| (B) construct congruent segments, congruent angles,
a segment bisector, an angle bisector, perpendicular lines, the perpendicular
bisector of a line segment, and a line parallel to a given line through
a point not on a line using a compass and a straightedge;
(C) use the constructions of congruent segments, congruent
angles, angle bisectors, and perpendicular bisectors to make conjectures
about geometric relationships; and
(D) verify the Triangle Inequality theorem using constructions
and apply the theorem to solve problems.
(6) Proof and congruence. The student uses the process
skills with deductive reasoning to prove and apply theorems by using
a variety of methods such as coordinate, transformational, and axiomatic
and formats such as two-column, paragraph, and flow chart. The student
is expected to:
(A) verify theorems about angles formed by the intersection
of lines and line segments, including vertical angles, and angles
formed by parallel lines cut by a transversal and prove equidistance
between the endpoints of a segment and points on its perpendicular
bisector and apply these relationships to solve problems;
(B) prove two triangles are congruent by applying the
Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side,
and Hypotenuse-Leg congruence conditions;
(C) apply the definition of congruence, in terms of
rigid transformations, to identify congruent figures and their corresponding
sides and angles;
(D) verify theorems about the relationships in triangles,
including proof of the Pythagorean Theorem, the sum of interior angles,
base angles of isosceles triangles, midsegments, and medians, and
apply these relationships to solve problems; and
(E) prove a quadrilateral is a parallelogram, rectangle,
square, or rhombus using opposite sides, opposite angles, or diagonals
and apply these relationships to solve problems.
(7) Similarity, proof, and trigonometry. The student
uses the process skills in applying similarity to solve problems.
The student is expected to:
(A) apply the definition of similarity in terms of
a dilation to identify similar figures and their proportional sides
and the congruent corresponding angles; and
(B) apply the Angle-Angle criterion to verify similar
triangles and apply the proportionality of the corresponding sides
to solve problems.
(8) Similarity, proof, and trigonometry. The student
uses the process skills with deductive reasoning to prove and apply
theorems by using a variety of methods such as coordinate, transformational,
and axiomatic and formats such as two-column, paragraph, and flow
chart. The student is expected to:
(A) prove theorems about similar triangles, including
the Triangle Proportionality theorem, and apply these theorems to
solve problems; and
(B) identify and apply the relationships that exist
when an altitude is drawn to the hypotenuse of a right triangle, including
the geometric mean, to solve problems.
(9) Similarity, proof, and trigonometry. The student
uses the process skills to understand and apply relationships in right
triangles. The student is expected to:
(A) determine the lengths of sides and measures of
angles in a right triangle by applying the trigonometric ratios sine,
cosine, and tangent to solve problems; and
(B) apply the relationships in special right triangles
30°-60°-90° and 45°-45°-90° and the Pythagorean
theorem, including Pythagorean triples, to solve problems.
(10) Two-dimensional and three-dimensional figures.
The student uses the process skills to recognize characteristics and
dimensional changes of two- and three-dimensional figures. The student
is expected to:
(A) identify the shapes of two-dimensional cross-sections
of prisms, pyramids, cylinders, cones, and spheres and identify three-dimensional
objects generated by rotations of two-dimensional shapes; and
(B) determine and describe how changes in the linear
dimensions of a shape affect its perimeter, area, surface area, or
volume, including proportional and non-proportional dimensional change.
(11) Two-dimensional and three-dimensional figures.
The student uses the process skills in the application of formulas
to determine measures of two- and three-dimensional figures. The student
is expected to:
(A) apply the formula for the area of regular polygons
to solve problems using appropriate units of measure;
(B) determine the area of composite two-dimensional
figures comprised of a combination of triangles, parallelograms, trapezoids,
kites, regular polygons, or sectors of circles to solve problems using
appropriate units of measure;
(C) apply the formulas for the total and lateral surface
area of three-dimensional figures, including prisms, pyramids, cones,
cylinders, spheres, and composite figures, to solve problems using
appropriate units of measure; and
(D) apply the formulas for the volume of three-dimensional
figures, including prisms, pyramids, cones, cylinders, spheres, and
composite figures, to solve problems using appropriate units of measure.
(12) Circles. The student uses the process skills to
understand geometric relationships and apply theorems and equations
about circles. The student is expected to:
(A) apply theorems about circles, including relationships
among angles, radii, chords, tangents, and secants, to solve non-contextual
problems;
(B) apply the proportional relationship between the
measure of an arc length of a circle and the circumference of the
circle to solve problems;
(C) apply the proportional relationship between the
measure of the area of a sector of a circle and the area of the circle
to solve problems;
(D) describe radian measure of an angle as the ratio
of the length of an arc intercepted by a central angle and the radius
of the circle; and
(E) show that the equation of a circle with center
at the origin and radius r is x2 + y2 = r2 and determine
the equation for the graph of a circle with radius r and center (h,
k), (x - h)2 + (y - k)2 = r2 .
(13) Probability. The student uses the process skills
to understand probability in real-world situations and how to apply
independence and dependence of events. The student is expected to:
(A) develop strategies to use permutations and combinations
to solve contextual problems;
(B) determine probabilities based on area to solve
contextual problems;
(C) identify whether two events are independent and
compute the probability of the two events occurring together with
or without replacement;
(D) apply conditional probability in contextual problems;
and
(E) apply independence in contextual problems.
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