Equal legs AB = AC (matched ticks) force equal base angles ∠B = ∠C (matched arcs). The apex line straight down to the midpoint M of BC is the altitude, the median, and the angle bisector — all at once.
Bring last lesson's fold to a triangle. Give a triangle two equal sides and it gains an axis of symmetry straight down the middle — and once a figure is symmetric, consequences come pouring out. The two base angles must be equal. The single line dropped from the apex turns out to be three lines in one: the height, the median, and the angle bisector, all sharing the same path to the base. The isosceles triangle is where the congruence of Lesson 15.3 and the reflection of Lesson 15.4 finally pay off, giving us a tool we will lean on for the rest of geometry. And it points the way to the most balanced triangle of all — the perfectly even equilateral.
15.5.1 The symmetry of an isosceles triangle
An isosceles triangle is a triangle with (at least) two equal sides. Those two equal sides are the legs; the corner where they meet is the apex; and the remaining side — the one the legs do not share — is the base. The two angles sitting on the base are the base angles.
Here is the picture that explains everything that follows. Fold the triangle along the line that runs from the apex A straight down to the midpoint of the base. Because the two legs have exactly the same length, the left half lands perfectly on the right half: the leg AB falls onto the leg AC, and vertex B lands on vertex C. So an isosceles triangle has an axis of symmetry — exactly the kind of fold line we studied in Lesson 15.4.
Fold along the dashed axis through the apex and the midpoint of the base: the left half mirrors the right. The fold sends leg onto leg and base angle onto base angle.
Key idea
Two equal sides give a triangle a line of symmetry down the apex. Everything in this lesson is just that symmetry, read out loud.
15.5.2 Equal sides face equal angles — the Base Angles Theorem
Once the triangle can fold onto itself, the two base angles have nowhere to hide: the fold carries ∠B exactly onto ∠C, so they must be equal. This is the most-used fact about isosceles triangles.
Base Angles Theorem
If a triangle has two equal sides, then the angles opposite those sides are equal. In symbols: if AB = AC, then ∠B = ∠C.
We can prove it without any folding, using congruence from Lesson 15.3. Drop the bisector of the apex angle to the point M where it meets the base, and compare the two halves:
Statement
Reason
AB = AC
Given (the two legs)
∠BAM = ∠CAM
AM bisects the apex angle
AM = AM
Common side (shared by both halves)
△ABM ≅ △ACM
SAS
∠B = ∠C
Matching parts of congruent triangles
The two half-triangles are perfect copies, so all their matching parts agree — and in particular the base angles ∠B = ∠C. Reasoned out loud, the conclusion line is simply: “AB = AC, so ∠B = ∠C (base angles of an isosceles triangle).”
The bisector AM splits △ABC into two congruent halves △ABM ≅ △ACM (SAS). Matching parts then force the base angles equal.
Try it Equal legs, equal base angles
Raise or lower the apex. The legs stay equal by construction — watch the two base angles stay equal too, however tall the triangle gets.
Apex height
Worked example
An isosceles triangle has apex angle ∠A = 40°. The three angles total 180°, and the two base angles are equal, so each base angle is (180° − 40°) ÷ 2 = 70°.
15.5.3 Three lines in one
From the apex you could draw three different special segments to the base — and in Lesson 15.1 we saw that in a general triangle these three usually head off in three different directions:
the altitude — the perpendicular dropped to the base (the height);
the median — the segment to the midpoint of the base;
the angle bisector — the line that splits the apex angle in two.
In an isosceles triangle something special happens: all three are the very same segment. They lie exactly on the axis of symmetry, and they hit the base at its midpoint M, meeting it at a right angle. The proof in 15.5.2 already shows why — the congruence △ABM ≅ △ACM forces BM = CM (so AM is the median), forces ∠AMB = ∠AMC = 90° (so AM is the altitude), and started from a bisected apex angle (so AM is the bisector).
In the isosceles triangle (left) the altitude, median, and bisector from the apex are one segment to the midpoint M. In a scalene triangle (right) the same three cevians split apart into three different lines.
Try it The apex line is all three
Switch between altitude, median, and bisector. Notice the highlighted segment never moves — it is the same line each time.
Key idea
In an isosceles (or equilateral) triangle, the apex's altitude = median = angle bisector, all riding the axis of symmetry. This is a favorite shortcut in proofs: prove one and you have all three.
15.5.4 Equal angles face equal sides — the converse
Every property in geometry is worth turning around. The Base Angles Theorem says equal sides ⇒ equal base angles. Its converse reads the arrow the other way:
Converse (the test)
If two angles of a triangle are equal, then the sides opposite them are equal. In symbols: if ∠B = ∠C, then AB = AC — so the triangle is isosceles.
This is the difference between a property and a test, the same two-way door we met with parallel lines in Stage 14. A property runs from sides to angles: once you know a triangle is isosceles, you may conclude its base angles are equal. The test runs the other way, from angles to sides: when you see two equal angles, you may conclude the triangle is isosceles. Both directions are true for isosceles triangles — but you must say which one you are using.
Worked example
In △ABC you measure ∠B = ∠C = 65°. By the converse, the sides opposite those angles are equal: AB = AC. The triangle is isosceles — even though nobody told you so directly.
Try it Fill in the angles
Set the apex angle. The two base angles must split what is left of 180° — and they split it evenly, because the legs are equal.
Apex ∠A40°
15.5.5 Equilateral triangles
Push the idea to its limit. Make all three sides equal and you have an equilateral triangle. Now there is nothing to single out as “the base” — any side will do. Reading the Base Angles Theorem off each pair of equal sides in turn forces all three angles equal, and three equal angles that add to 180° must each be exactly 60°.
The converse is true as well: a triangle with three equal angles (an equiangular triangle) has three equal sides. So for triangles, equiangular = equilateral. An equilateral triangle is isosceles in three different ways at once — pick any vertex as the apex — and it carries three axes of symmetry, one through each vertex.
An equilateral triangle: all three sides equal (matched ticks), all three angles equal at 60° (matched arcs), and three axes of symmetry, one from each vertex to the opposite midpoint.
Watch out
The base angles are the two angles on the base — never the apex angle. In an isosceles triangle the apex angle and a base angle are usually different; only in the equilateral case are all three the same.
★ Recap
What to carry away
• An isosceles triangle has two equal legs meeting at the apex; the third side is the base, and it has an axis of symmetry down the apex.
• Base Angles Theorem: equal legs ⇒ equal base angles (AB = AC ⇒ ∠B = ∠C), proved by SAS congruence of the two halves.
• The apex's altitude = median = angle bisector — one segment to the midpoint of the base, perpendicular to it.
• Converse / test: equal base angles ⇒ equal sides — use it to prove a triangle isosceles.
• An equilateral triangle has all sides and all angles equal; each angle is 60°; it is isosceles three ways with three axes of symmetry.
✎ Exercises
An isosceles triangle has apex angle 40°. Find each base angle.
Answer
The three angles total 180°, and the base angles are equal, so each is (180° − 40°) ÷ 2 = 70°.
A base angle of an isosceles triangle is 50°. Find the apex angle.
Answer
Both base angles are 50°, so they use 100°. The apex angle is 180° − 100° = 80°.
Each angle of an equilateral triangle is ____.
Answer
All three angles are equal and add to 180°, so each is 180° ÷ 3 = 60°.
In △ABC you find ∠B = ∠C = 65°. What can you conclude about the sides?
Answer
By the converse of the Base Angles Theorem, equal angles face equal sides, so AB = AC — the triangle is isosceles.
In an isosceles triangle, the altitude drawn from the apex to the base is also the ____ and the ____. Explain why.
Answer
It is also the median and the angle bisector. The two halves △ABM ≅ △ACM are congruent, so the foot lands at the midpoint (median) and the apex angle is split in two (bisector).
The two base angles of an isosceles triangle are 2x and 50°. Solve for x.
Answer
The base angles are equal, so 2x = 50°, giving x = 25.
🎯 Quick check
Six questions to lock it in. Tap the answer you think is right.
§ For teachers and parents
The big idea of this lesson is that symmetry produces equality. Two equal sides give a triangle a line of symmetry, and from that single fact every result here follows: the equal base angles, and the collapse of the altitude, median, and angle bisector into one segment. We prove the Base Angles Theorem with an SAS congruence of the two halves rather than waving at the picture, so students see how a careful argument replaces “it looks equal.”
The misconception to watch is twofold. First, students mislabel which angles are the “base angles” — they are the two angles on the base, not the apex angle, and a common error is to set the apex angle equal to a base angle. Second, students forget the theorem runs both ways: the property (equal sides ⇒ equal angles) lets you conclude angles from known sides, while the converse / test (equal angles ⇒ equal sides) lets you prove a triangle isosceles. Ask which direction is being used in any given problem.
This lesson supports CCSS HS G-CO.C.10 (prove theorems about triangles, including that base angles of isosceles triangles are congruent) and reinforces 8.G.A.5 (informal arguments about angle relationships in triangles).
eastmath.com · Stage 15 · 15.5 Isosceles Triangles · Reasoning, one step at a time
eastmath.com · 15.5 Isosceles Triangles · 15.5.3 Three lines in one