The angle sum, the exterior angle shortcut, and the isosceles rule.
The problem
In triangle ABC, ∠A = 50° and ∠B = 60°. Find ∠C.
The three angles total 180°, so ∠C = 180 − 50 − 60 = 70°. B is the exterior angle at C (180 − 70 = 110), which also equals 50 + 60 — the two remote interior angles added together.
∠A + ∠B + ∠C = 180°
true for every triangle
50 + 60 + ∠C = 180
drop the values in
∠C = 70°
180 − 110
Tear the three corners off a paper triangle and lay them side by side — they form a straight line. The proof uses a line parallel to one side and the alternate angles it creates.
An exterior angle sits on a straight line with its interior angle.
exterior = 180 − 70
linear pair with ∠C
= 110°
two steps in total
An exterior angle equals the sum of the two far interior angles.
50 + 60 = 110°
∠A and ∠B — the ones not touching it
∠C is the one it sits beside
It follows from the other two rules: the three interior angles make 180°, and so does the linear pair. Subtract and the adjacent angle cancels.
apex = 40°, two equal base angles
the setup
180 − 40 = 140
left for the two of them
each = 140 ÷ 2 = 70°
split evenly
| Triangle | Sides | Angles |
|---|---|---|
| equilateral | all three equal | 60, 60, 60 |
| isosceles | two equal | two equal |
| scalene | all different | all different |
| right | — | one 90°, other two sum to 90° |
If a question gives 40° in an isosceles triangle without saying where, there are two answers: 40 as the apex gives 70-70-40, and 40 as a base angle gives 40-40-100.
Same rule, one equation to solve.
angles x, 2x and 3x
the setup — angles in the ratio 1 : 2 : 3
x + 2x + 3x = 180
the angle sum
6x = 180, x = 30
collect and divide
30°, 60°, 90°
a right triangle
x = 30 is not the answer unless the question asked for x. Most ask for the angles, and stopping at x is the trap the choices are built around.
An exterior angle is (5x − 8)°; its remote interior angles are (2x + 10)° and (x + 6)°. Find the exterior angle.
5x − 8 = (2x + 10) + (x + 6)
exterior = the two far ones
5x − 8 = 3x + 16, x = 12
− 3x, + 8, ÷ 2
exterior = 5(12) − 8 = 52°
remotes 34° and 18°: 34 + 18 = 52 ✓
Different: no 180 appears — the exterior rule is an equation between three expressions.
One angle of an isosceles triangle is 104°. Find the other two.
104 + 104 = 208 > 180
it cannot be a base angle
(180 − 104) ÷ 2 = 76 ÷ 2
so it is the apex
base angles = 38° and 38°
only one answer this time
Different: an angle of 90° or more can only be the apex, so the two-answer case of step 4 collapses to one.
A triangle's angles are x, 2x and 4x. Find the largest.
x + 2x + 4x = 180, 7x = 180
the angle sum
x = 1807
7 does not divide 180 — keep the fraction
4x = 7207°
about 102.9°; the others 1807° and 3607°
Different: the answer is not a whole number. Multiply the fraction through rather than rounding x first.
In △ABC, ∠B = 40° and ∠C = 80°. AD bisects ∠A and meets BC at D. Find ∠ADC.
∠A = 180 − 40 − 80 = 60°
big triangle first
∠BAD = 60 ÷ 2 = 30°
the bisector halves it
∠ADC = ∠B + ∠BAD = 40 + 30
exterior angle of △ABD
∠ADC = 70°
one step, no third triangle needed
Check through △ADC: 30 + 80 + 70 = 180 ✓. The angle on the other side of D, ∠ADB, is 110° — that is the classic wrong pick.
Two segments AC and BD cross at E, making two triangles that meet at E. The angles at E are vertical, so they are equal — which means the other two angles of each triangle must have the same total.
∠A + ∠B = ∠C + ∠D
both equal 180 − ∠E
35 + 75 = 50 + x
example values
x = 60
110 − 50
Expect the angle sum hidden inside a figure: an extended side, a bisector, a bowtie, or angles given as expressions. Find x first, then read the question again — most ask for an angle, and often the linear-pair partner of the one you just found. Ten-second check: the three angles you end with must total 180°.
The problem
An exterior angle of a triangle is 125° and one remote interior angle is 60°. Find the other remote interior angle.
An exterior angle equals the sum of the two remote interior angles, so 125 = 60 + x and x = 65°. C computes 180 − 125 = 55, which is the adjacent interior angle, not a remote one.
