GLCE/HSCE: Geometry Assessment
GLCE/HSCE: Geometry Assessments
L4.1.1 Distinguish between inductive and deductive reasoning, identifying and providing examples of each.
1. The process of using facts, rules, definitions, or properties in logical order to reach a conclusion is called
C. deductive reasoning
D. detachment reasoning
2. Which of the following is an example of inductive reasoning?
A. Two altitudes of a right triangle are perpendicular to each other. Jamie drew a triangle with two perpendicular altitudes. Therefore, she drew a right angle.
B. The sum of the interior angles of a triangle is 180. Leroy drew a triangle where the sum of two of the angles was 100. Leroy concluded that the third angle of his triangle was 80.
C. The diagonals of a square bisect its right-angle vertices. Kelly drew a square with diagonals. Therefore, the angles formed at each vertex are complementary.
D. Squares, rectangles, and rhombuses have four sides and are classified as quadrilaterals. Chan drew a four-sided figure. Chan concluded that his figure is a quadrilateral.
L4.1.2 Differentiate between statistical arguments (statements verified empirically using examples or data) and logical arguments based on the rules of logic.
1.Assuming the following statements are true, which of the following is a valid conclusion?
Some musicians are happy people.
All happy people like music.
A. Some musicians like music.
B. Some happy people do not like music.
C. All musicians like music.
D. All happy people are musicians.
2. Which of the following groups of statements represents a valid argument?
A. Given: All four sided figures are quadrilaterals. All parallelograms are quadrilaterals.
Conclusion: All parallelograms are quadrilaterals.
B. Given: All rectangles have angles. All squares have four sides.
Conclusion: All rectangles are squares.
C. Given: All quadrilaterals have four sides. All squares have four sides.
Conclusion: All quadrilaterals are squares.
D. Given: All squares have congruent sides. All rhombuses have congruent sides.
Conclusion: All rhombuses are squares.
3. Daniel wants to send Jasmine flowers for her birthday. At the flower store, he can choose between roses, irises, or carnations. The salesperson tells Daniel that 50% of the customers buy roses, 30% buy carnations, and 20% buy irises. Which of the following is a valid conjecture?
A.More customers buy roses than carnations.
B. The salesperson likes carnations.
C. Jasmine will be excited to receive flowers for her birthday.
D. Daniel will buy Jasmine irises.
L4.1.3 Define and explain the roles of axioms (postulates), definitions, theorems, counterexamples, and proofs in the logical structure of mathematics. Identify and give examples of each.
1. A _________ is a statement that describes a fundamental relationship between the basic terms of geometry and is accepted as true.
2. In the figure, points A, B, and C lie in plane Z. Which of the following postulates can be used to show that A and B are collinear?
A. If two planes intersect, then their intersection is a line.
B. If two lines intersect, then their intersection is exactly one point.
C. Through any three points not on the same line, there is exactly one plane.
D. Through any two points, there is exactly one line.
3. State the counterexample that demonstrates that the converse of the following statement is false:
If an angle measures 48, then it is acute.
A. An angle measures 56 and is acute.
B. All acute angles have measures between 0 and 90.
C. An angle measures 48 if and only if it is acute.
D. If an angle is acute, then it measures 48.
L4.3.3 Explain the difference between a necessary and a sufficient condition within the statement of a theorem. Determine the correct conclusions based on interpreting a theorem in which necessary or sufficient conditions in the theorem or hypothesis are satisfied.
1.The quadrilateral ABCD is a parallelogram.
Which of the following pieces of information would suffice to prove that ABCD is a rectangle?
A.AB = AD
B.angle A and angle B are supplementary
C.measure of angle B = measure of angle D
D. AC = BD
2. Which is a necessary and sufficient condition for a parallelogram to be classified as a rectangle?
A. Opposite sides are congruent.
B. Opposite sides are parallel.
C. Diagonals bisect each other.
D. All angles are right angles.
G1.1.1 Solve multistep problems and construct proofs involving vertical angles, linear pairs of angles, supplementary angles, complementary angles, and right angles.
1. The measures of some angles are given in the figure.
What is the value of x?
2. George used a decorative fencing to enclose his deck.
Using the information on the diagram and assuming the top and bottom are parallel, the measure of angle x is
3. The Department of Transportation wants to extend the intersecting road across the highway, as indicated by the dotted line.
What should x be to ensure that the intersecting road and the new construction form a straight line?
G1.1.3 Perform and justify constructions, including midpoint of a line segment and bisector of an angle, using straightedge and compass.
1. The drawing shows a compass and straightedge construction of
A. a perpendicular to a given line at a point on the line
B. the bisector of a given angle
C. an angle congruent to a given angle
D. a perpendicular to a given line from a point not on the line
2. Use a compass, straightedge, and the drawing below to answer the question.
Which point lies on the line that bisects angle CAB?
3. Which pair of points determines the perpendicular bisector of line segment AB?
A. Y, W
B. Y, Z
C. X, W
D. X, Z
G1.1.4 Given a line and a point, construct a line through the point that is parallel to the original line using straightedge and compass. Given a line and a point, construct a line through the point that is perpendicular to the original line. Justify the steps of the constructions.
1. Which point would be on a line perpendicular to l through T?
2. Use your compass and straightedge to construct a line that is perpendicular to ST and passes through point O.
Which other point lies on this perpendicular?
3. To which point should a line segment from A be drawn so that the resulting figure is a rectangle?
G1.1.5 Given a line segment in terms of its endpoints in the coordinate plane, determine its length and midpoint.
1. The coordinates of the midpoint of AB are
A. (-2, 5)
B. (5, 3)
C. (2, 5)
D. (-5, 3)
2. Which point is the greatest distance from the origin?
A. (3, 4)
B. (9, 2)
C. (-9, 1)
D. (-8, -5)
3. The coordinates of the midpoint of AB are (-2, 1), and the coordinates of A are (2, 3). What are the coordinates of B?
A. (0, 2)
B. (-1, 2)
C. (-6, -1)
D. (-3, -4)
G1.1.6 Recognize Euclidean geometry as an axiom system. Know the key axioms and understand the meaning of and distinguish between undefined terms, axioms, definitions, and theorems.
1. Plane geometry is based on several undefined terms. Which of the following is an undefined term?
L3.1.1 Convert units of measurement within and between systems; explain how arithmetic operations on measurements affect units, and carry units through calculations correctly.
1. John needs 180 square feet of tile. The tiles are 3-inch squares and are packed 50 to a box. What is the minimum number of boxes of tiles John must purchase?
G1.2.1 Prove that the angle sum of a triangle is 180 and that an exterior angle of a triangle is the sum of the two remote interior angles.
1. What is the measure of angle 3?
2. The figure has angle measures as shown.
What is the measure of angle ABD?
3. In the figure, the measure of angle CAD is twice the measure of angle CAB.
What is the measure of CAB?
G1.2.2 Construct and justify arguments and solve multistep problems involving angle measure, side length, peri