Congruence, Similarity, Right Triangles, and Trigonometry

Congruence, Similarity, Right Triangles, and Trigonometry
2018 Geometry Bootcamp 2018 Congruence, Similarity, Right Triangles, and Trigonometry 2018 Geometry Bootcamp

MAFS.912.G-CO.1.1 Drag the words to the correct descriptions. Not all options will be used. the set of points extending in one direction in a plane the set of points between two endpoints in a plane the set of all points the same distance from a given point in a plane the set of points extending in opposite directions in a plane Circle Ray Line Line Segment Angle Groups 1, 2, and 3 Circle Line Segment Ray Line

2018 Geometry Bootcamp MAFS.912.G-CO.1.1 Classify each statement in the table as correct or incorrect. Select one cell per row. Correct Incorrect If two lines in the same plane do not intersect, the lines must be parallel. If two line in the space do not intersect, the lines must be parallel. If two lines are parallel, the lines must lie in the same plane. Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.1 Points 𝐽, 𝐾, and 𝐿 are distinct points, and 𝐽𝐾=𝐾𝐿. Which of these statements must be true? Select all that apply. 𝐽, 𝐾, and 𝐿 are coplanar. 𝐽, 𝐾, and 𝐿 are collinear. K is the midpoint of 𝐽𝐿 . 𝐽𝐾 ≅ 𝐾𝐿 The measure of ∠𝐽𝐾𝐿 is 90°. Groups 2 and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.2 Pentagon ABCDE is shown in the 𝑥𝑦−coordinate plane. Pentagon ABCDE will be rotated 90° clockwise about the point (1, 1) to form pentagon A’B’C’D’E’. Choose the graph that shows the correct placement of A’B’C’D’E’ after the transformation. A B C D Groups 1 and 2 B 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.2 Triangle 𝐴𝐵𝐶 is shown in the 𝑥𝑦-coordinate plane. It will be rotated 90 degrees clockwise about the origin to form triangle 𝐴’𝐵’𝐶’. Select the correct orientation of 𝐴’𝐵’𝐶’ and place it correctly in the coordinate plane. Groups 1 and 2 2018 Geometry Bootcamp

MAFS.912.G-CO.1.2 A triangle is shown on the coordinate grid.
2018 Geometry Bootcamp MAFS.912.G-CO.1.2 A triangle is shown on the coordinate grid. Use the Connect Line tool to draw the triangle after a transformation following the rule 𝑥, 𝑦 → 𝑥−4, 𝑦+3 Groups 1 and 2 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.2 The pre-image of △ABC and its image △A′B′C′ are shown on the coordinate plane. Which rule describes the transformation represented in the graph? 1 2 𝑥+2, 1 2 𝑦−3 2𝑥+2, 2𝑦−3 1 2 𝑥−2, 1 2 𝑦+3 2𝑥−2, 2𝑦+3 Groups 1, 2, and 3 B 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.4 A rotation about Point D maps Point 𝐵 to 𝐵′ and Point 𝐶 to 𝐶′. Which statement must be true? Select all that apply. 𝐴𝐶= 𝐴 ′ 𝐶 ′ 𝐵𝐷= 𝐵 ′ 𝐷 𝑚∠𝐴’𝐷𝐴 =𝑚∠𝐶’𝐷𝐶 𝑚∠𝐶’ DB ′ =𝑚∠𝐵’𝐷𝐶 𝑚∠𝐶’𝐷𝐶=𝑚∠𝐵’𝐷𝐵 If Point 𝐶 is 𝑥, 𝑦 , then Point 𝐶’ is −𝑥, 𝑦 If Point B is (𝑥, 𝑦), then Point 𝐵’ is (𝑦, 𝑥) Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.4 On a coordinate plane, each point on 𝑆𝑇 is defined by coordinates 𝑥, 𝑦 . The segment is rotated 90° counterclockwise to create 𝑆′𝑇′ . Which of the following defines a point on 𝑆′𝑇′ that corresponds to a point on 𝑆𝑇 . 𝑥, −𝑦 −𝑥, 𝑦 𝑦, 𝑥 −𝑦, 𝑥 Groups 2 and 3 D 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.3 Which figure always has exactly four lines of reflection that map the figure onto itself? square rectangle regular octagon equilateral triangle Groups 1, 2, and 3 A 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.3 Select the values that correctly complete the sentence about the symmetry of a regular pentagon. 5 A regular pentagon has _____ lines of symmetry and 1 2 5 6 72 degree has ___________ rotational symmetry. Groups 2 and 3 60 degree 72 degree 108 degree 540 degree 2018 Geometry Bootcamp

MAFS.912.G-CO.1.3 Trapezoid RSTU is shown.
2018 Geometry Bootcamp MAFS.912.G-CO.1.3 Trapezoid RSTU is shown. Write the equation for the line that would map the trapezoid onto itself. 𝒚=𝟐 Groups 2 and 3 2018 Geometry Bootcamp

MAFS.912.G-CO.1.5 In the diagram below, ∆𝐴𝐵𝐶≅∆𝐷𝐸𝐹.
Which sequence of transformations maps ∆𝐴𝐵𝐶 onto ∆𝐷𝐸𝐹? a reflection over the 𝑥−axis followed by a translation a reflection over the 𝑦−axis followed by a translation a rotation of 180° about the origin followed by a translation a counterclockwise rotation of 90° about the origin followed by a translation Groups 1 and 2 B

MAFS.912.G-CO.1.5 Quadrilateral 𝐵𝐶𝐷𝐸 is shown on the coordinate grid.
2018 Geometry Bootcamp MAFS.912.G-CO.1.5 Quadrilateral 𝐵𝐶𝐷𝐸 is shown on the coordinate grid. Keisha reflects the figure across the line 𝑦=𝑥 to create 𝐵 ′ 𝐶 ′ 𝐷 ′ 𝐸 ′ . Use the Connect Line tool to draw quadrilateral 𝐵 ′ 𝐶 ′ 𝐷 ′ 𝐸 ′ . Groups 1 and 2 2018 Geometry Bootcamp

MAFS.912.G-CO.1.5 The right triangle in the coordinate plane is rotated 270° clockwise about the point (2, 1) and then reflected across the 𝑦−axis to form △ 𝐴 ′ 𝐵 ′ 𝐶 ′ . Drag and drop the appropriate orientation for triangle 𝐴’𝐵’𝐶’ into the correct position on the coordinated plane. Groups 1, 2, and 3

MAFS.912.G-CO.1.5 △ABC is shown on the coordinate plane.
After rotating △ABC 180° about the origin and then reflecting it over the x-axis, what are the coordinates of △A″B″C″? A″(2, 6), B″(5, 4), C″(2, 1) A″(6, 2), B″(4, 5), C(1, 2) A″(−2, 6), B″(5, 4), C″(−2, 1) A″(6, −2), B″(4, −5), C″(1, −2) Groups 1, 2, and 3 B

MAFS.912.G-CO.1.5 Given quadrilateral ABCD, what are the coordinates for the resulting image, A″B″C″D″, after the two transformations listed? First transformation: Rotate 90° clockwise about the origin. Second transformation: Translate 𝑥, 𝑦 →(𝑥 + 1, 𝑦 − 2). Enter the coordinates for the resulting image A″B″C″D″ in the boxes. A” = B” = C” = D” = (𝟏, 𝟕) (𝟕, 𝟏) (𝟓, −𝟐) (𝟑, 𝟎) Groups 2 and 3

MAFS.912.G-CO.1.5 Triangle EGF is graphed below.
2018 Geometry Bootcamp MAFS.912.G-CO.1.5 Triangle EGF is graphed below. Triangle EGF will be rotated 90° counterclockwise around the origin and will then be reflected across the y-axis, producing an image triangle. Which additional transformation will map the image triangle back onto the original triangle? rotation 270° counterclockwise around the origin rotation 180° counterclockwise around the origin reflection across the line 𝑦 = –𝑥 reflection across the line 𝑦 = 𝑥 Groups 2 and 3 D 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.1.5 Parallelogram 𝑅𝑆𝑇𝑈 has midpoints 𝐾, 𝐿, 𝑀, 𝑁 marked on the sides as shown. Which rigid motion could be applied to ∆𝑅𝑆𝑈 to show that ∆𝑅𝑆𝑈≅∆𝑇𝑈𝑆? reflection over 𝑆𝑈 reflection over 𝐿𝑁 rotation 90° clockwise about the intersection point of 𝐾𝑀 and 𝐿𝑁 . rotation 180° clockwise about the intersection point of 𝑆𝑈 and 𝑅𝑇 . Groups 2 and 3 D 2018 Geometry Bootcamp

the perimeter of ∆ 𝑨 ′ 𝑩 ′ 𝑪 ′
2018 Geometry Bootcamp MAFS.912.G-CO.2.6 Triangle 𝐴𝐵𝐶 is shown in the 𝑥𝑦-coordinate plane. The triangle will be rotated 180° clockwise around the point (3, 4) to create triangle 𝐴’𝐵’𝐶’. Indicate whether each of the listed features of the image will or will not be the same as the corresponding feature in the original triangle by selecting the appropriate box in the table. the coordinates of A’ the coordinates of C’ the perimeter of ∆ 𝑨 ′ 𝑩 ′ 𝑪 ′ the area of ∆ 𝑨 ′ 𝑩 ′ 𝑪 ′ the measure of ∠B’ the slope of 𝑨 ′ 𝑪′ will be the same will not be the same Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.2.6 In the illustration, figure I and figure II are isosceles trapezoids. Tran makes a conjecture that figure I is congruent to figure II. Select each transformation or combination of transformations that can help Tran prove his conjecture. Select all that apply. Rotate figure I 60° clockwise around point P. Rotate figure I 120° clockwise around point P. Reflect figure I across line 𝑟, and then reflect the image across line 𝑠. Reflect figure I across line 𝑠, and then reflect the image across line 𝑡. Rotate figure I 180° around point P, and then reflect the image across line 𝑡. Rotate figure I 120° counterclockwise around point P, and then reflect the image across line 𝑟. Groups 1, 2, and 3 2018 Geometry Bootcamp

MAFS.912.G-CO.2.7 All corresponding sides and angles of △𝑅𝑆𝑇 and △𝐷𝐸𝐹 are congruent. Select all the statements that must be true. There is a reflection that maps 𝑅𝑆 to 𝐷𝐸 . There is a dilation that maps △𝑅𝑆𝑇 to △𝐷𝐸𝐹. There is a translation followed by a rotation that maps 𝑅𝑇 to 𝐷𝐹 . There is a sequence of rigid motions that maps △𝑅𝑆𝑇 to △𝐷𝐸𝐹. There is not necessarily a sequence of rigid motions that maps △𝑅𝑆𝑇 to △𝐷𝐸𝐹. Groups 1, 2, and 3

2018 Geometry Bootcamp MAFS.912.G-CO.2.7 △ABC and △DEF are plotted on the coordinate plane shown. Which conclusions can be made about △ABC and △DEF if △ABC is mapped onto △DEF by reflecting △ABC over the 𝑦-axis and reflecting it over the 𝑥-axis? Select all that apply. △ABC ≅ △DEF The corresponding sides are proportional: 𝐴𝐵 𝐷𝐸 = 𝐴𝐶 𝐷𝐹 = 𝐵𝐶 𝐸𝐹 . Reflecting △ABC across the 𝑦-axis and then the 𝑥-axis yields the same transformation as rotating △ABC 90° counterclockwise around the origin. Groups 1, 2, and 3 △ABC ∼△DEF △ABC is acute, but △DEF is obtuse. All corresponding sides are congruent. 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.2.8 Aniyah has repeatedly stenciled the triangle shape shown on her bedroom wall. Her best friend, Daniela, wants to copy the exact same shape on her bedroom wall. Which statement has sufficient information about the triangle for Aniyah to give to Daniela to guarantee Daniela will have the exact same triangle? Construct a triangle with angles whose measures are 33°, 42°, and 105°. Construct a triangle with sides of measure 22 inches and 27 inches where the included angle is 105°. Construct a triangle with sides of measure 39 inches and 22 inches and a nonincluded angle of measure 33°. Construct a triangle with a 105° angle opposite from a side of length 39 inches where the remaining two sides differ in length by 5 inches. Groups 1, 2, and 3 B 2018 Geometry Bootcamp

MAFS.912.G-CO.3.9 Line 𝑚 is parallel to line 𝑛.
2018 Geometry Bootcamp MAFS.912.G-CO.3.9 Line 𝑚 is parallel to line 𝑛. What is the measure of ∠𝑋𝑌𝑍? 36° 42° 78° 102° Groups 1, 2, and 3 C 2018 Geometry Bootcamp

MAFS.912.G-CO.3.9 Quadrilateral 𝐴𝐵𝐶𝐷 is shown.
2018 Geometry Bootcamp MAFS.912.G-CO.3.9 Quadrilateral 𝐴𝐵𝐶𝐷 is shown. For what value of 𝑥 will line 𝐵𝐷 be the perpendicular bisector of segment 𝐴𝐶? 5 9 14 17 Groups 1, 2, and 3 B 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.9 Mikayla is using the following information to prove that ∠𝑇𝑈𝑆 and ∠𝑃𝑈𝑄 are complementary angles in the diagram shown. Part of her proof is shown below. Given: The ray 𝑈𝑆 bisects ∠𝑇𝑈𝑅 and the ray 𝑈𝑄 bisects ∠𝑃𝑈𝑅. Which statements could be used to complete Mikayla’s proof? Statements Reasons 1 ∠𝑇𝑈𝑅 and ∠𝑃𝑈𝑅 are supplementary angles TP is a line 2 𝑚∠𝑇𝑈𝑅+m∠𝑃𝑈𝑅=180° Definition of supplementary angles 3 𝑚∠𝑇𝑈𝑅=2∙m∠𝑇𝑈𝑆 𝑚∠𝑃𝑈𝑅=2∙m∠𝑃𝑈𝑄 Property of angle bisectors 4 Substitution 5 Division property of equality 6 ∠𝑇𝑈𝑆 and ∠𝑃𝑈𝑄 are complementary angles Definition of complementary angles 4. 2∙m∠𝑇𝑈𝑆=2∙m∠𝑃𝑈𝑄 5. 𝑚∠𝑇𝑈𝑆=𝑚∠𝑃𝑈𝑄 5. 𝑚∠𝑇𝑈𝑆+m∠𝑃𝑈𝑄=90° 4. 2∙𝑚∠𝑇𝑈S+2∙m∠𝑃𝑈𝑄=180° A. B. C. D. Groups 1, 2, and 3 D 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.9 In the figure shown, 𝐶𝐹 intersects 𝐴𝐷 and 𝐸𝐻 at points 𝐵 and 𝐹, respectively. Part A Given:∠𝐶𝐵𝐷≅∠𝐵𝐹𝐸 Prove:∠𝐴𝐵𝐹≅∠𝐵𝐹𝐸 Select from the drop-down menus to support each line of the proof. Groups 1, 2, and 3 Statement: ∠𝐶𝐵𝐷≅∠𝐵𝐹𝐸 Reason: Statement: ∠𝐶𝐵𝐷≅∠𝐴𝐵𝐹 Reason: Statement: ∠𝐴𝐵𝐹≅∠𝐵𝐹𝐸 Reason: 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.9 In the figure shown, 𝐶𝐹 intersects 𝐴𝐷 and 𝐸𝐻 at points 𝐵 and 𝐹, respectively. Part B Given:m∠𝐶𝐵𝐷=𝑚∠𝐵𝐹𝐸 Prove:m∠𝐵𝐹𝐸+𝑚∠𝐷𝐵𝐹=180° Select from the drop-down menus to support each line of the proof. Groups 1, 2, and 3 Statement: 𝑚∠𝐶𝐵𝐷=𝑚∠𝐵𝐹𝐸 Reason: Statement: m∠𝐶𝐵𝐷+𝑚∠𝐷𝐵𝐹=180° Reason: Statement: m∠𝐵𝐹𝐸+𝑚∠𝐷𝐵𝐹=180° Reason: 2018 Geometry Bootcamp

MAFS.912.G-CO.3.9 Given: 𝐽𝑀 is the perpendicular bisector of 𝐿𝐾 .
2018 Geometry Bootcamp MAFS.912.G-CO.3.9 Given: 𝐽𝑀 is the perpendicular bisector of 𝐿𝐾 . Prove: J is equidistant from L and K Statements Reasons 𝐽𝑀 is the perpendicular bisector of 𝐿𝐾 Given ∠𝐿𝑀𝐽 and ∠𝐽𝑀𝐾 are right angles All right angles are congruent 𝐿𝑀 ≅ 𝐾𝑀 Definition of bisector Reflexive property of equality ∆𝐿𝑀𝐽≅∆𝐾𝑀𝐽 SAS 𝐿𝐽 ≅ 𝐾𝐽 J is equidistant from L and K Definition of equidistant 𝐽𝑀 ≅ 𝐽𝑀 Definition of right angle ∠𝐿𝐽𝑀≅∠𝐾𝐽𝑀 ∠𝐿𝑀𝐽≅∠𝐽𝑀𝐾 ∠𝐿𝑀𝐽≅∠𝐽𝑀𝐾 Group 3 𝐽𝑀 ≅ 𝐽𝑀 Definition of right angle Definition of perpendicular Corresponding parts of congruent triangles are congruent Corresponding parts of congruent triangles are congruent 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.10 Roads connecting the towns of Oceanside, River City, and Lake View form a triangle. The distance from Oceanside to River City is 38 kilometers. The distance from River City to Lake View is 26 kilometers. What is the smallest possible whole number of kilometers between Lake View and Oceanside? Enter your answer in the box. 13 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.10 The figure shows triangle 𝐴𝐵𝐶. Segment 𝐷𝐸 connects the midpoints of respective sides 𝐴𝐵 and 𝐵𝐶 . Which of the statements about the figure cannot be proven? DE ∥ AC 2 DE =AC BD DA = DE AC ∆BDE~∆BAC Groups 1, 2, and 3 C 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.10 Which of the following sets of lengths could form the sides of a triangle? Choose all that apply. 5, 9, 7 4, 8, 3 11, 11, 22 2, 5, 4 7, 9, 12 10, 8, 2 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.10 Complete the proof by filling in the missing reasons from the “Reasons Bank” below. Given: 𝐷𝐻 ∥ 𝐹𝐺 Prove: ∆𝐻𝐸𝐷~∆𝐹𝐸𝐺 Reasons Bank AA Similarity Postulate Statements Reasons 𝐷𝐻 ∥ 𝐹𝐺 Given ∠𝐻≅∠𝐹 ∠𝐻𝐸𝐷≅∠𝐹𝐸𝐺 ∆𝐻𝐸𝐷~∆𝐹𝐸𝐺 SAS Similarity Theorem SSS Similarity Theorem If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent Vertical angles are congruent Groups 2 and 3 If 2 parallel lines are cut by a transversal, then the alternate interior angles are congruent. Vertical angles are congruent If 2 parallel lines are cut by a transversal, then the corresponding angles are congruent AA Similarity Postulate 2018 Geometry Bootcamp

MAFS.912.G-CO.3.11 Parallelogram ABCD is shown.
2018 Geometry Bootcamp MAFS.912.G-CO.3.11 Parallelogram ABCD is shown. What are the values of 𝑥 and 𝑦? Enter the correct values in the boxes. 𝑥= 70° 𝑦= 110° Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.11 In quadrilateral 𝑅𝑆𝑇𝑈, 𝑅𝑇 and 𝑆𝑈 intersect at point 𝑀, so that 𝑅𝑀 ≅ 𝑇𝑀 and 𝑅𝑆 is parallel to 𝑇𝑈 . Which additional information is needed to prove that figure 𝑅𝑆𝑇𝑈 is a rectangle? 𝑅𝑈 ≅ 𝑅𝑆 𝑈𝑀 ≅ 𝑆𝑀 ∠𝑅𝑆𝑇≅∠𝑆𝑇𝑈 ∠𝑅𝑈𝑇≅∠𝑇𝑆𝑅 Groups 1, 2, and 3 C 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.3.11 Which set of statements would describe a parallelogram that can always be classified as a rhombus? I. Diagonals are perpendicular bisectors of each other. II. Diagonals bisect the angles from which they are drawn. III. Diagonals form four congruent isosceles right triangles. I and II I and III II and III I, II, and III Groups 1, 2, and 3 D 2018 Geometry Bootcamp

MAFS.912.G-CO.3.11 Given: Quadrilateral 𝑃𝑄𝑅𝑆 is a parallelogram.
2018 Geometry Bootcamp MAFS.912.G-CO.3.11 Given: Quadrilateral 𝑃𝑄𝑅𝑆 is a parallelogram. Prove: 𝑃𝑇=𝑅𝑇 and 𝑆𝑇=𝑄𝑇 Statements Reasons Quadrilateral 𝑃𝑄𝑅𝑆 is a parallelogram Given 𝑃𝑄 ∥ 𝑆𝑅 and 𝑃𝑆 ∥ 𝑄𝑅 Definition of a parallelogram ∠𝑃𝑄𝑆≅∠𝑅𝑆𝑄 ∠𝑄𝑃𝑅≅∠𝑆𝑅𝑃 Opposite sides of a parallelogram are congruent ∆𝑆𝑅𝑇≅∆𝑄𝑃𝑇 𝑃𝑇 ≅ 𝑅𝑇 𝑆𝑇 ≅ 𝑄𝑇 Corresponding parts of congruent triangles are congruent 𝑃𝑇=𝑅𝑇 and 𝑆𝑇=𝑄𝑇 Definition of congruent segments S.S.S 𝑆𝑄 ≅ 𝑃𝑅 S.A.S 𝑃𝑄 ≅ 𝑆𝑃 When two parallel lines are cut by a transversal, alternate interior angles are congruent A.S.A 𝑃𝑄 ≅ 𝑅𝑆 When two parallel lines are cut by a transversal, corresponding angles are congruent 𝑃𝑄 ≅ 𝑅𝑆 Groups 1, 2, and 3 A.S.A When two parallel lines are cut by a transversal, same side interior angles are congruent When two parallel lines are cut by a transversal, alternate interior angles are congruent 2018 Geometry Bootcamp

MAFS.912.G-CO.3.11 Given: ABCD is a parallelogram
2018 Geometry Bootcamp MAFS.912.G-CO.3.11 m∠1=𝑚∠2 m∠2=𝑚∠4 Given: ABCD is a parallelogram m∠1=𝑚∠3 m∠2=𝑚∠3 Prove: ∠1 and ∠2 are supplementary. m∠1+𝑚∠2=180° Proof: m∠1+𝑚∠1+𝑚∠2+𝑚∠2=360° It is given that ABCD is a parallelogram with angles named, in consecutive order, ∠1, ∠2, ∠3, and ∠4. m∠3+𝑚∠4+𝑚∠3+𝑚∠4=360° Opposite angles of a parallelogram are congruent. Thus __________ and __________ by the ______________. m∠1=m∠3 𝑚∠2=𝑚∠4 Definition of congruence definition of congruence The sum of the interior angles of any quadrilateral is 360°, so m∠1+∠2+∠3+∠4=360° . Definition of equality Groups 1, 2, and 3 Using the _______________, _______________. substitution property 𝑚∠1+𝑚∠1+𝑚∠2+𝑚∠2=360° Division property of equality Therefore, 2 m∠1+∠2 =360° Substitution property of equality By the _______________, _______________. division property 𝑚∠1+𝑚∠2=180° Transitive property of equality Therefore, ∠1 and ∠2 are supplementary. 2018 Geometry Bootcamp

MAFS.912.G-CO.3.11 The figure shows parallelogram 𝐴𝐵𝐶𝐷 with 𝐴𝐸=16.
2018 Geometry Bootcamp MAFS.912.G-CO.3.11 The figure shows parallelogram 𝐴𝐵𝐶𝐷 with 𝐴𝐸=16. Let 𝐵𝐸= 𝑥 2 −48 and let 𝐷𝐸=2𝑥. What are the lengths of 𝐵𝐸 and 𝐷𝐸 ? 𝐵𝐸= 16 𝐷𝐸= 16 Groups 2 and 3 Parallelogram ABCD is a __________________ because the ____________________. rectangle diagonals are congruent 2018 Geometry Bootcamp

MAFS.912.G-CO.3.11 Three vertices of parallelogram PQRS are shown:
2018 Geometry Bootcamp MAFS.912.G-CO.3.11 Three vertices of parallelogram PQRS are shown: 𝑄 8, 5 , 𝑅 5,1 , 𝑆 2,5 Place the statements and reasons in the table to complete the proof that shows that parallelogram PQRS is a rhombus. Statements Reasons Pythagorean Theorem 𝑆𝑅=𝑄𝑅 Substitution 𝑆𝑅 = 𝑄𝑅 Definition of congruent lines 𝑃𝑆 = 𝑄𝑅 Property of a parallelogram Parallelogram PQRS is a rhombus Definition of a rhombus 𝑆𝑅=5 𝑃𝑄=5 𝑄𝑅=5 𝑆𝑅=5 𝑆𝑅= 7 𝑃𝑄= 7 𝑄𝑅= 7 𝑄𝑅=5 Pythagorean Theorem ∠𝑃𝑆𝑅=90° 𝑆𝑅 ≅ 𝑃𝑄 Groups 2 and 3 Pythagorean Theorem Definition of perpendicular lines 𝑆𝑅 ≅ 𝑃𝑄 Property of a parallelogram Property of a parallelogram Definition of parallel lines 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.4.12 Which of the following best describes the construction? 𝐴𝐵<𝐶𝐷 𝐷 is the midpoint of 𝑚. 𝐶𝐷 ≅ 𝐴𝐵 𝐶 is the midpoint of 𝑚. Groups 1, 2, and 3 C 2018 Geometry Bootcamp

MAFS.912.G-CO.4.12 A student is working on a geometric construction.
2018 Geometry Bootcamp MAFS.912.G-CO.4.12 A student is working on a geometric construction. If 𝐴𝐷 is drawn, what geometric construction is shown? angle bisector copying an angle perpendicular bisector measuring an angle Groups 1, 2, and 3 A 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.4.12 The figure above shows the construction of the angle bisector of angle AOB using a compass. Which of the following statements must always be true in the construction of the angle bisector? Classify each statement in the table as correct or incorrect. Select one cell per row. Correct Incorrect 𝑂𝐴=𝑂𝐵 𝐴𝑃=𝐵𝑃 𝐴𝐵=𝐵𝑃 𝑂𝐵=𝐵𝑃 𝑂𝐴=𝐵𝑃 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.4.12 The figure shows line 𝑟, points 𝑃 and 𝑇 on line 𝑟, and point 𝑄 not on line 𝑟. Also shown is ray 𝑃𝑄. Part A Part B Once the construction is complete, which of the reasons listed contribute to proving the validity of the construction? Consider the partial construction of a line parallel to r through point Q. What would be the final step in the construction? When two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. When two lines are cut by a transversal and the vertical angles are congruent, the lines are parallel. definition of segment bisector definition of an angle bisector Groups 1, 2, and 3 draw a line through P and S draw a line through Q and S draw a line through T and S draw a line through Wand S B A 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-CO.4.13 Steven constructs an equilateral triangle inscribed in circle 𝑃. His first three steps are shown. He creates radius 𝑃𝑄 using a point 𝑄 on the circle. Using point 𝑄 as the center and the length of 𝑃𝑄 as a radius, he uses a compass to construct an arc that intersects the circle at 𝑅. Using point 𝑅 as the center and the length of 𝑃𝑄 as a radius, he uses a compass to construct an arc that intersects the circle at 𝑆. What should be Steven’s next step in constructing the equilateral triangle? Draw line segment connecting the points 𝑄, 𝑅, and 𝑆 to construct △𝑄𝑅𝑆. Draw line segment connecting the points 𝑃, 𝑅, and 𝑆 to construct △𝑃𝑅𝑆. Construct an arc intersecting the circle by using point 𝑆 as the center and the length of 𝑃𝑄 as a radius. Construct an arc intersecting the circle by using point 𝑃 as the center and the length of 𝑃𝑄 as a radius. Groups 2 and 3 C 2018 Geometry Bootcamp

MAFS.912.G-SRT.1.1 The diagram represents a dilation with a center at P. If the scale factor of the dilation is 𝑘, which statement about the diagram is true? If 𝑘>1, then the image of 𝑆𝑇 could be 𝑄𝑅 . If 0<𝑘<1, then the image of 𝑄𝑅 could be 𝑈𝑉 . If 𝑘>1, then the image of 𝑆𝑇 could be 𝑈𝑉 . If 𝑘=2, then the image of 𝑄𝑅 is itself. Groups 1, 2, and 3 C

MAFS.912.G-SRT.1.1 Triangle ABC has been dilated from center D.
If the length of segment 𝐵𝐶=𝑥, which is the length of segment 𝑆𝑇? 2x 3x 4x 5x Groups 1, 2, and 3 A

2018 Geometry Bootcamp MAFS.912.G-SRT.1.1 In the figure, point P will be the center of dilation of triangle ABC. Point P is collinear with vertices B and C. The scale factor of the dilation will be 3. Consider the relationship between the sides of triangle ABC and the sides of the dilation image, triangle A’B’C’. Select from the drop-down menus to correctly complete each sentence. Side A’B’ will side AB. Groups 1, 2, and 3 Side A’C’ will side AC. Side B’C’ will side BC. 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.1.1 A line segment is dilated by a scale factor of 2 centered at a point not on the line segment. Which statement regarding the relationship between the given line segment and its image is true? The line segments are perpendicular, and the image is one-half of the length of the given line segment. The line segments are perpendicular, and the image is twice the length of the given line segment. The line segments are parallel, and the image is twice the length of the given line segment. The line segments are parallel, and the image is one-half of the length of the given line segment. Groups 1, 2, and 3 C 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.1.1 Square ABCD is shown in the 𝑥𝑦−coordinate plane. The square will be dilated with the center O by a scale factor of 2 to create square A’B’C’D’. Which statements are true? Select all that apply. 𝐵𝐶 ∥ 𝐵 ′ 𝐶′ 𝐴𝐶 ≅ 𝐴 ′ 𝐶′ 𝐴𝐷 ⊥ 𝐶 ′ 𝐷′ Point D’ has the same coordinates as point C. Point C’ lies on the line containing points O and C. Groups 2 and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.1.1 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? Line 𝑟 has slope 5 and 𝑦−intercept (0, 2). Line 𝑟 has slope 8 and 𝑦−intercept (0, 5). Line 𝑟 has slope 8 and 𝑦−intercept (0, 9). Line 𝑟 has slope 11 and 𝑦−intercept (0, 8). Groups 2 and 3 C 2018 Geometry Bootcamp

MAFS.912.G-SRT.1.2 Two triangles are shown.
2018 Geometry Bootcamp MAFS.912.G-SRT.1.2 Two triangles are shown. Which is a true statement about the two triangles? The triangles are not similar. △ABC ∼△EDF by AA Similarity Postulate △ABC ∼△FDE by SAS Similarity Postulate △ABC ∼△FDE by AA Similarity Postulate Groups 1, 2, and 3 D 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.1.3 Two angle measures for both △ABC and △XYZ are given. Using the given information about the triangles, is △ABC ∼ △XYZ? Yes, the triangles are similar by AA. No, because only 1 pair of corresponding angles are congruent. No, we cannot determine similarity without knowing the third angles. No, we cannot determine similarity without knowing the side ratios. Groups 1, 2, and 3 A 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.1.2 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. Triangle 𝐾𝐿𝑀 is similar to △ 𝐾 ′ 𝐿 ′ 𝑀 ′ . Triangle 𝐾𝐿𝑀 is not similar to △ 𝐾 ′ 𝐿 ′ 𝑀 ′ . There was a dilation of scale factor 0.5 centered at the origin. There was a dilation of scale factor 1 centered at the origin. There was a dilation of scale factor 1.5 centered at the origin. There was a translation left 0.5 and up 1.5. There was a translation left 1.5 and up 0.5. Groups 1, 2, and 3 2018 Geometry Bootcamp

MAFS.912.G-SRT.1.2 The coordinate plane shows △𝐹𝐺𝐻 and △𝐹"G"𝐻".
2018 Geometry Bootcamp MAFS.912.G-SRT.1.2 The coordinate plane shows △𝐹𝐺𝐻 and △𝐹"G"𝐻". Which sequence of transformations can be used to show that △𝐹𝐺𝐻~△𝐹"G"𝐻“? A dilation about the origin with a scale factor of 2, followed by a 180° clockwise rotation about the origin. A dilation about the origin with a scale factor of 2, followed by a reflection over the line 𝑦=𝑥. A translation 5 units up and 4 units left, followed by a dilation with a scale factor of about point 𝐹”. A 180° clockwise rotation about the origin, followed by a dilation with a scale factor of about point 𝐹”. Groups 1, 2, and 3 B 2018 Geometry Bootcamp

MAFS.912.G-SRT.1.2 𝑇(𝑥, 𝑦)→(−𝑥, −𝑦) 𝑇(𝑥, 𝑦)→(𝑥+2, 2𝑦)
2018 Geometry Bootcamp MAFS.912.G-SRT.1.2 Triangle 𝐴𝐵𝐶 is defined in the coordinate plane by the points 𝐴=(1, 1), 𝐵=(3, 4), and 𝐶=(4, 2), as shown. Under which transformations will the image of triangle 𝐴𝐵𝐶 be similar to the preimage? Select all that apply. 𝑇(𝑥, 𝑦)→(−𝑥, −𝑦) 𝑇(𝑥, 𝑦)→(𝑥+2, 2𝑦) 𝑇(𝑥, 𝑦)→(0.5𝑥, 0.5𝑦) 𝑇(𝑥, 𝑦)→(𝑥+4, 𝑦−2) 𝑇(𝑥, 𝑦)→(2𝑥, 3𝑦) Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.1.3 The figure shows ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹 with side lengths as indicated. What is the value of 𝑥? Groups 1, 2, and 3 Enter your answer in the box. 15 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.2.4 In the diagram of ∆𝐴𝐵𝐶 below, 𝐷𝐸 is parallel to 𝐴𝐵 , 𝐶𝐷 = 15, 𝐴𝐷= 9, and 𝐴𝐵= 40. The length of 𝐷𝐸 is 15 24 25 30 Groups 1, 2, and 3 C 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.2.4 In the diagram, 𝐿𝑊 is parallel to 𝑆𝑇 . 𝑅𝐿=6 centimeters, 𝐿𝑆=3 centimeters, and 𝑊𝑇=4 centimeters. What is the value of 𝑥? 5 7 8 12 Groups 1, 2, and 3 C 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.2.4 In the diagram below of ∆𝐴𝐵𝐶, ∠𝐴𝐵𝐶 is a right angle, 𝐴𝐶 = 12, 𝐴𝐷 = 8, and altitude 𝐵𝐷 is drawn. What is the length of 𝐵𝐶 ? 4 2 4 3 4 5 4 6 Groups 2 and 3 B 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.2.5 A billboard at ground level has a support length of 26 feet that extends from the top of the billboard to the ground. A post that is 5 feet tall is attached to the support and is 4 feet from where the base of the support is attached to the ground. In the figure shown, the distance, in feet, from the base of the billboard to the base of the support is labeled 𝑥. What is the value of 𝑥? 16.24 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.2.5 Given ∆𝑀𝑅𝑂 shown below, with trapezoid 𝑃𝑇𝑅𝑂, 𝑀𝑅=9, 𝑀𝑃=2, and 𝑃𝑂=4. What is the length of 𝑇𝑅 ? 6 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.2.5 A 9-foot (𝑓𝑡) ladder and a 4-foot ladder are leaning against a house. The two ladders create angles of the same measure with the ground. The 4-foot ladder has a height of 3.8 feet against the house. What is the height, in feet, of the 9-foot ladder against the house? 8.55 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.2.5 Given △ABC ∼ △FDE, what are the values of 𝑥 and 𝑦? Select all that apply. 𝑥 = −1 𝑥 = 2 𝑥 = 4 𝑦 = −2 𝑦 = 2 𝑦 = 23 Groups 1, 2, and 3 2018 Geometry Bootcamp

MAFS.912.G-SRT.3.6 𝐴𝐵 𝐴𝐶 𝐴𝐵 𝐵𝐶 𝐵𝐶 𝐴𝐶 𝐷𝐸 𝐷𝐹 𝐷𝐸 𝐸𝐹 𝐸𝐹 𝐷𝐹
Triangle ABC and DEF are right triangles, as shown. Triangle ABC is similar to triangle DEF. Which ratios are equal to 𝑠𝑖𝑛𝐶? Select all that apply. 𝐴𝐵 𝐴𝐶 𝐴𝐵 𝐵𝐶 𝐵𝐶 𝐴𝐶 𝐷𝐸 𝐷𝐹 𝐷𝐸 𝐸𝐹 𝐸𝐹 𝐷𝐹 Groups 1, 2, and 3

MAFS.912.G-SRT.3.6 Which statement is true given ∆𝐴𝐵𝐶 ~ ∆𝐷𝐸𝐹 and 𝑚∠𝐴=𝑚∠𝐷= 90°? cos⁡B ≅ cos⁡E cos⁡B ~ cos⁡E cos⁡B = cos⁡E cos⁡B ≠ cos⁡E Groups 1, 2, and 3 C

MAFS.912.G-SRT.3.6 Right triangle ABC is shown.
2018 Geometry Bootcamp MAFS.912.G-SRT.3.6 Right triangle ABC is shown. What must be true about ∠A and ∠B? Select all that apply. ∠A ≅ ∠B ∠A and ∠B are complementary ∠A and ∠B are supplementary cos A = cos B cos A = sin B sin A = cos B sin A = sin B Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.3.7 In right triangle 𝐴𝐵𝐶, 𝑚∠𝐶=90°. If 𝑐𝑜𝑠𝐵= 5 13 , which function also equals ? tan 𝐴 tan 𝐵 sin 𝐴 sin 𝐵 Groups 1, 2, and 3 C 2018 Geometry Bootcamp

MAFS.912.G-SRT.3.7 The figure shows triangle 𝐴𝐵𝐶.
2018 Geometry Bootcamp MAFS.912.G-SRT.3.7 The figure shows triangle 𝐴𝐵𝐶. Select all expressions that must be equivalent to 𝑐𝑜𝑠𝐴. sin 𝑥° sin 𝑦° cos 𝑦 ° cos 90−𝑦 ° cos 90−𝑥 ° sin 90−𝑦 ° sin 90−𝑥 ° Groups 1, 2, and 3 2018 Geometry Bootcamp

MAFS.912.G-SRT.3.7 cos 𝐴 = sin 𝐴 cos 𝐴 = sin 𝐵 cos 𝐴 = cos 𝐵
2018 Geometry Bootcamp MAFS.912.G-SRT.3.7 Triangle 𝐴𝐵𝐶 is shown. Which statement must be true? cos 𝐴 = sin 𝐴 cos 𝐴 = sin 𝐵 cos 𝐴 = cos 𝐵 sin 𝐴 = sin 𝐵 Groups 1, 2, and 3 B 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 Standing by a lighthouse at point 𝐿, you locate two buoys, at points 𝐽 and 𝐾. You know that point is 𝐽 is 72 yards from the lighthouse, 𝐾 is 75 yards from the lighthouse, and point 𝐾 is 21 yards to the right of point 𝐽. Which of the following inverse trigonometric ratios can be used to find 𝑚∠𝐿? sin − cos − tan − sin − cos − tan − Groups 1, 2, and 3 2018 Geometry Bootcamp

MAFS.912.G-SRT.3.8 Write the expression that can be used to find 𝐴𝐶?
2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 Write the expression that can be used to find 𝐴𝐶? 𝟕.𝟖∙ sin 𝟔𝟕° Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 The right triangle shown is missing the lengths of two sides. Enter the lengths of the two missing sides in the boxes below. Round your answers to the nearest tenth. Length of the hypotenuse: Length of the leg: cm 10.4 6.7 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 A window washer 45 meters up the side of an office building can look down at an angle of depression of 20° at his truck parked on street. What is the horizontal distance d from the truck to the office building, rounded to the nearest tenth of a meter? 15.4 m 16.4 m 42.3 m 123.6 m Groups 1, 2, and 3 D 2018 Geometry Bootcamp

MAFS.912.G-SRT.3.8 A submarine dives as shown in the diagram.
2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 A submarine dives as shown in the diagram. To the nearest degree, determine the dive angle whose measure is 𝑋. Enter your answer in the box. Groups 1, 2, and 3 14 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 Twelve students are lined up to have their class picture taken. The photographer’s camera has a picture angle of 52°. The picture angle limits the width of the photo that can be taken. The line of students is approximately 26 feet long. About how far must the photographer be from the line of students in order to center all 12 students in the picture? 15 feet 27 feet 30 feet 53 feet Groups 1, 2, and 3 B 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 Bob places an 18-foot ladder 6 feet from the base of his house and leans it up against the side of his house. Find, to the nearest degree, the measure of the angle the bottom of the ladder makes with the ground. Enter your answer in the box. 71 Groups 1, 2, and 3 2018 Geometry Bootcamp

MAFS.912.G-SRT.3.8 Triangle RST is shown.
2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 Triangle RST is shown. ∆𝐽𝐾𝐿 ~ ∆𝑅𝑆𝑇 with a scale factor of 1.5. What is tan 𝐿? 𝟓 𝟖 or 𝟕.𝟓 𝟏𝟐 Groups 1, 2, and 3 2018 Geometry Bootcamp

2018 Geometry Bootcamp MAFS.912.G-SRT.3.8 Two boats are traveling toward a lighthouse that is 200 feet (𝑓𝑡) above sea level at its top. When the two boats and the lighthouse are collinear, the boats are exactly 250 feet apart and the closest to the lighthouse has an angle of elevation to the top of the lighthouse of 15°, as shown. What is the value of 𝑥, rounded to the nearest hundredth? Groups 2 and 3 Enter your answer in the box. 11.35 2018 Geometry Bootcamp

Congruence, Similarity, Right Triangles, and Trigonometry

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Congruence, Similarity, Right Triangles, and Trigonometry

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