Isosceles Triangles

Try this

At the start $A B \neq = A C$ : the median $A M$ (solid), the altitude $A H$ (dashed) and the angle bisector $A D$ (dotted) are three different lines, and $M$ , $H$ , $D$ sit in a row on the base.
Drag $A$ to the right, toward the spot straight above the midpoint of $B C$ : the three feet squeeze closer and closer. Notice that $D$ is always between $M$ and $H$ .
Click “Move A so that AB = AC”: the three lines coincide exactly, and at the same moment the $∠ B = ∠ C$ light comes on.
Drag $A$ a little further: the three lines separate again, and their left-to-right order is reversed.

Two basic properties of an isosceles triangle: equal sides face equal angles ( $A B = A C \Rightarrow ∠ B = ∠ C$ ), and the three lines coincide: the bisector of the apex angle, the median to the base and the altitude to the base are the same line. This line is the axis of symmetry of the isosceles triangle.

Predict first

An isosceles triangle has an angle of $8 0^{\circ}$ . What is its apex angle?

Only

8 0^{\circ}

Only

2 0^{\circ}

8 0^{\circ}

2 0^{\circ}

5 0^{\circ}

Choose an answer first; then you can see the result.

Why: fold along the axis of symmetry

Proof that the three lines coincide

Step 1 / 4

Not isosceles: three separate lines
The median aims at the midpoint of $B C$ , the altitude aims at the foot of the perpendicular, and the angle bisector splits $∠ A$ into two equal halves. Three different requirements usually give three different points.
Now make AB = AC
The left and right triangles are congruent
Read off two conclusions

This property is often used backwards: if one line in a triangle is both “altitude and median”, or “altitude and angle bisector”, or “median and angle bisector”, then the triangle is isosceles. When you see “perpendicular + bisects”, think of completing an isosceles triangle.

In the same way, “equal sides face equal angles” works backwards: equal angles face equal sides ( $∠ B = ∠ C \Rightarrow A B = A C$ ). To prove two segments equal when they are in the same triangle, look at the angles opposite them first.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: The median, altitude and angle bisector from $A$ are the same line

$A$ can only move along the perpendicular bisector of $B C$ , so $A B = A C$ always. Drag it up and down: the three lines always coincide.

Practice

Exercise 1Basic

The apex angle of an isosceles triangle is $10 0^{\circ}$ . Find one of its base angles in degrees.

Exercise 2Basic

In $△ A B C$ , $A B = A C = 13$ and $B C = 10$ . Find the altitude to $B C$ .

Exercise 3Advanced

In an isosceles triangle, the altitude to one leg makes an angle of $4 0^{\circ}$ with the other leg. Find the apex angle in degrees (give every possibility, separated by commas).

Exercise 4Challenge

In $△ A B C$ , $A B = A C$ and $∠ A = 3 6^{\circ}$ . $B D$ bisects $∠ A B C$ and meets $A C$ at $D$ . Find $∠ B D C$ in degrees, and think about how many isosceles triangles there are in the figure.

Try this

Why: fold along the axis of symmetry

Proof that the three lines coincide

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Assuming a given angle is the apex angle (or a base angle)

Using “the three lines coincide” for lines to a leg