Let ABCD be a quadrilateral.
- (i) List all sides of ABCD adjacent to side AB. List all sides opposite to AB.
- (ii) List all angles of ABCD adjacent to ∠A. List all angles opposite to ∠A.
- (iii) Define a pair of opposite sides and a pair of opposite angles without using the names of the vertices.
Step-by-step solution
Idea: The name ABCD tells us the sides: AB, BC, CD, DA. Adjacent means sharing an endpoint (for sides) or sharing a side (for angles). Opposite means not adjacent.
(i) List all sides of ABCD adjacent to side AB. List all sides opposite to AB.
- The sides are AB, BC, CD, DA. A side adjacent to AB shares an endpoint with it: BC (shares B) and DA (shares A).½ mark
- The only side with no endpoint in common with AB is CD. So CD is opposite to AB.½ mark
(ii) List all angles of ABCD adjacent to ∠A. List all angles opposite to ∠A.
- Adjacent angles are at the two ends of one side. A is joined to B (side AB) and to D (side DA), so ∠B and ∠D are adjacent to ∠A.½ mark
- C is not joined to A by a side (AC is a diagonal), so ∠C is opposite to ∠A.½ mark
(iii) Define a pair of opposite sides and a pair of opposite angles without using the names of the vertices.
- Opposite sides: two sides of a quadrilateral that have no endpoint (vertex) in common.½ mark
- Opposite angles: two internal angles of a quadrilateral whose vertices are not the two endpoints of the same side (that is, the vertices are the two ends of a diagonal).½ mark
Answer to write in the exam
(i)
Sides of ABCD: AB, BC, CD, DA
Adjacent to AB: BC (common end B), DA (common end A)
∴ Opposite to AB: CD
(ii)
A and B are ends of side AB; A and D are ends of side DA ⇒ ∠B, ∠D adjacent to ∠A
A and C are joined by a diagonal, not a side
∴ Opposite to ∠A: ∠C
(iii)
Opposite sides: two sides of the quadrilateral with no common endpoint
Opposite angles: two internal angles at vertices that are not the endpoints of a common side
Common mistakes that cost marks
- Listing CD as adjacent to AB because it is drawn “next to” it. Adjacent sides must share an endpoint.
- Listing ∠C as adjacent to ∠A. A and C are joined by a diagonal, not a side.
- Defining opposite sides as “parallel sides”. That is true only in special quadrilaterals such as parallelograms.
How this can come in the exam
In quadrilateral PQRS, the angle opposite to ∠Q is
- ∠P
- ∠R
- ∠S
- ∠Q itself
Show answer
(C) ∠S
The sides are PQ, QR, RS, SP. Q is joined by sides to P and R, so ∠P and ∠R are adjacent; ∠S is opposite.
Try one yourself
In quadrilateral WXYZ, name the side opposite to YZ and the angles adjacent to ∠Z.
Show answer
Side opposite to YZ: WX. Angles adjacent to ∠Z: ∠Y and ∠W.
More questions like this
- 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.)
- 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?
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