Download Curriculum Outline for Geometry Chapters 1

Curriculum Outline for Geometry
Text: Geometry, Edward B. Burger, et al, Holt, Rinehart, Winston, 2008
Accompanying website: www.go.hrw.com
Weeks 1 and 2
Chapter 1: Foundations for Geometry
The student will be able to:
1. draw and name points, lines, and planes
2. describe and name collinear, noncollinear, coplanar, and non-coplanar points
3. describe and name segments, endpoints, rays, and opposite rays
4. draw diagrams to represent line and plane postulates
5. determine the length of a line segment
6. define congruent line segments
7. construct a segment congruent to a given segment
8. explain the concept of “betweenness” and solve numeric applications
9. define and solve problems involving midpoints and segment bisectors
10. define, draw, and name an angle and its vertex
11. label points in the interior and exterior of an angle
12. name an angle in different ways
13. measure an angle in degrees by using a protractor
14. classify angles as acute, right, obtuse, straight, or reflex
15. define and construct congruent angles
16. state and apply the angle addition postulate
17. construct the bisector of an angle
18. solve numeric problems involving angle bisectors
19. define, draw, and name adjacent angles
20. define a linear pair of angles
21. define complementary and supplementary angles and solve numerical problems
involving them
22. define vertical angles and solve numerical problems involving them
23. write the formulas area and perimeter of a rectangle, square, and triangle
24. solve numerical problems involving the area and perimeter of a rectangle, square, and
triangle
25. define the terms radius, diameter, and circumference of a circle
26. define pi as the ratio of the circumference of a circle to its diameter
27. solve numerical problems involving the circumference and area of the circle
28. draw and label coordinate axes and scale on a graph
29. develop and apply the midpoint formula
30. solve numeric problems to calculate the coordinates of an endpoint when given a
midpoint and the other endpoint
31. use the Pythagorean Theorem to find the distance between two points
32. develop the distance formula and apply it to numeric problems
33. define the terms transformation, preimage, and image
34. identify reflections, rotations, and translations when given a diagram of each
35. find the coordinates of the image of a figure after applying a translation
36. graph the preimage and image of a given figure, when given the coordinates of the
preimage
____________________________________________________________________________
Weeks 3 to 6
Chapter 2: Geometric Reasoning
The student will be able to:
1. use inductive reasoning to identify patterns based on previous information
2. prove a conjecture is true or find a counterexample to disprove it
3. specify the hypothesis and conclusion of a conditional statement
4. write any statement in conditional form
5. identify, write, and analyze the truth value of conditional statements
6. write the inverse, converse, and contrapositive of conditional statements
7. recognize and write logically equivalent statements
8. apply the law of detachment in logical reasoning
9. apply the law of syllogism in logical reasoning
10. separate a biconditional into a statement and its converse
11. determine the truth value of biconditional statements
12. write definitions as biconditionals
13. understand the properties of equality
14. justify each step in solving equations using the properties of equality
15. identify the reflexive, symmetric, and transitive properties of congruence
16. understand and apply these theorems in geometric proofs:
a.
b.
c.
d.
If two angles form a linear pair, then they are supplementary.
If two angles are supplementary to the same angle or to two congruent angles, then the two
angles are congruent.
All right angles are congruent.
If two angles are complementary to the same angle or to congruent angles, then the two angles
are congruent.
17. use the definitions (right angles, segment bisectors, angle bisectors, midpoints,
perpendicular lines, triangles), properties (equality and congruence), postulates, and
theorems (line segments and angles) presented so far to complete statements and/or
reasons in a geometric proof
18. understand and apply the theorems:
a.
b.
Vertical angles are congruent.
If two congruent angles are supplementary, then each angle is a right angle.
19. construct a flowchart proof when given a two-column statement and reason proof
20. write a paragraph proof when given a two-column statement and reason proof
21. write symbolic logic symbols to represent negations, conditionals, conjunctions, and
disjunctions
22. complete truth tables for negations, conditionals, conjunctions, and disjunctions
23. construct a truth table for a compound statement
____________________________________________________________________________
Weeks 7 and 8
Chapter 3: Parallel and Perpendicular Lines
The student will be able to:
1. identify and define parallel lines, perpendicular lines, skew lines, and parallel planes
2. lines perpendicular to planes, perpendicular planes, and parallel planes. (pp. 897905)
I. state the conditions for a line to be perpendicular to a plane
b. If a line is perpendicular to each of two intersecting lines at their point of intersection, then the
line is perpendicular to the plane determined by them.
c. Through a given point there passes one and only one plane perpendicular to a given line.
d. Through a given point there passes one and only one line perpendicular to a given plane.
II. discover, state, and apply conditions for planes to be perpendicular
a. If two planes are perpendicular to each other if and only if one plane contains a line
perpendicular to the second plane.
b. If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the
given plane.
III. discover, state, and apply the theorems (perpendicular planes):
a. Two lines perpendicular to the same plane are coplanar.
b. If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of
intersection with the given plane (known as its foot), is in the given plane.
IV. investigate and state theorems (parallel planes):
a. If a plane intersects two parallel planes then the lines of intersection are parallel.
b. If two planes are perpendicular to the same line then they are parallel.
c. If two planes are parallel to a third plane, they are parallel.
d. A line perpendicular to one of two parallel planes is perpendicular to the other.
3. identify corresponding, alternate interior, alternate exterior, and same-side interior
angles
4. understand the following postulate and theorems:
a.
If two parallel lines are cut by a transversal, then the pairs of corresponding angles are
congruent.
b.
c.
d.
If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are
congruent.
If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are
congruent.
If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are
supplementary.
5. apply the postulate and theorems listed above to numeric problems and to proofs
6. write the converses of the postulate and theorems involving parallel lines and angles:
a. If two coplanar lines are cut by a transversal so that a pair of corresponding
angles is congruent, then the two lines are parallel.
b. If two coplanar lines are cut by a transversal so that a pair of alternate interior
angles is congruent, then the two lines are parallel.
c. If two coplanar lines are cut by a transversal so that a pair of alternate exterior
angles is congruent, then the two lines are parallel.
d. If two coplanar lines are cut by a transversal so that a pair of same-side
interior angles is supplementary, then the two lines are parallel.
7. understand Euclid’s Parallel Postulate:
Through a point P not on line l, there is exactly one line parallel to l.
8. construct a line parallel to a given line through a given point, using a compass and a
straightedge
9. apply the converse postulate and theorems to numeric problems and to proofs
10. define the perpendicular bisector of a line segment
11. construct the perpendicular bisector of a segment using a compass and a straightedge
12. identify the distance from a point to a line as the shortest segment from a point to the
line
13. understand the theorems about perpendicular lines:
a.
b.
c.
If two intersecting lines form a linear pair of congruent angles, then the lines are
perpendicular.
In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular
to the other line.
If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each
other.
14. apply these theorems to numeric problems and to proofs
15. find slope from the graph of a line by counting rise over run
16. calculate the slope of a line using the slope formula
17. identify the slope of a line as positive, negative, zero, or no slope
18. state the relationship between the slopes of parallel lines
19. state the relationship between the slopes of perpendicular lines
20. write the equation of a line using the point-slope form of a line
21. write the equation of a line using the slope-intercept form of a line
22. write the equation of a vertical line
23. write the equation of a horizontal line
24. graph a line when given the equation of the line
25. determine if a pair of lines is parallel, intersecting, or coinciding
_______________________________________________________________
Weeks 9 and 10
Chapter 4: Triangle Congruence
The student will be able to:
1. classify triangles according to angle measure: acute, equiangular, obtuse, and right
2. classify triangles by side lengths: equilateral, isosceles, and scalene
3. apply triangle classification to numeric problems
4. understand and apply the theorem:
The sum of the angle measures of a triangle is 180°
5. apply the corollaries of this theorem:
a.
b.
The acute angles of a right triangle are complementary.
The measure of each angle of an equiangular triangle is 60º.
6. understand and apply the theorem:
The measure of an exterior angle of a triangle is equal to the sum of the measures of its remote
interior angles.
7. understand and apply the theorem:
If two angles of one triangle are congruent to two angles of another triangle, then the third pair of
angles is congruent.
8. name corresponding angles and sides when given a pair of congruent triangles
9. solve numeric problems involving corresponding sides and angles
10. write a two-column proof to prove triangles congruent (proving 3 angles and 3 sides
congruent)
11. understand and apply the theorem:
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are
congruent.
12. label the included angle of a triangle when given two sides
13. understand and apply the theorem:
If two sides and the included angle of one triangle are congruent to two sides and the included
angle of another triangle, then the triangles are congruent.
14. write proofs involving these theorems
15. understand and apply the theorem:
If two angles and the included side of one triangle are congruent to two angles and the included
side of another triangle, then the triangles are congruent.
16. understand and apply the theorem:
If two angles and a non-included side of one triangle are congruent to two angles and a nonincluded side of another triangle, then the triangles are congruent.
17. understand and apply the theorem:
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another
right triangle, then the triangles are congruent.
18. write proofs involving these theorems
19. understand the statement:
Corresponding parts of congruent triangles are congruent.
20. use this theorem in two column proofs
21. use this theorem in coordinate geometry proofs
22. position figures in the coordinate plane for use in coordinate proofs
23. prove geometric concepts when given numeric and algebraic expressions for the
coordinates of figures
24. understand and prove the theorem:
If two sides of a triangle are congruent, then the angles opposite the sides are congruent.
25. understand and prove the converse of this theorem:
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
26. find the measure of the vertex angle of an isosceles triangle, when given one base
angle
27. find the measure of a base angle of an isosceles triangle, when given the vertex angle
28. prove the corollary:
If a triangle is equilateral then it is equiangular.
29. prove the converse of this corollary:
If a triangle is equiangular, then it is equilateral.
30. apply the theorems and corollaries to numeric problems and to coordinate geometry
proofs
_____________________________________________________________________________
Weeks 11 and 12
Chapter 5: Properties and Attributes of Triangles
The student will be able to:
1. prove and apply theorems about perpendicular bisectors:
a.
b.
If a point is on the perpendicular bisector of a segment, then it is equidistant from the
endpoints of the segment.
(Converse) If a point is equidistant from the endpoints of a segment, then it is on the
perpendicular bisector of the segment.
2. identify the locus of the endpoints of a segment
3. prove and apply theorems about angle bisectors:
a.
b.
If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
(Converse) If a point in the interior of an angle is equidistant from the sides of the angle, then
it is on the bisector of the angle.
4. write the equation for the perpendicular bisector of a line segment in the coordinate
plane, when given the endpoints of the segment
5. define and apply the words concurrent, circumcenter, circumscribe, incenter
6. prove and apply the properties of the perpendicular bisectors of a triangle. “The
circumcenter of a triangle is equidistant from the vertices of the triangle.”
7. solve numeric and coordinate geometry problems involving perpendicular bisectors
8. prove and apply properties of angle bisectors of a triangle. “The incenter of a triangle
is equidistant from the sides of the triangle.”
9. solve problems involving the angle bisectors of a triangle
10. define the terms median, centroid, altitude, and orthocenter of a triangle
11. apply the properties of the median of a triangle
The centroid of a triangle is located ⅔ of the distance from each vertex to the midpoint of the opposite
side.
12. apply properties of altitudes of a triangle to numeric and coordinate geometry
problems
13. define and draw the 3 midsegments of a triangle
14. prove and apply the triangle midsegment theorem:
A midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.
15. solve numeric and coordinate examples using the triangle midsegment theorem
16. write an indirect proof (Enrichment)
17. apply inequality theorems to numeric problems using one triangle:
a.
b.
c.
If two sides of a triangle are not congruent, then the larger angle is opposite the longer side.
If two angles of a triangle are not congruent, then the longer side is opposite the larger angle.
The sum of any two side lengths of a triangle is greater than the third side.
18. apply inequality theorems to two triangles:
a.
b.
If two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent,
then the longer third side is across from the larger included angle.
(Converse) If two sides of one triangle are congruent to two sides of another triangle and the third sides are not
congruent, then the larger included angle is across from the longer third side.
19. simplify and rationalize radicals
20. use the Pythagorean Theorem and its converse to solve problems
21. use Pythagorean inequalities to classify triangles as right, acute, or obtuse
22. justify and apply properties of 45°-45°-90° triangles
23. justify and apply properties of 30°-60°-90° triangles
____________________________________________________________________________
Weeks 13 and 14
Chapter 6 – Polygons and Quadrilaterals
The student will be able to:
1.
2.
3.
4.
5.
define polygon, regular polygon, concave, and convex polygon
classify polygons based on their sides and angles
find and use the measures of interior and exterior angles of polygons
define parallelogram
prove and apply properties of parallelograms:
a.
b.
c.
d.
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
If a quadrilateral is a parallelogram, then its consecutive angles are supplementary.
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
6. use properties of parallelograms to solve numeric problems and coordinate geometry
problems
7. prove that a quadrilateral is a parallelogram using the theorems:
a.
b.
c.
d.
e.
If one pair of opposite sides of a quadrilateral is parallel and congruent, then the quadrilateral is a parallelogram.
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
If an angle of a quadrilateral is supplementary to both its consecutive angles, then the quadrilateral is a parallelogram.
If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
8. solve numeric and coordinate problems using these theorems
9. prove and apply properties of rectangles, rhombi, and squares
10. use properties of rectangles, rhombi, and squares to solve problems
11. use properties of rectangles, rhombi, and squares in proofs
12. prove that a quadrilateral is a special parallelogram using the theorems:
a.
b.
c.
d.
e.
If one angle of a parallelogram is a right angle then the parallelogram is a rectangle.
If the diagonals of a parallelogram are congruent, then the parallelogram s a rectangle.
If one pair of consecutive sides of a parallelogram is congruent then the parallelogram is a rhombus.
If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.
If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.
13. apply the theorems listed to proofs and to coordinate geometry problems
14. define a kite, a trapezoid, an isosceles trapezoid, and the midsegment of a trapezoid
15. use properties of kites to solve problems (Enrichment)
a.
b.
If a quadrilateral is a kite, then its diagonals are perpendicular.
If a quadrilateral is a kite, the exactly one pair of opposite angles is congruent.
16. specify the base, legs, and base angles of a trapezoid
17. use properties of trapezoids to solve problems
a.
b.
c.
If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.
If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles.
A trapezoid is isosceles if and only if its diagonals are congruent.
d.
The midsegment of a trapezoid is parallel to each base, and its length is one-half the sum of the lengths of the bases
____________________________________________________________________________
Weeks 15 and 16
Chapter 7 – Similarity
The student will be able to:
1. define ratio, proportion, means, extremes, cross products
2. investigate, justify, and apply the theorem: "In a proportion, the product of the means
equals the product of the extremes."
3. Solve numerical, algebraic and verbal problems involving ratios and proportions
4. determine if a proportion is valid
5. given the cross products define a ratio or proportion
6. Define similar, similar polygons, similarity ratio
7. Identify corresponding angles and corresponding sides in similar polygons
8. Determine if two polygons are similar
9. Apply the concept of similar polygons in verbal problems
10. Apply the Angle-Angle, Side-Side-Side, Side-Angle-Side theorem of proving two
triangles similar



If two angles of a triangle are congruent to two angles of another triangle, then the triangles are similar.
If the three sides of one triangle are proportional to three corresponding sides of another triangle, then the triangles are
similar.
If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then
the triangles are similar.
11. Verify that two triangles are similar using given data
12. Find the lengths of unknown sides in two similar triangles
13. Write a formal proof involving similar triangles
14. Prove and apply the following theorems in formal proofs to show line segments are
in proportion:





If a line is parallel to one side of a triangle and intersects the other two sides, then the line divides the two sides
proportionally.
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.
An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of
the other two sides.
If a line segment joins the midpoints of two sides of a triangle, then it is parallel to the third side and has length equal to
one-half the length of the third side.
15. Prove and apply the following theorems in formal proofs:


If two triangles are similar, then their corresponding sides are in proportion.
In a proportion, the product of the means equals the product of the extremes.
16. Set up and solve proportions to find the height or distance of too large to measure
17. Set up and solve proportions to find the dimensions of an object on a scale drawing
18. Use ratios to find the perimeter and areas of similar polygons
19. Define the following terms: dilation, scale factor
20. Dilate a polygon given the scale factor
21. Find the coordinates of similar triangles
22. Prove two triangles are similar that are drawn on the coordinate plan