1.
How many counterexamples do you need to disprove a conjecture about a geometric relationship?
Correct Answer
B. 1
Explanation
To disprove a conjecture about a geometric relationship, you only need one counterexample. A counterexample is a single example that contradicts the conjecture, proving it to be false. By providing just one counterexample, you can demonstrate that the conjecture does not hold true in all cases, thus disproving it.
2.
Amy drew the diagram below and made a conjecture. "Midsegments in a triangle _____ form(s) 4 equal triangles. Which word completes Amy's conjecture?
Correct Answer
A. Always
Explanation
Amy's conjecture is that midsegments in a triangle always form 4 equal triangles. This means that regardless of the type of triangle, the midsegments will always divide the triangle into 4 equal parts.
3.
Which of these statements is true about a triangle?
Correct Answer
A. A triangle may have a line longer than the other two.
Explanation
A triangle may have a line longer than the other two because the lengths of the sides of a triangle can vary. In a scalene triangle, all three sides have different lengths, so one side can be longer than the other two. In an isosceles triangle, two sides are equal in length, while the third side can be longer or shorter. Only in an equilateral triangle are all three sides equal in length. Therefore, it is possible for a triangle to have a line longer than the other two.
4.
The midsegments of a quadrilateral always form a
Correct Answer
D. Parallelogram
Explanation
The midsegments of a quadrilateral always form a parallelogram. A midsegment is a line segment that connects the midpoints of two sides of a quadrilateral. In a quadrilateral, the opposite sides are parallel, and the midsegments connect the midpoints of these parallel sides. Since a parallelogram has opposite sides that are parallel, the midsegments will also be parallel to each other. Therefore, the correct answer is parallelogram.
5.
Which conjecture is correct?
Correct Answer
B. The diagonals of a quadrilateral always form angles that are supplementary when adjacent.
Explanation
The given answer is correct because it accurately describes the relationship between the diagonals of a quadrilateral. When the diagonals of a quadrilateral are drawn, the angles formed between them are always supplementary, meaning they add up to 180 degrees. This property holds true for all quadrilaterals, regardless of their shape or size. Therefore, the statement that the diagonals of a quadrilateral always form angles that are supplementary when adjacent is a valid conjecture.
6.
A conjecture that is sometimes true should not be
Correct Answer
C. Accepted
Explanation
If a conjecture is sometimes true, it means that there are instances where it holds true. Therefore, it would not be appropriate to reject the conjecture outright. Instead, it should be accepted, and further examination should be conducted to determine the conditions under which it is true and the conditions under which it is false. By accepting the conjecture, researchers can investigate and analyze it further to gain a deeper understanding of its validity.
7.
Colin wanted to make a variety of pennants by sewing a smaller triangle on top of a larger triangle, aligning the vertices of the smaller triangle with the midpoints of the larger triangle.
He argued that the length of the side of the smaller triangle is exactly one-half the length of an intersecting side of the larger triangle. Colin's conjecture needs to be revised because it is only correct for ________ triangles.
Correct Answer
D. Equilateral
Explanation
Colin's conjecture needs to be revised because it is only correct for equilateral triangles. In an equilateral triangle, all sides are equal in length and all angles are equal. Therefore, the length of the side of the smaller triangle would indeed be exactly one-half the length of an intersecting side of the larger triangle. However, this would not hold true for right, scalene, or isosceles triangles, as their side lengths and angles differ.
8.
Which conjecture is true for a kite?
Correct Answer
A. Its midsegments form a rectangle.
Explanation
The correct answer is "Its midsegments form a rectangle." A kite is a quadrilateral with two pairs of adjacent congruent sides. The midsegments of a kite are the segments connecting the midpoints of its sides. Since a kite has two pairs of congruent sides, its midsegments will also be congruent. The midsegments of a kite form a rectangle because opposite sides of a rectangle are congruent and its angles are right angles. Therefore, the statement that the midsegments of a kite form a rectangle is true.
9.
Jafar said the midpoints of a rhombus always form a square. His conjecture is
Correct Answer
D. Incorrect because it is sometimes false
Explanation
The answer is "incorrect because it is sometimes false." This means that Jafar's statement is not always true. While it is true that the midpoints of a rhombus can form a square in some cases, it is not always the case. There are instances where the midpoints of a rhombus do not form a square. Therefore, Jafar's conjecture is not always true.
10.
Kevin created a true conjecture to predict the measure of the exterior angle of a triangle when the two angles opposite the adjacent interior angle are known. Which formula would support his conjecture?
Correct Answer
B. D = b + c
Explanation
The formula d = b + c supports Kevin's conjecture because when the two angles opposite the adjacent interior angle are known, the measure of the exterior angle of a triangle is equal to the sum of those two angles.