Geometry Major Content Review
Topic / Standards1 / G-CO: Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
2 / G-CO: Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
3 / G-CO: Prove geometric theorems
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
4 / G-SRT: Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
5 / G-SRT: Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
6 / G-SRT: Define trigonometric ratios and solve problems involving right triangles
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
7. Explain and use the relationship between the sine and cosine of complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
7 / G.GPE: Use coordinates to prove simple geometric theorems algebraically
4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
8 / G.GPE: Use coordinates to prove simple geometric theorems algebraically
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
9 / G-C: Understand and apply theorems about circles
1. Prove that all circles are similar.
2. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral
10 / G-GMD: Explain volume formulas and use them to solve problems
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use Cavalieri’s principle.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Visualize relationships between two-dimensional and three-dimensional objects
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Name ______Geometry Review #1
G-CO: Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
1. The image of ∆ABC after a rotation of 90° clockwise about the origin is ∆DEF, as shown below.
Which statement is true?
(1) /(2) /
(3) /
(4) /
2. Given right triangles ABC and DEF where and are right angles, and . Describe a precise sequence of rigid motions which would show .
3. Triangle ABC is shown in the xy-coordinate plane.
The triangle will be rotated 180° clockwise around the point (3,4) to create ∆A'B'C'. Which characteristics of ∆A'B'C' will be the same for the corresponding characteristic of ∆ABC?
Select all that apply.
4. Triangle ABC has vertices at A1,2, B4,6, and C(4,2) in the coordinate plane. The triangle will be reflected over the x-axis and then rotated 180° about the origin to form ∆A'B'C'. What are the vertices of ∆A'B'C'?
5. In the diagram below, PG≅HT, EG≅AT, and ∠G≅∠T. Which method could you use to prove
the triangles are congruent?
6. Given: Circles with centers A and B intersect at C and D.
Prove: ∠CAB≅∠DAB.
7.
8.
9.
10.
Name ______Geometry Review #2
G-CO: Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.
1. In the figure shown, CF intersects AD and EH at points B and F, respectively.
Given: ∠CBD≅∠BFE
Prove: ∠ABF≅∠BFE
2. AB and CD in intersect at E. m∠AEC=6x+20, and m∠DEB=10x. What is the value of x?
3. In the diagram below, line p intersects line m and line n. If m∠1=7x and m∠2=5x+30, lines m and n are parallel when x equals
(1) 12.5
(2) 15
(3) 87.5
(4) 105
4. Determine the value of f in the diagram below.
5. In the figure below, GHis a line of reflection. State and justify two conclusions about distances in this figure. At least one of your statements should refer to perpendicular bisectors.
6.
Name ______Geometry Review #3
G-CO: Prove geometric theorems
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
1. Vertex angle A of isosceles triangle ABC measures 20° more than three times m∠B. Find m∠C.
2. Which statement is not always true about a parallelogram?
1) / The diagonals are congruent.2) / The opposite sides are congruent.
3) / The opposite angles are congruent.
4) / The opposite sides are parallel.
3. In the accompanying diagram of parallelogram ABCD, diagonals AC and DB intersect at E, AE=3x-4, andEC=x+12 . What is the value of x?
4. In the diagram below of ∆ABC, DE is a midsegment of ∆ABC, DE=7, AB=10, and BC=13.
Find the perimeter of ∆ABC.
5. Triangle ABC is graphed on the set of axes below. What are the coordinates of the point of intersection of the medians of ∆ABC?
(1) (–1, 2)
(2) (–3, 2)
(3) (0, 2)
(4) (1,2)
6. Prove the sum of the exterior angles of a triangle is 360°.
7. Given: , , and bisects
Prove that is a right angle.
8. One method that can be used to prove that the diagonals of a parallelogram bisect each other is shown in the given partial proof.
Given: Quadrilateral PQRS is a parallelogram
Prove: PT=RT
ST=QT
9.
10.
Name ______Geometry Review #4
G-SRT: Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
1. In the xy-coordinate plane, ∆ABC has vertices A-4,6, B2,6, and C(2,2). ∆DEF is shown in the plane.
What is the scale factor and the center of dilation that maps ∆ABC to ∆DEF?
(1) The scale factor is 2, and the center of dilation is point B.
(2) The scale factor is 2, and the center of dilation is the origin.
(3) The scale factor is 12, and the center of dilation is point B.
(4) The scale factor is 12, and the center of dilation is the origin.
2. The figure shows line AC and line PQ intersecting at point B. Lines A'C' and P'Q' will be the images of lines AC and PQ, respectively, under a dilation with center P and scale factor 2.
Which statement about the image of lines AC and PQ would be true under the dilation?
(1) Line A'C' will be parallel to line AC, and the line P'Q' will be parallel to line PQ.
(2) Line A'C' will be parallel to line AC, and the line P'Q' will be the same line as line PQ.
(3) Line A'C' will be perpendicular to line AC, and the line P'Q' will be parallel to line PQ.
(4) Line A'C' will be perpendicular to line AC, and the line P'Q' will be the same as line PQ.
3. In the coordinate plane, line p has slope 8 and y-intercept (0,5). Line r is the result of dilating line p by a factor of 3 with center (0,3). What is the slope and y-intercept of line r?
(1) Line r has a slope of 5 and y-intercept (0,2).
(2) Line r has a slope of 8 and y-intercept (0,5).
(3) Line r has a slope of 8 and y-intercept (0,9).
(4) Line r has a slope of 11 and y-intercept (0,8).
4. Line segment AB with endpoints A(4,16) and B(20,4) lies in the coordinate plane. The segment will be dilated with a scale factor of 34 and a center at the origin to create A'B'. What will be the length of A'B'?
5. The line y = 2x – 4 is dilated by a scale factor of 32 and centered at the origin. Which equation represents the image of the line after the dilation?
(1) y = 2x – 4
(2) y = 2x – 6
(3) y = 3x – 4
(4) y = 3x – 6
6. The equation of line h is 2x+y=1. Line m is the image of line h after a dilation of scale factor 4 with respect to the origin. What is the equation of the line m?
(1) y = –2x + 1
(2) y = –2x + 4
(3) y = 2x + 4
(4) y = 2x + 1
7. Triangle KLM is the pre-image of ∆K'L'M', before a transformation. Determine if these two figures are similar. Which statements are true? Select all that apply.
8. In the diagram below, triangles XYZ and UVZ are drawn such that ∠X≅∠U and ∠XZY≅∠UZV. Describe a sequence of similarity transformations that shows ∆XYZ is similar to ∆UVZ.
Name ______Geometry Review #5
G-SRT: Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
1. In isosceles ∆MNP, line segment NO bisects vertex ∠MNP, as shown below. If MP = 16, find the length of MO and explain your answer.