Isosceles triangles and perpendicular bisectors

Warm-up

Three quick questions from earlier lessons.

Discover

An isosceles triangle has two equal sides. Fold it along the line from the apex to the midpoint of the base. What do you notice about the base angles?

They are equal
One is twice the other

Change the height of the apex. Compare the base angles.

They are always equal
One grows faster than the other

Key idea

Symmetry gives equal parts.

In an isosceles triangle the base angles are equal. A point is on the perpendicular bisector of

AB

exactly when it is equally far from

A

and

B

.

Worked example

Apex angle $40^\circ$ .

Base angles together:

140^\circ

180 − 40.

Each base angle:

70^\circ

Equal halves.

Your turn, guided

Your turn: base angles $35^\circ$ each. The apex angle.

110^\circ

180 − 70.

Practice ★

An isosceles triangle has apex angle $114^\circ$ (between the two equal sides). Find each base angle, in degrees.
An isosceles triangle has apex angle $38^\circ$ (between the two equal sides). Find each base angle, in degrees.

Each base angle of an isosceles triangle measures $52^\circ$ . Find the apex angle, in degrees.

Watch out!

Practice ★★

Point $P$ lies on the perpendicular bisector of $AB$ , and $PA = 13$ . Find $PB$ .Not to scale
Point $P$ lies on the perpendicular bisector of $AB$ , and $PA = 9$ . Find $PB$ .Not to scale

Exit ticket

Three questions to check you have got it.

An isosceles triangle has apex angle $114^\circ$ (between the two equal sides). Find each base angle, in degrees.
An isosceles triangle has apex angle $38^\circ$ (between the two equal sides). Find each base angle, in degrees.
An isosceles triangle has apex angle $129^\circ$ (between the two equal sides). Find each base angle, in degrees.

Summary card

Isosceles triangles and bisectors

Base angles equal; perpendicular bisector = points equidistant from the ends.

Watch out: halve 180° − apex for each base angle.

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