You have used internal angles of quadrilaterals, but they too require an exact definition, just like how we gave one for a quadrilateral. Precisely define the internal angle of a quadrilateral at a given vertex. Your answer should work for a non-convex quadrilateral too. (Hint: use the opposite vertex as well.)
Step-by-step solution
Idea: Two rays AB and AD always make two angles: one less than 180° and a reflex one. We must say which one faces the inside of the quadrilateral. A dent happens exactly at a vertex that sits inside the triangle formed by the other three vertices.
- At vertex A the two sides are AB and AD. The rays AB and AD divide the plane into two angles: ∠BAD (less than 180°) and the reflex angle 360° − ∠BAD. The internal angle is one of these.½ mark
- Bring in the opposite vertex C and look at triangle BCD formed by the other three vertices.½ mark
- If A lies outside triangle BCD, the quadrilateral near A lies between AB and AD on the side of the smaller angle. Internal angle at A = ∠BAD (less than 180°).1 mark
- If A lies inside triangle BCD, the vertex A is the dent: the quadrilateral wraps round A. Internal angle at A = 360° − ∠BAD (a reflex angle). This is the non-convex case.1 mark
Check: Convex square: no vertex is inside the triangle of the other three, so each internal angle is ∠BAD = 90°. Dart A(4, 3), B(0, 0), C(8, 3), D(0, 6): A is inside ∆BCD and ∠BAD ≈ 73.7°, so the internal angle at A ≈ 286.3°; the other angles are about 16.3° (at B), 41.1° (at C) and 16.3° (at D), and the four add to 360° ✓.
Answer to write in the exam
Let ABCD be a quadrilateral, vertex A, neighbours B and D, opposite vertex C
∠BAD = angle between rays AB and AD, less than 180°
If A lies outside ∆BCD: internal angle at A = ∠BAD
If A lies inside ∆BCD: internal angle at A = 360° − ∠BAD (reflex)
Common mistakes that cost marks
- Saying “the internal angle is always the angle less than 180°”. That fails at the dent of a non-convex quadrilateral.
- Using “C lies inside ∠BAD” as the test. It can fail when the dent is at B or D; the reliable test is whether A lies inside triangle BCD.
- Mixing up the internal angle with the exterior angle (180° − internal angle).
How this can come in the exam
In quadrilateral PQRS, vertex P lies inside triangle QRS and ∠QPS = 140°. Find the internal angle at P and state whether PQRS is convex.
Show answer
P inside ∆QRS ⇒ the internal angle at P is the reflex angle 360° − 140° = 220° (1 mark). It is more than 180°, so PQRS is non-convex (1 mark).Try one yourself
Can two vertices of a quadrilateral both lie inside the triangle formed by the other three? Give a reason using angles.
Show answer
No. Then two internal angles would each be more than 180°, adding to more than 360°, but all four internal angles add to only 360°.
More questions like this
- In a quadrilateral ABCD, suppose AB ‖ DC. Can ABCD be non-convex? What if we instead assume AB = CD? What if we instead assume ∠A = ∠C?
- Consider three non-collinear points A, B, C and draw the lines AB, BC, CA. For every possible location of point D in the plane outside these lines, decide if ABCD is self-intersecting, non-convex, or convex. (Hint: the three lines divide the plane into 7 regions.)
- Can a quadrilateral be both self-intersecting and non-planar?
- To test if a given quadrilateral is a parallelogram, do we have to check that the opposite sides are parallel? Are there other ways to test this?
- Recall the following properties of a parallelogram that we proved last year.
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