What is the converse of the isosceles triangle theorem?

If two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles.

How is the converse proved?

Draw the bisector of the third angle. It cuts the triangle into two triangles with two pairs of equal angles and a common side not between them, so they are congruent by AAS, and the two sides opposite the equal angles are equal.

Is a triangle with two equal angles always isosceles?

Yes. Equal base angles force equal sides opposite them; the theorem and its converse together say that a triangle is isosceles exactly when two of its angles are equal.

Equal Angles Make an Isosceles Triangle

How to use

▶ Play runs the proof, Next → and ← Back step one scene at a time, Reset restores the opening scene and the original figure. Learn more about the controls
Drag $A$ up or down to change the base angles, or $B$ or $C$ ; the readout under the canvas shows both base angles and both sides. Learn more about why it holds
Labels toggles the point names, Measures writes each marked angle's size beside its arc. The Step explanations log keeps every step so far. Learn more about reading a scene

Proof?The base angles at B and C are equal. Draw the bisector AD of the angle at A. The triangles ABD and ACD have equal angles at B and C, equal halves at A, and the common side AD, which is not between those angles; by AAS they are congruent, so AB = AC. Drag A up or down, or B or C.Isosceles converse

DisplayLabelsMeasures

Drag A to change the base angles, or B or C

∠B = 55°,  ∠C = 55°,  AB = 4.39,  AC = 4.39

Step 1 / 5

Step explanations

1A triangle with two equal angles

Given
In triangle ABC the angles at B and C are equal, both β. To prove: AB = AC, the sides opposite the equal angles are equal, so the triangle is isosceles.

Drag A: the base angles stay equal. Learn more about the statement

Getting Started with the Visualizer

Getting Started with the Visualizer

DemoA run through the proof player

Step 0 of 6

Step through the proof, then drag

A

up or down to change the two equal base angles, or drag

B

C

, and watch the two sides

AB

and

AC

in the readout stay equal. The picture below is the player with its parts numbered; the other tabs list what can be dragged, what each control does and what the step log shows.

Opening scene: the controls

1234567

1Proof — the theorem loaded in the player; the ? gives its idea in one line

2Display — Labels shows the point names, Measures writes each marked angle's size beside its arc

3Draggable points — A slides up and down the axis of the triangle, B and C move the base; the hollow point D follows

4Readout — the line under the figure: both base angles and both sides, live while you drag

5Back, Play, Next, Reset — step through the proof or run it; Reset returns to step 1 and restores the original figure

6Speed and step counter — how fast Play advances, and which of the 5 steps is on screen

7Step explanations — every step reached so far, with its justification tag and a link to the section that explains it; click an entry to jump back to it

Proof complete: step 5 of 5

Proof?Isosceles converseDisplayLabelsMeasuresDrag A to change the base angles, or B or C

∠B = 55°,  ∠C = 55°,  AB = 4.39,  AC = 4.39

← Back↻ ReplayNext →ResetNormal ▾Step 5 / 5

Step explanations

1A triangle with two equal angles

Given
In triangle ABC the angles at B and C are equal, both β. To prove: AB = AC, the sides opposite the equal angles are equal, so the triangle is isosceles.

2The bisector AD of the angle at A

Construction
Draw the bisector of the angle at A; it meets BC at D. It cuts the angle at A into two equal halves: ∠BAD = ∠CAD = φ.

3ABD ≅ ACD

AAS
The triangles ABD and ACD have ∠ABD = ∠ACD = β, ∠BAD = ∠CAD = φ, and the common side AD. AD is not between the two angles (it is opposite the angle at B in one triangle and at C in the other), so this is angle, angle, side: by AAS the triangles are congruent.

4AB = AC

Corresponding parts
In congruent triangles the sides opposite equal angles are equal. AB in ABD and AC in ACD are both opposite the angle at D, so AB = AC (one tick each).

5Equal base angles make an isosceles triangle

Combining
A triangle with two equal angles has equal sides opposite them. Together with the isosceles base angles theorem: a triangle is isosceles exactly when two of its angles are equal. The same congruence also gives BD = DC and AD ⟂ BC: the bisector is the median and the altitude. ∎

12345

1The bisector AD — from the apex to the base, with two halves φ at A

2The two triangles — ABD and ACD, shaded either side of AD

3The equal sides — AB and AC, one tick each

4Justification tags — Given, Construction, AAS, Corresponding parts, Combining

5Replay — at the last step Play becomes Replay and Next is off

The Statement

The isosceles base angles theorem says that equal sides give equal angles. This page proves the converse (Euclid I.6): in a triangle

ABC

\angle B = \angle C \implies AB = AC

the sides opposite two equal angles are equal. Together the two theorems say that a triangle is isosceles exactly when two of its angles are equal, which is how isosceles triangles are usually recognised in practice: by their angles.

In the player

B

and

C

are free and

A

slides along the perpendicular to

BC

at its midpoint, which keeps the base angles equal; the readout shows both base angles and both sides.

Opening scene, frozen

The triangle ABC with equal angles β at B and C.

The Bisector

Draw the bisector of the angle at

A

; it meets

BC

at a point

D

. It cuts the angle at

A

into two equal halves:

\angle BAD = \angle CAD = \varphi

Now

AD

splits the triangle into the two triangles

ABD

and

ACD

The bisector, frozen

AD bisects the angle at A: two halves φ.

Angle, Angle, Side

Compare the triangles

ABD

and

ACD

\angle ABD = \angle ACD = \beta, \qquad \angle BAD = \angle CAD = \varphi, \qquad AD = AD

Two pairs of equal angles and a pair of equal sides. The common side

AD

is not the side between the two angles (it lies opposite the angle at

B

in one triangle and at

C

in the other), so this is the angle-angle-side pattern, and by AAS the triangles are congruent.

Angle, angle, side, frozen

ABD and ACD shaded: β, φ and the common side AD.

Equal Sides

Corresponding parts of congruent triangles are equal. The side

AB

of the first triangle and the side

AC

of the second both lie opposite the angle at

D

, so they correspond:

AB = AC

Equal sides, frozen

AB = AC, one tick each.

Why It Holds

A triangle with two equal angles has equal sides opposite them:

\angle B = \angle C \implies AB = AC

The end-of-proof mark ∎ in the log closes the argument. The tags name the facts used: the given figure, a construction, AAS, corresponding parts and combining.

The same congruence gives more:

BD = DC

and the angles at

D

are equal and add up to

180°

, so they are right angles. The bisector from the apex is also the median and the altitude, and it lies on the perpendicular bisector of the base. Euclid proved I.6 differently, by contradiction: if

AB

were longer, cut off a piece equal to

AC

and get a triangle equal to the whole, which is impossible.

The conclusion, frozen

Equal base angles make an isosceles triangle.

Dragging the Figure

Drag

A

up or down: the two base angles change together and stay equal, the bisector stays on the axis of the triangle, and the readout shows

AB

and

AC

equal at every height. Drag

B

C

to change the base;

A

keeps its place along the perpendicular at the new midpoint.

The player keeps the base angles between

20°

and

80°

so the marks stay readable, and keeps the triangle in the frame; a drag outside that range is refused.

After dragging A, frozen

A lowered: smaller base angles, still equal, and the two sides still equal.

Consequences

• Recognising isosceles triangles. In many proofs an isosceles triangle appears through its angles: two equal angles are enough to conclude two equal sides.
• Equiangular means equilateral. Applying the theorem twice to a triangle with three equal angles gives three equal sides, the converse half of the equilateral triangle page.
• Larger angle, larger side. The unequal version is the larger side, larger angle theorem and its converse: the larger of two angles faces the larger side.
• Tangents and chords. Two tangents from a point make equal angles with the chord of contact, so the tangent lengths are equal, as on the two tangents page.
• Angles in a circle. A triangle inscribed in a circle with two equal angles has two equal sides; equal inscribed angles stand on equal chords.