reasoning in geometry
Define inductive reasoning
Using patterns or observations to make a general conclusion.
What is a ray?
A part of a line that has one endpoint and extends forever in one direction.
A flip over a line is called a ________.
Reflection
Name the triangle congruence rule: two sides and the included angle match.
SAS (Side-Angle-Side).
A quadrilateral with exactly one pair of parallel sides is called a ________.
Trapezoid
Give an example of a conditional statement in “if–then” form.
If two angles form a linear pair, then they are supplementary.
Name the angle pair: two angles that add up to 90°.
Complementary angles
What transformation slides a figure without turning it?
Translation
List two ways to prove triangles are congruent that use angles.
ASA and AAS.
How many sides does a nonagon have?
9 sides
What is the contrapositive of the statement: “If a figure is a square, then it has four sides”?
If a figure does not have four sides, then it is not a square.
A segment measures 12 cm. Point B is the midpoint. What is the length of AB?
6cm
Describe the rotation: turning a figure 90° counterclockwise around the origin.
(x, y) → (–y, x)
If two angles of a triangle are 40° and 55°, what is the third angle?
85°
In a parallelogram, opposite angles are ________.
Congruent
Explain why one counterexample is enough to disprove a conjecture.
Because a conjecture claims something is always true, and one counterexample shows it is not always true.
In the diagram, lines l and m are parallel, and t is a transversal. If one alternate interior angle is 70°, find the other alternate interior angle.
70° (alternate interior angles are congruent).
A figure is reflected over the y-axis, then translated right 4 units. Describe the transformation.
Reflection over y-axis: (x, y) → (–x, y)
Translation: (–x + 4, y)
Which congruence theorem is valid for right triangles only?
HL (Hypotenuse-Leg).
The interior angles of a pentagon add up to what?
(5 – 2) × 180 = 540°
Use deductive reasoning to prove:
“Vertical angles are congruent.”
When two lines intersect, they form two pairs of vertical angles.
Each vertical angle shares a linear pair with the same adjacent angle.
Linear pairs are supplementary.
If two angles are supplementary to the same angle, they are congruent.
Therefore, vertical angles are congruent.
Using the Angle Addition Postulate, solve:
If ∠ABC is split into ∠ABD and ∠DBC.
m∠ABD = 3x + 5, m∠DBC = x – 1, and m∠ABC = 60°.
(3x + 5) + (x – 1) = 60
4x + 4 = 60
4x = 56
x = 14
A triangle undergoes a rotation of 180° about the origin and then a reflection over the x-axis. What is the final rule?
180° rotation: (x, y) → (–x, –y)
Reflection over x-axis: (–x, –y) → (–x, y)
Final rule: (x, y) → (–x, y)
Prove triangles ΔABC and ΔDEF are congruent if AB = DE, AC = DF, and ∠A = ∠D.
Given AB = DE, AC = DF, and ∠A = ∠D (included angle).
Therefore, ΔABC ≅ ΔDEF by SAS.
In a rectangle, diagonals are 26 units long. If one diagonal is split into segments 10 and x by the intersection point, find x.
Diagonals of a rectangle bisect each other.
So each half = 26 ÷ 2 = 13
10 + x = 13 → x = 3