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Geometry Major Content Review

Topic Standards

1 G-CO: Understand congruence in terms of rigid motions6. 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 theorems9. 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 theorems10. 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 transformations1. 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 similarity4. 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 triangles6. 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 algebraically4. 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 algebraically6. 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 circles1. 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 problems1. 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 objects4. 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 ∆≝¿, 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 A (1,2 ) ,B (4,6 ) ,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 .


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=6 x+20, and m∠DEB=10 x. What is the value of x?

3. In the diagram below, line p intersects line m and line n. If m∠1=7 x and m∠2=5 x+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, GH is 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.


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=3 x−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 parallelogramProve: PT=RT ST=QT


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 ) , B (2,6 ) , and C (2,2). ∆≝¿ is shown in the plane.

What is the scale factor and the center of dilation that maps ∆ ABC to ∆≝¿?

(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. LinesA '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 2 x+ 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 .


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.

2. The figure shows ∆ ABC ∆≝¿ with side lengths as indicated. What is the value of x?

3. In the diagram below of ∆ ABC, TV ∥BC , AT=5, TB=7, and AV=10. What is the length of VC?

4. The figure shows two semicircles with centers P and M. The semicircles are tangent to each other at point B, and DE is tangent to both semicircles at F and E.

5. In right triangle ABD, AB=53, and altitude DC=14. Find the lengths of BC and AC.


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.

1. A man who is 5 feet 9 inches tall casts a shadow of 8 feet 6 inches. Assuming that the man is standing perpendicular to the ground, what is the angle of elevation from the end of the shadow to the top of the man’s head, to the nearest tenth of a degree?

(1) 34.1 (2) 34.5 (3) 42.6 (4) 55.9

2. In right triangle ABC with the right angle at C, sin A = 2x + 0.1 and cos B = 4x – 0.7. Determine and state the value of x. Explain your answer.

3. In right triangle ABC, m∠B≠m∠C. Let sin B=r and cosB=s. What is sinC−cosC?

(1) r + s(2) r – s (3) s – r

(4) rs

4. The degree measure of an angle in a right triangle is x, and sin x=13 . Which of these expressions

are also equal to 13? Select all that apply.

5. In this figure, triangle GHJ is similar to triangle PQR. Which ratio represents tanH?

6. Mariela is standing in a building and looking out of a window at a tree. The tree is 20 feet away from Mariela. Mariela’s line of sight to the top of the tree creates a 42 ° angle of elevation, and her line of sight to the base of the tree creates a 31 ° angle of depression. What is the height, in feet, of the tree?

7. Explain why cos(x) = sin(90 – x) for x such that 0 < x < 90.

8. An archaeological team is excavating artifacts from a sunken merchant vesel on the ocean floor. To assist the team, a robotic probe is used remotely. The probe travels approximately 3,900 meters at an angle of depression of 67.4 degrees from the team’s ship on the ocean surface down to the sunken vessel on the ocean floor. The figure shows a representation of the team’s ship and the probe.

How many meters below the surface of the ocean will the probe be when it reaches the ocean floor? Give your answer to the nearest hundred meters.

9. Right triangle WXY is similar to triangle DEF. The foloowing are measurements in right triangle DEF:

m∠F=90°DE=√113DF=7EF=8

Which expression represents cosW ?

(1) cosW= 7√113

(2) cosW= 8√113

(3) cosW=7√113(4) cosW=8√113

10. An unmanned aerial vehicle (UAV) is equipped with cameras used to monitor forest fires. The figure represents a moment in time at which a UAV, at point B, flying at an altitude of 1,000 meters is directly above point D on the forest floor. Point A represents the location of a small fire on the forest floor.

A. At the moment in time represented by the figure, the angle of depression from the UAV to the fire has ameasure of 30 °. What is the distance from the UAV to the fire?

B. What is the distance, to the nearest meter, from the fire to point D?

11. A spring is attached at one end to support B and at the other end to collar A, as represented in the figure. Collar A slides along the vertical bar between points C and D. In the figure, the angle θ is the angle created as the collar moves between points C and D.

A. When θ=28 °, what is the distance from point A to point B to the nearest tenth of a foot?

B. When the spring is stretched and the distance from point A to point B is 5.2 feet, what is the value of θ to the nearest tenth of a degree?

12. The map below shows the three tallest mountain peaks in New York State: Mount Marcy, Algonquin Peak, and Mount Haystack. Mount Haystack, the shortest peak, is 4960 feet tall. Surveyors have determined the horizontal distance between Mount Haystack and Mount Marcy is 6336 feet and the horizontal distance between Mount Marcy and Algonquin Peak is 20,493 feet.

The angle of depression from the peak of Mount Marcy to the peak of Mount Haystack is 3.47 degrees. The angle of elevation from the peak of Algonquin Peak to the peak of Mount Marcy is 0.64 degrees. What are the heights, to the nearest foot, of Mount Marcy and Algonquin Peak? Justify your answer.

13. As shown below, a canoe is approaching a lighthouse on the coastline of a lake. The front of the canoe is 1.5 feet above the water and an observer in the lighthouse is 112 feet above the water.

At 5:00, the observer in the lighthouse measured the angle of depression to the front of the canoe to be 6°. Five minutes later, the observer measured and saw the angle of depression to the front of the canoe had increased by 49°. Determine and state, to the nearest foot per minute, the average speed at which the canoe traveled toward the lighthouse.


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).

1. The graphs of the equations y=2x−7 and y−kx=7 are parallel when k equals

(1) –2(2) 2(3) –7(4) 7

2. What is the equation of a line that is parallel to the line whose equation is y=x+2 and passes through the point (2, –4)?

3. Which equation represents a line perpendicular to the line whose equation is 2 x+3 y=12 and passes through the point (–1, 2)?

4. Determine if the point ¿) lies on the circle centered at the origin and containing the point (0, 2). Justify your reasoning.

5. Write the equation of the circle shown in the graph below.

6. The slope of QR is x−1

4 and the slope of ST is 83 . If QR⊥ ST , determine and state the value of x.

7. What are the coordinates of the midpoint of the line segment with endpoints (2, –5) and (8,3)?

8. Given: quadrilateral PQRS with vertices P(–5, 2), Q(–4, 5), R(2, 3), S(1, 0). Prove that PQRS is a rectangle.


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.

1. In the coordinate plane shown, point C (not shown) lies on AB. If the ratio of the length of AC to the length of CB is 3:1, what is the y-coordinate of point C?

2. Line segment JK in the xy-coordinate plane has endpoints with coordinates (−4,11) and (8 ,−1). What are two possible locations for point M so that M divides JK into two parts with lengths in a ratio of 1:3? Indicate both locations.

3. What are the coordinates of the point on the directed line segment from K(–5,–4) to L(5,1) that partitions the segment into a ratio of 3 to 2?

(1) (–3,–3)(2) (–1,–2)

(3) (0 ,−32)

(4) (1,–1)

4. Calculate the area and perimeter of the rectangle with vertices P(–5, 2), Q(–4, 5), R(2, 3), S(1, 0).

5. Calculate the area and perimeter of the triangle with vertices 𝐴(−2,3), 𝐵(4,4), and 𝐶 (−1,−2).


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

1. As shown in the diagram below, circle A as a radius of 3 and circle B has a radius of 5. Use transformations to explain why circles A and B are similar.

2. Point B is in the center of a circle, and AC is a diameter of the circle. Point D is a point is a point on the circle different from A and C. If m∠BDA=20 °, what is m∠CBD?

3. In a circle, diameter AB is perpendicular to chord CD at L. Which statement will always be true about this circle?

(1) CL=LD(2) AL>LB(3) (CL )× (LD )=AB(4) BL>LA

4. Determine the value of x.

5. Determine m∠CAE .

6. In the accompanying diagram of circle O, chord AY is parallel to diameter DOE, AD is drawn, and m AD=35. What is M∠DAY ?

7. In the diagram below secants PQS and PRT intersect at P. If m∠P=40 and mQR=86, find m ST .

8. A toy truck is located within a circular play area. Alex and Dominic are sitting on opposite endpoints of a chord that contains the truck. Alex is 4 feet from the truck, and Dominic is 3 feet from the truck. Meira and Tamara are sitting on opposite endpoints of another chord containing the truck. Meira is 8 feet from the truck. How many feet, to the nearest tenth of a foot, is Tamara from the truck? Draw a diagram to support your answer.

9. In the diagram below, secant ACD and tangent AB are drawn from external point A to circle O.

Prove the theorem: If a secant and a tangent are drawn to a circle from an external point, the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. (AC ∙ AD=AB2)


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.

1. Two stacks of 23 quarters each are shown below. One stack forms a cylinder but the other stack does not form a cylinder. Use Cavalieri’s principle to explain why the volumes of these two stacks of quarters are equal.

2. The table shows the approximate measurements of the Great Pyramid of Giza in Egypt and the Pyramid of Kukulcan in Mexico.

Approximately what is the difference between the volume of the Great Pyramid of Gaza and the volume of the pyramid of Kukulcan?

(1) 1,945,000 cubic meters(2) 2,562,000 cubic meters(3) 5,835,000 cubic meters(4) 7,686,000 cubic meters

3. Two cylinders each with a height of 50 inches are shown. Which statements are true? Select all that apply.

4. A rectangle will be rotated 360 ° about a line which contains the point of intersection of its diagonals and is parallel to a side. What three-dimensional shape will be created as a result of the rotation?

(1) a cube(2) a rectangular prism(3) a cylinder(4) a sphere

5. In the diagram below, a right circular cone with a radius of 3 inches has a slant height of 5 inches, and a right cylinder with a radius of 4 inches has a height of 6 inches. Determine and state the number of full cones of water needed to completely fill the cylinder with water.

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