big picture
  14. 13 Track quiz

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