Reasoning in Geometry
What is a conjecture?
A conjecture is an unproven statement based on observation or pattern that is believed to be true. It’s a claim you form from examples, but it needs a proof to be stated as a theorem.
What is a line segment?
A line segment is part of a line consisting of two endpoints and all points between them.
Define a translation.
A translation slides every point of a figure the same distance in the same direction. It keeps shape and size.
What does SSS stand for?
Side–Side–Side
What is a quadrilateral?
It is a four sided shape.
Give an example of deductive reasoning in geometry.
The sum of angles in any triangle is 180°. This uses known theorems and logic to show a specific result.
If two lines are cut by a transversal, what are vertical angles?
Vertical angles are the pairs of opposite angles formed when two lines intersect; they are congruent.
A point moves (–3, +2). What transformation is this?
Every point (x,y)(x,y)(x,y) goes to (x−3, y+2)(x-3, y+2)(x−3,y+2).
If two triangles have all three sides equal, what can you conclude?
By SSS , the triangles are congruent. So corresponding sides correspond to corresponding angles exactly.
In parallelogram JKLM, LM = 17cm, which means that KJ = this.
17cm
If a pattern holds for the first five numbers, can you make an inductive guess? Explain.
Yes, an example of this would be numbers 2, 4, 6, 8, 10 → inductive guess: next is 12 (pattern: add 2). Induction uses observed examples to form a conjecture, but it’s not certain. You must later prove it if you want certainty.
Line AB = 6 cm and BC = 4 cm on a line. What is AC?
If A, B, and C are collinear with B between A and C, then AC = AB + BC = 6 + 4 = 10 cm.
The point (4, -3) is transformed to the point (-3, -4).
90-degree rotation about the origin
Given ∠A = 60°, ∠B = 70°, find ∠C in triangle ABC.
Sum of interior angles = 180°. ∠C = 180° − (60° + 70°) = 180° − 130° = 50°.
Name 4 of the 6 ways to show a quadrilateral must be a parallelogram.
1) 2 pairs of // sides 2) both opposite sides ≅
3) both opposite <≅ 4) 1 pair of sides // & ≅
5) Diagonals bisect
6) One < is supp. to both consecutive angles
Explain why a true statement remains true using logic rules.
Logic rules hold when used correctly. In geometry, proven theorems from true premises show true conclusions. A correctly deduced statement remains true.
Prove that vertical angles are congruent.
Two lines intersect at point O, forming angles ∠1, ∠2, ∠3, ∠4 in order.
Suppose ∠1 and ∠3 are vertical.∠1 and ∠2 are a linear pair, so ∠1 + ∠2 = 180°.
Also, ∠2 and ∠3 are a linear pair, so ∠2 + ∠3 = 180°. Subtract the equal expressions: (∠1 + ∠2) − (∠2 + ∠3) = 180° − 180° ⇒ ∠1 − ∠3 = 0 ⇒ ∠1 = ∠3.
Therefore, vertical angles are congruent. □
Transformation involves reflecting over the x-axis and then the y-axis, which results in changing the signs of both the x- and y- coordinates.
180-degree rotation about the origin
Use ASA to prove two triangles are congruent.
Side AB between ∠A and ∠B equals DE between ∠D and ∠E, so by the ASA the triangles are congruent: △ABC≅△DEF.
Prove that the diagonals of a rectangle are congruent.
Let rectangle ABCD with right angles at each corner. Diagonals are AC and BD. Consider triangles △ABC and △CDA.
Using SAS on triangles △ABC and △CDA, we deduce △ABC≅△CDA. Corresponding parts give AC = BD. Diagonals are congruent.
AB=CD
BC=DABC = DABC=DA
∠B=∠D=90
A student assumes something is true from a pattern but it fails. Explain why inductive reasoning can be misleading.
Induction generalizes from examples; it can produce false conjectures if the pattern changes later. Induction suggests believing conjectures but requires proof or counterexample.
Two parallel lines create multiple angle pairs.
Corresponding angles are equal; alternate interior angles are equal; consecutive interior angles are supplementary. Use this to justify right/parallel relationships.
The coordinate notation when the points A(-3, -5),
B(2, 7) and C (-9, 12) are transformed to A' (-10, 2),
B'(-5, 14) and C' (-16, 19).
(x, y) ---> (x -7, y +7)
Name the 2 non congruence postulates
SSA and AAA
If quadrilateral LUTZ has LU // TZ and UT = LZ, is it for sure a parallelogram?
no, see counter example (trapezoid)