How to Find the Missing Angle in an Isosceles Triangle

An isosceles triangle carries a gift: its two equal sides sit opposite two equal angles. Know one angle and you effectively know two, which is why one measurement plus the 180° rule settles the whole triangle. The only care needed is direction. Going from base angle to apex is a straight subtraction, but going from apex to base angles ends with a divide by two, and skipping that division is the single most common error on this topic.

Which angles match, and why

The equal angles are the ones opposite the equal sides, usually called the base angles. The odd one out, between the two equal sides, is the apex. The matching is not a convention, it is forced: fold the triangle along the line from the apex to the midpoint of the base and the two halves land exactly on each other, carrying one base angle onto the other.

That fold line is also the altitude and the perpendicular bisector of the base, all at once. It is why so many isosceles problems split cleanly into two congruent right triangles, and why the height lesson for this shape halves the base before doing anything else.

Both directions in one picture

The diagram works the 70° base angle case forward to the 40° apex, then works a 40° apex back to 70° base angles, and shows the failed shortcut in red: 180 − 40 = 140 is the total of the two base angles, not each one.

A worked check you can repeat on paper

A gable truss has equal rafters meeting the bottom chord at 70° on each side. The peak angle is 180 − 70 − 70 = 40°. Quick, and the check 70 + 70 + 40 = 180 closes.

Now the reverse, which is where hands slip. The peak of a different truss measures 52°. The remainder is 180 − 52 = 128, and each heel angle is 128 / 2 = 64°. Confirm: 64 + 64 + 52 = 180.

One subtlety worth a sentence. If the known angle is a base angle greater than or equal to 90°, stop: two of them would already spend the whole budget, so the given angle must be the apex or the measurement is wrong.

When the missing angle is not the one you expect

Sometimes a problem gives one base angle and asks for the other. The answer is the same number, no arithmetic at all, and recognizing that saves time. The equal sides guarantee it.

An equilateral triangle is the extreme case: all three sides equal, so all three angles equal, so each is 180 / 3 = 60° with no measurement needed. Every equilateral triangle is isosceles, and the rules here apply to it, just with nothing left to find.

Mistakes that catch people out

• Skipping the divide by two. 180 minus the apex is the combined share of both base angles. Halve it before reporting.
• Doubling the wrong angle. The formula apex = 180 − 2 × base only doubles a base angle. Doubling the apex has no meaning in this triangle.
• Misreading which angles are equal. The equal angles sit opposite the equal sides. If the tick marks are on the slanted sides, the equal angles are at the bottom, whatever the drawing orientation suggests.
• Accepting a base angle of 90° or more. Two right or obtuse base angles cannot fit in one triangle. A stated base angle of 95° means the labels are swapped or the data is bad.
• Trusting the sketch over the ticks. Diagrams are often drawn deliberately out of proportion. Work from the marked equalities, not from what looks equal on paper.

Try these

Common questions

What is the formula for the missing angle in an isosceles triangle?

If a base angle is known, the apex is 180 minus twice the base angle. If the apex is known, each base angle is 180 minus the apex, divided by two.

Which two angles of an isosceles triangle are equal?

The two angles opposite the equal sides, called the base angles. The angle between the equal sides, the apex, is the one that can differ.

Why are the base angles equal?

Folding the triangle along the line from the apex to the midpoint of the base maps one half exactly onto the other, carrying one base angle onto the other. Equal sides force equal opposite angles.

Can a base angle of an isosceles triangle be 90 degrees or more?

No. Two base angles of 90 degrees or more would total at least 180 on their own, leaving nothing for the apex. Base angles are always acute.

Can the apex be obtuse?

Yes. An apex up to but not including 180 degrees is possible. An apex of 120 degrees leaves 30 degrees for each base angle, which is a perfectly good triangle.

Is an equilateral triangle isosceles?

Yes. It meets the definition with room to spare, and every angle rule here applies to it. All three of its angles are 60 degrees.

Related lessons and calculators

• How to find the third angle of any triangle, the general 180 rule this page is a special case of.
• How to find the height of an isosceles triangle, which uses the same fold line as an altitude.
• How to find all three angles from three sides, for when you have lengths rather than any angle at all.
• Triangle solver, to confirm any of these results from whatever combination you know.
• NCTM, the National Council of Teachers of Mathematics, for the underlying congruence proofs.
• Area formulas for every common shape, a calculator covering fifteen shapes and an index of every geometry lesson here.