Triangle angle properties
Use the angle sum and equal-side markings to justify unknown triangle angles.
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Three interior angles make 180°
The angles inside a triangle add to 180°. Follow the three corners of the same triangle before calculating; an angle outside it does not belong in that total. Turning the triangle or changing its side lengths does not change the rule.
Use all three corners
Angles a, b and c are the interior angles at A, B and C.
View figure data
| Given | Information |
|---|---|
| Geometry | ABC is a triangle. |
| Angle a | CAB |
| Angle b | ABC |
| Angle c | BCA |
| Angle a | 48° |
| Angle b | 67° |
State the total, then subtract both known angles:
Check 48° + 67° + 65° = 180°. Subtracting only 48° leaves the combined size of the other two angles, not c alone.
If a side is already extended through a triangle’s corner, the interior corner and its adjacent outside angle fill a straight angle of 180°. The outside angle does not belong in the triangle’s three-angle total.
For a separate example, triangle PQR has angles P = 40° and R = 85°, and PQ is already extended through Q to S. First find ∠ PQR = 180°-40°-85° = 55°. Since P, Q and S are collinear with Q in the middle, ∠ RQS = 180°-55° = 125°. Check that the outside 125° equals 40° + 85°. Use a stated extension; do not add a new line to a question that forbids construction.
Match equal sides to their opposite angles
In an isosceles triangle, two sides are equal. The angles opposite those sides are equal too. First locate the vertex where the equal sides meet; the other two corners are the equal base angles. The base need not be horizontal.
Check your understanding 1 Name the sides and their opposite angles
The two sides tagged e are equal. Write their names using =, then name the equal interior angles by their capital corner letters. Use the tags rather than the triangle’s direction on the page.
The equal-side vertex has turned
Triangle ABC is isosceles. Matching e tags identify the equal sides.
View figure data
| Given | Information |
|---|---|
| Shape | ABC is an isosceles triangle. The two sides tagged e are equal. |
| Use of the drawing | Positions identify sides and corners. Use only the stated class and marks; do not infer extra equal lengths or right angles from appearance. |
Show answer
AB = AC, and B = C. Side AB is opposite corner C; side AC is opposite corner B. The equal sides meet at A, but that does not make angle A equal to a base angle. Reversing a side name, such as BA = CA, states the same lengths.
The given angle is between the equal sides
CA and CB are equal. The vertex angle c is at C; a and b are opposite the equal sides.
View figure data
| Given | Information |
|---|---|
| Geometry | ABC is isosceles; CA and CB are equal sides. |
| Angle a | CAB |
| Angle b | ABC |
| Angle c | BCA |
| Angle c | 44° |
The base angles share 180°-44° = 136° equally, so a = b = 136°÷2 = 68°. Divide by two only after identifying two unknown equal angles. The total check is 68° + 68° + 44° = 180°.
Check your understanding 2 The given angle has moved
CA = CB again, but now the given 52° is angle a at A. Find b and c. Explain which two angles are equal before calculating.
A base angle is given
The equal sides CA and CB meet at C. The known angle is at A, not C.
View figure data
| Given | Information |
|---|---|
| Geometry | ABC is isosceles; CA and CB are equal sides. |
| Angle a | CAB |
| Angle b | ABC |
| Angle c | BCA |
| Angle a | 52° |
Show answer
The angles opposite CA and CB are at B and A, so b = 52°. Then c = 180°-52°-52° = 76°. Check 52° + 52° + 76° = 180°. Dividing 180°-52° by two would incorrectly make the vertex angle equal to the remaining base angle.
Three equal sides, three equal angles
An equilateral triangle has three equal sides. Each angle is opposite an equal side, so all three angles are equal. Their total is 180°: each is 180°÷3 = 60°.
Use equality and the total together
AB, BC and CA are all equal. The three marked interior angles are a, b and c.
View figure data
| Given | Information |
|---|---|
| Geometry | ABC is equilateral: all three sides are equal. |
| Angle a | CAB |
| Angle b | ABC |
| Angle c | BCA |
An equilateral triangle also has at least two equal sides, so the equal-base-angle property still applies. Do not assume that a triangle with only two stated equal sides is equilateral.
Check your understanding 3 Give every equality without measuring
DEF is equilateral. Write one equality chain for its three sides and one for its three interior corner angles. Are two sides that look equal enough to justify all these statements in another triangle?
Show answer
DE = EF = FD, and D = E = F. Each chain includes every pair of its equal terms. Three equal sides, or the stated equilateral class, justify the full chains. Two stated equal sides alone justify only the corresponding base-angle pair; appearance does not supply a third equality.
Check your understanding 4 Use a right angle without assuming equal sides
AB is perpendicular to AC. Angle b is 35°. State the triangle’s total and find a and c. Must b and c be equal?
One corner is a right angle
AB and AC are perpendicular, so the right corner is at A.
View figure data
| Given | Information |
|---|---|
| Geometry | CA is perpendicular to AB. |
| Angle a | CAB |
| Angle b | ABC |
| Angle c | BCA |
| Angle b | 35° |
Show answer
The total is 180°. Angle a = 90°, so c = 180°-90°-35° = 55°. The two acute angles total 90°, but they need not be equal. They would both be 45° only if the triangle were also isosceles with equal sides meeting at the right angle.
Check your understanding 5 Connect the corner to its sides
XYZ is right-angled at Y. Write the perpendicular side pair using perp or ⊥. Then write the sum of the interior angles at X and Z, using + and =. No equal sides are given.
Show answer
XY perp YZ, or XY ⊥ YZ. The right angle uses 90° of the triangle’s 180°, leaving X + Z = 90°. Neither XY = YZ nor X = Z follows from being right-angled alone.
Use the lines already in a composite figure
A figure may contain more than one shape. Name the triangle whose total you use, then connect its angle to the surrounding shape. Use only lines already shown; a nearby angle is not automatically part of that triangle.
Check your understanding 6 Keep the whole corner separate from its parts
ABCD is a rectangle and AC is already drawn. Angle a, BAC, is 35°. Find b, CAD, and c, BCA. Give a reason for each step.
A given diagonal splits a rectangle corner
AC is drawn inside rectangle ABCD. At A, a and b together fill DAB. Angle c is at C in triangle ABC.
View figure data
| Given | Information |
|---|---|
| Geometry | ABCD is a rectangle. AC is already drawn. |
| Angle a (BAC) | 35° |
Show answer
The rectangle corner DAB is 90°, so b = 90°-35° = 55°. In triangle ABC, angle ABC is also 90°. Thus c = 180°-90°-35° = 55°. The equal results come from these stated relationships; they are not evidence that every diagonal splits a rectangle corner equally.
Check your understanding 7 Notice a missing condition
A triangle has one angle of 44°, but no equal sides are stated or marked. Can you conclude that its other angles are both 68°? Explain what is known and what extra information would justify that answer.
Show answer
No. The other two angles total 180°-44° = 136°, but their individual sizes are not determined. For 68° each, they must be equal: for example, the triangle could be isosceles with the given 44° between its equal sides. A balanced-looking drawing does not supply that condition.
Check your understanding 8 Connect equal sides to a rectangle corner
ABCD is a rectangle. E is inside it, with AE and BE already drawn and AE = BE. Angle AEB is 50°. Find angles EAB, ABE and DAE. Explain why the first calculation is not yet the final answer.
Use the inside triangle, then the whole corner
AE and BE are already drawn. E is a named point inside rectangle ABCD.
View figure data
| Given | Information |
|---|---|
| ABCD | rectangle. |
| ABE | isosceles triangle. The equal sides meet at E. |
| Already drawn lines | AE, BE. Use the boundary and these lines; construct no extra line. |
| Angle a | AEB = 50°. |
| Angle b | BAE is requested. |
| Angle c | ABE is requested. |
| Angle d | EAD is requested. |
| Use of the drawing | The stated classes, alignments and angle values are exact. Use them rather than measuring the picture; appearance supplies no extra equalities or right angles. |
Show answer
In isosceles triangle ABE, the base angles EAB and ABE are equal. They share 180°-50° = 130°, so each is 65°. E lies inside the rectangle, so ray AE is inside the right angle DAB. The remaining angle DAE is 90°-65° = 25°. The 65° result belongs to the triangle; 25° is the other part of the rectangle corner.
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