1.2.6 · D5Basic Geometry
Question bank — Triangle properties — angle sum = 180°, exterior angle theorem
Before we start, one word we lean on: an interior angle is a corner angle inside the triangle; an exterior angle is made by extending one side past a vertex and measuring from that extension to the next side. They sit on a straight line together, so they are supplementary (add to ) — see Supplementary and Complementary Angles. A remote interior angle for a given exterior angle is one of the two interior angles the exterior angle does not touch.
True or false — justify
Every claim below is either always-true, always-false, or true-only-under-a-condition. Never accept a bare verdict — force the reason.
Every triangle's interior angles sum to exactly .
True in flat (Euclidean) geometry — the proof leans on a line through one vertex parallel to the opposite side, and parallels only behave this way on a flat plane. On a sphere the sum exceeds .
If three angles add to , they must be the angles of some triangle.
True — you can always build a triangle from any three positive angles summing to (fix a base, draw the two base angles, the sides meet at the apex which automatically has the third angle).
The angles , , can be the interior angles of a triangle.
True — they are all positive and sum to , which is the only requirement.
A triangle can have two right angles.
False — two angles already use the entire budget, leaving for the third corner, so the third "side" never closes into a vertex.
A triangle can have two obtuse angles.
False — each obtuse angle is more than , so two of them already exceed before the third angle is even counted.
An exterior angle of a triangle is always larger than either remote interior angle.
True — the exterior angle equals the sum of both remote interiors, and each interior is positive, so the sum beats either one alone.
The exterior angle equals the interior angle it sits next to.
False — the adjacent interior angle is supplementary to the exterior angle (they total ), so they are equal only in the special case where both are .
An equilateral triangle has three exterior angles of each.
True — each interior angle is , and the exterior is the supplement, ; equivalently it equals the two remote interiors summed.
The three exterior angles of any triangle sum to .
True — each exterior is minus its interior, so the three sum to ; this is the "one full turn" you make walking around the triangle (see Polygon Angle Sums).
A triangle with angles , , is impossible because it's too "thin".
False — thinness is allowed; the angles are positive and sum to , so it is a perfectly valid (very skinny) triangle.
Spot the error
Each item states a piece of "reasoning". Find the flaw and name it.
"Angles , , sum to , so my triangle has ."
The error is assuming three measured numbers must belong to one triangle. A real triangle's interiors always total ; a sum of means a measurement slip or three angles that aren't all interiors of the same triangle.
"Exterior angle at C is , so angle sum is ."
The is an exterior angle, which cannot enter the interior angle sum. The interior at C is its supplement, , and — flagging that the given data is inconsistent, not that the rule failed.
"Exterior angle theorem: the exterior angle equals a remote interior angle."
It equals the sum of both remote interiors, not one. Dropping the second term makes the exterior angle come out too small every time.
"The exterior angle at A is , so angle A is ."
Angle A (interior) is the supplement of its exterior angle: . The describes the angle outside the triangle, not the corner itself.
"In an isosceles triangle the base angles equal the vertex angle."
Only the two base angles are equal to each other (angles opposite the equal sides — see Properties of Isosceles Triangles). The vertex angle is generally different; it equals a base angle only in the equilateral case.
"I extended a side and got an exterior angle of , so the two remote interiors are each ."
The theorem gives the sum of the two remotes as , not that they are equal. They could be and , or any split totalling — equality is a separate, unstated assumption.
"A right triangle's two non-right angles must each be ."
They must only sum to (since ), not be equal. and is also a right triangle.
Why questions
These ask for the mechanism, the reason the theorem works — not just its statement.
Why does drawing a line parallel to a side (not any random line) prove the angle sum?
Parallel lines guarantee equal alternate interior angles when a transversal crosses them (see Parallel Lines and Transversals). That equality lets us relocate angles A and C to the third vertex, where they line up on a straight line and reveal the .
Why do the three interior angles specifically make a straight line (and not, say, a full circle)?
When A and C are copied to vertex B via parallels, angle B sits between them, and all three lie along one flat line through B — a straight line is exactly by definition (see Straight Lines and Angles).
Why is the exterior angle equal to the sum of the remote interiors rather than their difference?
The exterior and its adjacent interior fill a straight line (), and all three interiors also fill . Subtracting the shared adjacent interior from both leaves exterior the other two interiors added together.
Why can't we prove the angle sum by just measuring a few triangles with a protractor?
Measurement always carries error and only ever checks finitely many cases. A proof via parallel lines covers every triangle at once and is exact, which measurement can never be.
Why does the "walking around the triangle and turning" picture give of turning but of interior angle?
At each corner you turn by the exterior angle, not the interior one, and the three exteriors sum to a full turn (). The interiors are the supplements of those turns, giving the different total.
Why must the base angles of an isosceles triangle be equal — what forces it?
Reflecting the triangle across its axis of symmetry maps the two equal sides onto each other, carrying one base angle exactly onto the other, so they must be congruent (a congruence argument — see Triangle Congruence).
Why does the angle sum being fixed at mean you can never have three "independent" angles?
Once two interior angles are chosen, the third is completely determined by minus their sum — so a triangle really has only two free angle choices, not three.
Edge cases
The boundary and degenerate situations the theorems quietly assume away.
What happens to a triangle as one angle shrinks toward ?
The other two must approach a sum of , so the triangle flattens into a sliver; at exactly the two sides become collinear and it stops being a triangle (a degenerate case).
Can an exterior angle be ?
Only if the adjacent interior angle is , which is the degenerate flattened case — so for a genuine triangle the exterior angle is strictly less than and strictly greater than .
If a triangle's exterior angle equals , what is special about it?
The adjacent interior angle is also (supplement), so that vertex is a right angle — the triangle is right-angled at that corner.
At a vertex, how many distinct exterior angles are there, and are they equal?
Two — you can extend the side in either direction. They are vertically opposite and therefore equal, and both equal the same remote-interior sum.
Does the exterior angle theorem still hold at a –– (equilateral) triangle's every vertex?
Yes — at each vertex the exterior is , which is exactly the of the two remote interiors, so the theorem holds identically at all three corners.
What is the largest an interior angle of a triangle can approach without reaching?
It can approach (making the other two shrink toward ) but never equal it, since a corner collapses the triangle into a straight line.
Recall One-line self-test
Cover the answers and re-run any three "Spot the error" items — the trap always hides in mixing interior with exterior, or replacing sum with a single angle.