Lesson 3: Inductive Reasoning and Conjectures

Flashcards reviewing key definitions, constructions, and conjectures from Lesson 3 on Inductive Reasoning.

What is a conjecture in geometry?

A conjecture is an unproven statement or conclusion that is believed to be true based on observations or inductive reasoning.

What is an angle bisector?

An angle bisector is a ray, line, or line segment that divides an angle into two congruent angles.

What conjecture can be made regarding the distance from any point on an angle bisector to each side of the angle?

Any point located on the angle bisector of an angle is equidistant (at equal distance) from the two sides of the angle.

Does observing a pattern or conjecture in a few examples prove that it is true for all cases?

No, inductive reasoning based on specific examples does not prove a conjecture is true for all cases; a formal geometric proof is required to establish certainty.

In the diagram, ray $\vec{AB}$ bisects $\angle FAE$, $BF = 5x$, and $BE = 3(x + 2)$. What is the value of $x$?

$x = 3$. Since ray $\vec{AB}$ bisects $\angle FAE$, the distances $BF$ and $BE$ are equal. Setting $5x = 3(x + 2)$ gives $5x = 3x + 6$, which simplifies to $2x = 6$, so $x = 3$.

How can you construct a $45^\circ$ angle using Patty Paper?

Fold the Patty Paper to form perpendicular lines creating a $90^\circ$ angle, then fold the $90^\circ$ angle in half so its sides align, bisecting it into two $45^\circ$ angles.

What conjecture can be made about the angle bisectors of the four right angles formed by two intersecting perpendicular lines?

The angle bisectors of adjacent right angles formed by intersecting perpendicular lines are perpendicular to each other.

What conjecture can be made about the angle bisector of vertex $\angle A$ formed by the two congruent sides of an isosceles triangle?

The angle bisector of the vertex angle of an isosceles triangle is also the perpendicular bisector of the base, dividing the triangle into two congruent right triangles.