points,lines,plane
how many points determine a line
a line is determined by two distinct points
what is a line segment
a line segment is a part of line that has two endpoints and a definite length
what are complementary angles
complementary angles are two angles whose measures add up to 90 degrees
which statement is valid deductive reasoning
Given: All squares have four equal sides.
Statement: Shape A is a square, so Shape A has four equal sides.
Yes, this is valid deductive reasoning because it uses a general rule (“all squares…”) to make a specific conclusion about Shape A.
what is the midpoint of a segment
The midpoint is the point that is exactly halfway between the two endpoints of a segment, dividing it into two congruent parts.
how many points determine a plane
a plane is determined by three non-collinear points
how do you find the length of a line segment on a number line
subtract the coordinates and take the absolute value
what are supplementary angles
supplementary angles are two angles whose measures add up to 180 degrees
what is deductive reasoning
Deductive reasoning is a logical process where you use general rules, definitions, or known facts to reach a specific, guaranteed conclusion. If the premises are true, the conclusion must also be true.
What does it mean for a line to bisect a segment?
A line bisects a segment when it cuts the segment into two equal parts, meeting the segment at its midpoint.
what is the definition of a point
how do you find the midpoint of a segment with endpoints
use the midpoint formula
what are vertical angles
vertical angles are opposite angles formed by two intersecting lines, they are always congruent
If p → q is true, and p is true, what can we conclude using deductive reasoning?
We can conclude that q must be true. This is called modus ponens, a common deductive reasoning pattern.
A segment has endpoints X(–2, 5) and Y(6, 1). A line passes through the midpoint of XY. What does this tell us about the line?
Since it passes through the midpoint, the line is a segment bisector. It divides segment XY into two congruent segments.
what does it mean for points to be collinear
points are collinear if they all lie on the same line. If no single line can contain all of them, they are not collinear
what does it mean for segments to be congruent
congruent segments have the same length, even if they are in different possitions
what are adjacent angles
adjacent angles are two angles that share a common side and common vertex, and they do not overlap
Is the following deductive reasoning?
“If two angles form a linear pair, then they are supplementary. Angle A and Angle B form a linear pair, so they are supplementary.”
Yes, this is deductive reasoning because it applies a known geometric rule to a specific situation to reach a definite conclusion.
If point M is the midpoint of segment AB, and AM = 7 cm, what is the length of AB?
Since a midpoint divides a segment into two equal parts, AB = 2 × AM = 14 cm.
what is the intersection of two planes
when two planes interact they meet in exactly one line
what does it mean for a point to be the midpoint of a segment
a midpoint is a point that divides a segment into two congruent parts
what is a linear pair
a linear pair of adjacent angles whose non-shared sides form a straight line.
Why is deductive reasoning important in math proofs?
Deductive reasoning ensures that each step of a proof logically follows from accepted rules or previously proven statements, which guarantees the final conclusion is true.
Find the midpoint of the segment with endpoints A(4, 10) and B(8, 2).
Use the midpoint formula:
(x1+x22,y1+y22)\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)(2x1+x2,2y1+y2)
So the midpoint is
(4+82,10+22)=(6,6)\left( \frac{4+8}{2}, \frac{10+2}{2} \right) = (6, 6)(24+8,210+2)=(6,6)