Questions and Answers
Given PQRS is a parallelogram, which will complete the proof to show that T is the midpoint of both PR and QS?
What is true about side AC in the translation of triangle ABC such that side AB coincides with side DE and side BC coincides with side EF?
Side AC is congruent to side DF.
Which of the following establishes that triangles ADB and BDC are each similar to triangle ABC?
Which is equal to sin(1)?
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Which set of rigid motions will show that the two shapes are congruent by superimposing one onto the other?
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Which transformation was applied to figure 1 to arrive at figure 2?
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Two lines whose intersection forms four congruent angles are called __________.
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What could prove that Molly's house is the same distance from Sophia's and Cole's houses?
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If GH = 1 and a dilation with center H and a scale factor of 0.5 is applied, what will be the length of the image of segment GH?
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Which statements should be used to prove that the measures of angles 1 and 5 sum to 180 degrees?
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Which statement(s) should be used in proving that vertical angles 1 and 3 are congruent?
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What is the scale factor of the dilation that will carry the smaller triangle onto the larger triangle?
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Which transformation applied to point P is described as drawing a line perpendicular to a given line passing through P?
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The main triangular sail on a sailboat is 25 feet tall and 10 feet wide at the bottom. If the sides of another sail are proportional to the main sail, and the bottom of this sail is 8 feet wide, how tall is it?
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What information about the lines is needed to complete the definition of parallel lines?
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Which lists all the reflections carrying the square onto itself?
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Given the coordinates of the source points and their images after translations, select the two source-image pairs that correspond to the same translation.
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What transformation will map triangle QSU onto itself?
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What can be the image of line M under a dilation with center P?
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If tan x = 6/8, what are sin x and cos x? sin x = _______ and cos x = _______.
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An isosceles triangle with an angle of 10° was dilated by a factor of 0.5. What must be true about the resulting triangle?
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Which set of transformations could Pam use to prove the two triangles below are congruent?
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Which of the following gives enough information for finding all three angle measures of the triangle?
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A 35-foot long escalator forms a 55° angle with the floor. Which is closest to the height of the next floor?
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According to the length and angle measure expressions, which of the following must be true of triangle ABC?
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If triangle ABC is reflected onto triangle DEF, then the triangles are congruent. Based on this information, what other statement is true?
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Which rigid motion could be used to show that triangle ABC is congruent to triangle BAD?
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Two pairs of parallel lines form a parallelogram. If angles 2 and 6 are congruent, which pairs of angles could be used?
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If segment BD is parallel to segment CE and AB = 4, AC = 6, and AD = 7, what is AE?
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Which set of rigid motions could be used to show that triangle ABC is congruent to triangle DEF?
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What is the cosine ratio of angle X in the given triangle?
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A boat travels directly from dock A to dock B without stopping at dock C along the way. About how many miles will the boat travel?
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Study Notes
Parallelogram and Midpoint Proofs
- In a parallelogram, to prove that point t is the midpoint of both diagonals pr and qs, the angle-side-angle congruence is often used.
Translation and Congruence of Triangles
- A translation to the right aligning triangle ABC's sides with triangle DEF indicates that angles B and E are equal in measure. This implies side AC must equal DE since corresponding parts of congruent triangles are equal.
Similar Triangles and Pythagorean Theorem
- Demonstrating triangle similarity between ADB, BDC, and ABC is essential in establishing the Pythagorean theorem. Criteria for similarity include angle-angle similarity.
Trigonometric Values
- The value of sin(1) is equivalent to cos(89°), showcasing the co-function identity in right triangles.
Rigid Motions and Congruence
- A rigid motion set, including translations, rotations, or reflections, is necessary to prove two shapes are congruent by superimposing one onto the other.
Transformation Types
- Identifying the specific transformation (rigid motion) applied to a figure is crucial in understanding how shapes can overlap or change positions.
Perpendicular Lines
- Perpendicular lines are defined as two lines that intersect to create four congruent angles, emphasizing symmetry and equal partitioning of angles at the intersection.
Distance Equality in Geometry
- To prove that Molly’s house (point X) is equidistant from Sophia's and Cole's houses, one must demonstrate that both distances are equal through geometric or algebraic means.
Dilation and Image Lengths
- A dilation with a center at H and a scale factor of 0.5 applied to segment GH, originally 1, results in an image length of 0.5, illustrating the effect of scale factor on lengths.
Angles and Measurements
- Identifying relationships that sum angles, such as stating angles 1 and 5 sum to 180°, often relies on supplementary angles and linear pairs.
Vertical Angle Congruence
- Vertical angles, such as angles 1 and 3, are congruent due to their position opposite to each other when two lines intersect.
Scale Factor in Dilation
- The scale factor required to enlarge or reduce triangles (from smaller to larger) is critical in determining proportional lengths in similar figures.
Transformation Descriptions
- Drawing a perpendicular line from a given point and locating a point equidistant on the opposite side describes a reflection across that line.
Proportional Relationships in Triangles
- If the main triangular sail is 25 feet tall and 10 feet wide, then another sail with a base of 8 feet, proportionate in size, can be determined using cross-multiplication for similar triangles.
Definition of Parallel Lines
- Parallel lines need to be defined by their consistent distance apart and non-intersecting nature, supplemented by their directional vectors if specified.
Reflections in Geometry
- Identifying which reflections can superimpose shapes carries significance in establishing congruence through visual transformations.
Source-Image Pairing and Translations
- Understanding source points and their transformed images following translations can validate geometric transformations' congruence.
Equidistant Points on a Circle
- Knowing points spaced evenly on a circle can determine symmetries and rotational mappings of geometric shapes.
Dilation Outcomes
- A dilation applied to geometric figures alters their proportional dimensions while maintaining angle measures.
Triangle Congruence by Transformations
- Transformations such as rotations, translations, or reflections can be used to prove triangle congruence, showcasing the interconnectedness of geometric shapes.
Geometry of Angles in Triangles
- Sufficient information about triangle angles (e.g., two angles and one side) can be leveraged to find all measures due to properties of triangle sum.
Real-World Applications of Angles
- Estimating the height of an escalator forming a specific angle with the floor can involve trigonometric calculations to derive vertical heights from known lengths.
Triangle Congruence Through Reflection
- Reflecting triangle ABC onto triangle DEF ensures congruence, highlighting properties of reflection as a congruence-preserving transformation.
Parallel Segments and Lengths
- Given parallel segments, knowing lengths is vital for determining other segment relationships, applying properties of proportional triangles.
Rigid Motions for Congruent Triangles
- Identifying the appropriate rigid motion set for congruence between two triangles reinforces the fundamental geometric principle of congruence through transformation.
Cosine Ratios
- The cosine ratio for any angle in a right triangle is defined as the adjacent side length over the hypotenuse, providing a valuable tool in solving for unknown triangle dimensions.
Distance Calculation for Boats
- Calculation of distance traveled by a boat directly between two points involves knowledge of the coordinates and geometry to quantify travel distances in real-world scenarios.
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Prepare for your Geometry A final exam with this comprehensive set of flashcards. Cover key concepts involving parallelograms, translations, and angle properties. Test your understanding and ensure you're ready for exam day.
