Equal angles give equal sides
Equal sides give equal base angles. Do equal base angles give equal sides too?
- The two base angles are given equal.
- The bisector splits the apex into two equal halves, and is shared.
- The two sides from the apex come out equal.
The rule
Yes. Two equal angles in a triangle give equal opposite sides.
Swap the two parts of a claim to get its converse. A converse needs its own reason. It is not always true.
Here is one that fails. "A square has four right angles." The converse is "a shape with four right angles is a square." A rectangle breaks it.
For this claim, the converse is true. In triangle ABC, take angle B = angle C. Draw the bisector AD of angle A.
- Angle B = angle C, given.
- Angle BAD = angle CAD, by the bisector.
- AD is shared.
Two angles and a side that is not between them: AAS. So triangles ABD and ACD match. Matching sides are equal, so AB = AC.
Example: base angles of 70° total 140°. So the apex is 180 − 140 = 40°.
Slide the two equal angles. Read the sides opposite them.
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