Geometric Twins · Lesson 9 of 10
Angles of Isosceles and Equilateral Triangles
“Derive equal base angles from RHS and use them in triangle calculations and familiar patterns.”
• Identify equal sides, the apex, and the base of an isosceles triangle. • Prove that angles opposite equal sides are equal using RHS. • Calculate unknown angles in isosceles and equilateral triangles. • Recognise equal radii as a reason a triangle is isosceles. • Describe how congruent triangles appear in buildings and designs.
Equal sides create a useful triangle
An isosceles triangle has at least two equal sides. For AB = AC, the common vertex A is called the apex, and BC is the base. The angles at B and C are the base angles. The base is a side name, not a position: it remains the base even if the triangle is turned sideways.
A triangle with at least two equal sides. The angles opposite those equal sides are equal.
Why should equal sides give equal angles? Draw a perpendicular AD from A to BC. This perpendicular is an altitude, meaning a segment from a vertex perpendicular to the opposite side or the line containing it. Here D lies on BC. It divides the triangle into two right triangles, ADB and ADC. We can prove these two pieces congruent without assuming their base angles are equal.
Problem
Given AB = AC and AD perpendicular to BC, prove angle ABC = angle ACB.
- 1.Angles ADB and ADC are 90°. AB and AC are the hypotenuses of the two right triangles and are equal.
- 2.AD = AD is a shared non-hypotenuse side. Thus ADB ≅ ADC by RHS.
- 3.The match is A ↔ A, D ↔ D, and B ↔ C. Corresponding angles ABD and ACD are equal.
- 4.Because D lies on BC, these are the original base angles ABC and ACB. Hence the angles opposite AB and AC are equal.
This same correspondence proves BD = DC and angle BAD = angle CAD. Thus, in this isosceles triangle, the altitude from the apex also divides the base into equal lengths and divides the apex angle equally. These are conclusions of the congruence proof, not assumptions needed to make it work.
Calculate the base and apex angles
Once the equal base angles are known, the triangle angle sum gives their values. Remove the apex angle from 180° and share the remainder equally between the two base angles. Conversely, if one base angle is given, the other has the same value, and subtracting both from 180° gives the apex.
Problem
AB = AC and angle A = 80°. Find angles B and C.
- 1.Angles B and C are opposite the equal sides, so they are equal.
- 2.Their sum is 180° − 80° = 100°.
- 3.Each angle is 100° ÷ 2 = 50°. Therefore angle B = angle C = 50°.
Problem
In triangle PQR, PQ = PR and angle Q = 67°. Find the other angles.
- 1.Q and R are the base vertices because the equal sides meet at P.
- 2.Angle R = angle Q = 67° by the isosceles property.
- 3.Angle P = 180° − 67° − 67° = 46°. Check: 67° + 67° + 46° = 180°.
All three sides equal
An equilateral triangle has all three sides equal. Applying the equal-side angle property to different side pairs shows that all three angles are equal too. Since they add to 180°, each must be 60°. This conclusion comes from equal sides and the angle sum; it does not depend on which way the triangle is drawn.
A triangle whose three sides are equal. Each of its three interior angles is 60°.
Problem
Why does an equilateral triangle have three 60° angles?
- 1.AB = AC gives angle B = angle C. AB = BC gives angle C = angle A.
- 2.Hence angles A, B, and C have one common value.
- 3.Three copies of that value add to 180°, so each is 180° ÷ 3 = 60°.
You can check these ideas with constructions. Draw an isosceles triangle using equal-radius arcs from the endpoints of its base, then measure the base angles. Draw an equilateral triangle by choosing the same radius as the base length for both arcs. Measurements illustrate the properties, while the congruence proof explains why they hold exactly.
Equal radii and real designs
A triangle can be isosceles without having equal-side marks drawn on it. If A is the centre of a circle and B and C lie on that circle, AB and AC are equal radii. They give an isosceles triangle ABC. This is a useful way to connect a circle diagram with the base-angle property.
Problem
A is the centre, B and C lie on the circle, and angle BAC = 120°. Find angles B and C.
- 1.AB = AC because both segments are radii of this circle.
- 2.The equal base angles add to 180° − 120° = 60°.
- 3.Each is 60° ÷ 2 = 30°. The answer follows from the radii, not from measuring the picture.
Congruent triangles appear in repeated building frames, dome panels, pyramid faces, rangoli patterns, and the metal framework of bridges such as Howrah Bridge. Triangular panels also appear in the Louvre pyramid and the pyramids at Giza. When describing a pattern, identify a repeated triangular unit and compare its dimensions. A photograph can suggest a match, but perspective can change the apparent lengths and angles.
In AB = AC, the equal angles are at B and C, opposite the equal sides. Angle A is not automatically equal to them. All three angles are equal only when all three sides are equal.
Quiz
AB = AC and angle A = 80°. What is angle B?
Which condition proves ADB ≅ ADC in the altitude proof?
PQ = PR and angle Q = 67°. What is angle P?
What is each interior angle of an equilateral triangle?
A is a circle’s centre and B, C lie on it. Why is AB = AC?
Practice Problems
- Draw AB = AC, drop AD perpendicular to BC, and prove the base angles equal by RHS.
- Using the same congruence, prove BD = DC and angle BAD = angle CAD.
- Find both base angles of an isosceles triangle whose apex angle is 42°.
- Find the apex angle when each base angle is 71°.
- A circle has centre O and points P, Q on it. If angle POQ = 96°, find angles OPQ and OQP.
- Construct an equilateral triangle of side 4 cm and explain why each angle is 60°.
- Sketch a repeating triangular pattern inspired by a bridge, dome, pyramid, or rangoli. State which equalities would prove two chosen triangles congruent.
Key Takeaways
• An isosceles triangle has equal angles opposite its equal sides. • An altitude from its apex gives two right triangles congruent by RHS. • The same proof shows equal base halves and equal apex-angle halves. • The angle sum gives numerical values once the equal angles are identified. • An equilateral triangle has three 60° angles. • Equal radii can supply the equal sides in a circle diagram.