Math I
Build connected understanding across quantities, equations, functions, coordinate geometry, congruence, and data.
- Problem types
- 659
- Practice variants
- 2,636
Page 14 of 19
Each problem type has four distinct practice variants. Open a preview to move among all four.
Decide congruence using a rigid-motion sequence
Use rigid motions to transform figures and decide whether two figures are congruent.
A rigid-motion sequence remains rigid, but its order still matters for whether it reaches the stated target. Compute and label the reflected intermediate triangle first, translate those intermediate coordinates next, …
Preview problemUse coordinate evidence to decide whether figures are congruent
Use rigid motions to transform figures and decide whether two figures are congruent.
Coordinate evidence should establish congruence through preserved measurements rather than visual resemblance. Pair corresponding horizontal and vertical sides, calculate each remaining slanted side with the distance formula, and compare all …
Preview problemIdentify corresponding vertices after a transformation
Use rigid motions to transform figures and decide whether two figures are congruent.
Correspondence follows each labeled point through the transformation; it should not be guessed from nearby labels. Apply the vertical-axis reflection rule to every source coordinate, locate the resulting image label, …
Preview problemDistinguish congruent figures from merely similar figures
Use rigid motions to transform figures and decide whether two figures are congruent.
Similarity and congruence both preserve shape, so size is the deciding feature here. Express the side-length change as one scale factor: proportional corresponding sides support similarity, while congruence additionally requires …
Preview problemFind a rigid motion mapping one segment to another
Use rigid motions to transform figures and decide whether two figures are congruent.
A segment mapping must send both labeled endpoints correctly, not merely place one endpoint. Compute the displacement from each source endpoint to its assigned image and compare the two vectors; …
Preview problemReject congruence using a specific measurement mismatch
Use rigid motions to transform figures and decide whether two figures are congruent.
To test congruence from side data, sort each triangle’s complete length list and compare corresponding entries. Two matching lengths do not rescue a failed third pair: one unequal measurement is …
Preview problemWrite congruence notation from corresponding vertices
Use rigid motions to transform figures and decide whether two figures are congruent.
Triangle names encode correspondence position by position. Write the source vertices in one row and their stated partners directly beneath them, then read each row in order; this prevents a …
Preview problemIdentify corresponding sides and angles from congruence notation
Use rigid motions to show triangle congruence through corresponding sides and angles.
Congruence notation is a correspondence code: first position matches first, second matches second, and third matches third. Build the vertex map before naming any parts, then form each angle pair …
Preview problemGive a rigid-motion sequence mapping one triangle to another
Use rigid motions to show triangle congruence through corresponding sides and angles.
Infer the rigid motion from coordinate patterns across the entire correspondence. Compute target minus source for all three vertex pairs and compare those results with the sign-change and swap patterns …
Preview problemDecide triangle congruence from complete side measures
Use rigid motions to show triangle congruence through corresponding sides and angles.
Complete side data can settle triangle congruence without separately measuring angles. Order both side-length lists, pair equal lengths consistently, and ask whether all three pairs match; position or reflected orientation …
Preview problemUse corresponding parts of congruent triangles
Use rigid motions to show triangle congruence through corresponding sides and angles.
Before transferring a measure, decode the vertex order in the congruence statement. Map both endpoints of the known side to their partners in the other triangle, then use the rigid-motion …
Preview problemCorrect a mismatched triangle congruence statement
Use rigid motions to show triangle congruence through corresponding sides and angles.
The letters in a congruence statement are not interchangeable; their positions assert specific vertex pairs. Compare the proposed names column by column with the stated map, mark the first positions …
Preview problemUse a coordinate transformation to prove triangle congruence
Use rigid motions to show triangle congruence through corresponding sides and angles.
Treat the coordinate rule as one motion acting on the entire triangle, not as three unrelated calculations. Compare the displacement from each original vertex to its proposed partner, make sure …
Preview problemDecide whether side-angle information proves triangle congruence
Use rigid motions to show triangle congruence through corresponding sides and angles.
Start by locating the marked angle relative to the two marked sides; its placement is what distinguishes a valid rigid triangle pattern from ambiguous side-side-angle information. When that angle lies …
Preview problemVerify triangle congruence under reflection
Use rigid motions to show triangle congruence through corresponding sides and angles.
Organize a reflection around three checks: the mirror line determines the coordinate rule, every vertex must obey that same rule, and corresponding points must land the same perpendicular distance from …
Preview problemVerify triangle congruence under rotation
Use rigid motions to show triangle congruence through corresponding sides and angles.
For a rotation, lock in the center, turn size, and direction before touching the coordinates; those three features determine one rule for every vertex. After mapping the full triangle, use …
Preview problemClassify triangle congruence evidence
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
Read congruence marks by the kind of part they identify, not by their color or location in the drawing. Count the corresponding side and angle facts, check whether any marked …
Preview problemUse the reflexive property for a shared triangle part
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
The reflexive property is about identity, so first trace the boundary that both triangles literally use. A segment that appears in only one triangle cannot supply the shared fact; the …
Preview problemUse a congruence criterion before using corresponding parts
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
Keep this proof in two distinct phases: use only the marked givens to establish triangle congruence, then read the congruence statement position by position to identify corresponding parts. CPCTC belongs …
Preview problemComplete a missing proof statement or reason
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
To fill a proof blank, classify each given as a side fact or an angle fact before naming any theorem. Then check the angle’s placement relative to the known sides; …
Preview problemDetect the first invalid claim in a triangle congruence proof
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
Audit the proof chronologically and distinguish an unhelpful given from an unsupported deduction: a stated premise may be true without proving the desired result. At the first derived claim, compare …
Preview problemChoose missing information to complete a congruence criterion
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
Work backward from the named target pattern instead of adding any fact that happens to prove congruence. With two corresponding side pairs already supplied, locate their common endpoint in each …
Preview problemSolve algebraic congruence data before applying a criterion
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
Separate the algebra from the geometry. First equate the expressions attached to corresponding sides and solve that equation, then substitute back to verify the lengths really match; only after that …
Preview problemEvaluate which congruence criterion is valid and efficient
Explain ASA, SAS, and SSS triangle congruence criteria from rigid-motion congruence.
Because the task asks for every valid three-fact set, treat it as an exhaustive inventory rather than a search for one convenient proof. Omit each marked fact in turn, classify …
Preview problemUse the distance formula to prove segments congruent
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
A coordinate congruence proof is really a comparison of displacement, not a visual judgment about the segments. Compute each horizontal and vertical change with a consistent endpoint order, compare the …
Preview problemUse slope to prove distinct lines are parallel
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Use slope as a direction test: calculate rise over run for each point pair with the subtraction order kept consistent. Matching slopes establish the same direction, but they prove parallel …
Preview problemUse slope to prove lines are perpendicular
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Find both slopes before deciding anything from the picture, paying special attention to the sign when a line falls as x increases. For two nonvertical lines, test whether the slopes …
Preview problemUse midpoint formula to show diagonals bisect each other
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Bisection is a shared-midpoint claim, so calculate the midpoint of each diagonal independently rather than averaging all four vertices at once. Compare both coordinates of the results: one common point …
Preview problemProve a quadrilateral is a parallelogram using coordinates
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Pick one complete coordinate test for a parallelogram and carry it through; here, slopes make the opposite-side structure easy to track. Pair opposite sides rather than adjacent ones, compute each …
Preview problemProve a quadrilateral is a rectangle using coordinates
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Build the rectangle proof in two layers: first establish a parallelogram from both opposite-side direction pairs, then verify one right angle between adjacent sides. When a side is vertical, classify …
Preview problemProve a quadrilateral is a rhombus using coordinates
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Use the defining side condition and check all four cyclic edges, not just one opposite pair. Squared distances are the efficient invariant here: if the four positive squared lengths agree, …
Preview problemUse coordinates to prove or disprove a right triangle
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
A right-triangle check needs one perpendicular pair of sides that meet at the same vertex. Scan the coordinate changes for an immediate horizontal-versus-vertical pair; if that shortcut is unavailable, compare …
Preview problemUse coordinates to decide whether three points form an isosceles triangle
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Translate the classification into its defining measurement: an isosceles triangle needs at least one pair of equal side lengths, not three equal sides. Compare squared distances for the three edges …
Preview problemUse coordinates to prove a segment is a median
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Start from the definition: a median must join a vertex to the midpoint of the opposite side, not merely to some point on that side. Identify the opposite side, average …
Preview problemUse coordinates to prove a segment is an altitude
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
An altitude check has two independent requirements: the proposed foot must lie on the line containing the side opposite the vertex, and the vertex-to-foot segment must be perpendicular to that …
Preview problemChoose the best coordinate method for a geometric claim
Use coordinates, distance, and algebra to prove or disprove simple geometric statements.
Match the geometric verb to the coordinate invariant it describes before calculating anything: length claims use distance, bisection claims use midpoints, and parallel or perpendicular claims use direction. For a …
Preview problem