Diagrams are powerful tools in geometry because they help you see relationships among points, lines, angles, and shapes. A well drawn diagram can suggest a path for a proof and organize the information in a problem. However, a diagram is not evidence by itself unless the feature is stated, marked, or follows from a theorem.
Learning what you may assume from a diagram helps prevent false conclusions.
Key Facts
- You may assume points shown on a line are collinear if the diagram clearly places them on the same straight line.
- You may assume betweenness when a point is drawn between two labeled endpoints on the same segment, such as A, B, C with B between A and C.
- You may assume two drawn lines or segments intersect at a labeled point if the diagram shows them crossing there.
- You may not assume AB = CD just because two segments look the same length, unless they are marked congruent or stated.
- You may not assume an angle is 90° just because it looks square, unless a right angle mark or statement is given.
- Segment addition: If B is between A and C, then AB + BC = AC.
Vocabulary
- Diagram
- A diagram is a drawing that represents geometric objects and their relationships in a problem.
- Betweenness
- Betweenness describes a point lying on a line segment between two other points.
- Collinear
- Collinear points are points that lie on the same straight line.
- Auxiliary line
- An auxiliary line is an extra line or segment added to a diagram to help prove a result.
- Congruent
- Congruent figures or parts have exactly the same size and shape.
Common Mistakes to Avoid
- Assuming two segments are equal because they look equal is wrong because drawings are not usually made to exact scale.
- Assuming an angle is a right angle from appearance alone is wrong because only a right angle mark, given information, or a theorem can justify 90°.
- Using a diagram feature that is not labeled or stated is wrong because proof steps must come from givens, definitions, or proven facts.
- Ignoring betweenness or order of points is wrong because formulas like AB + BC = AC only work when B is actually between A and C.
Practice Questions
- 1 In a diagram, points A, B, and C are collinear with B between A and C. If AB = 7 cm and BC = 11 cm, find AC.
- 2 In triangle ABC, point D lies on segment AC. If AD = 4x + 1, DC = 2x + 7, and AC = 32, find x, AD, and DC.
- 3 A diagram shows two segments that appear to be the same length, but the problem gives no congruence marks or equal length statements. Explain whether you may use the equality of those segment lengths in a proof and why.