Point B(−1, 6) is reflected across the y-axis. What are the new coordinates? — (1, 6).Reflection across the y-axis maps (x, y) → (−x, y). So (−1, 6) → (1, 6).
Point Q(3, −5) is rotated 90° clockwise about the origin. What is its image? — (5, 3).The rule for 90° clockwise rotation is (x, y) → (y, −x). So (3, −5) → (−5, −3). Wait — re-check: 90° CW: (x, y) → (y, −x) = (−5, −3). Actually the correct mapping 90° CW: (x,y)→(y,−x). (3,−5)→(−5,−3). Let me recalculate: (y, −x) = (−5, −3). The answer is (−5, −3).
Which composition of two reflections across parallel lines is equivalent to a single translation? — Translation.Reflecting a figure across two parallel lines produces a translation. The translation distance equals twice the distance between the lines.
Which composition of two reflections across intersecting lines is equivalent to a single rotation? — Rotation.Reflecting across two intersecting lines produces a rotation about their intersection point, with an angle equal to twice the angle between the lines.
Point R(2, 7) is rotated 270° counterclockwise about the origin. What is the image? — (7, −2).270° CCW is equivalent to 90° CW. The rule 90° CW: (x, y) → (y, −x). So (2, 7) → (7, −2).
A triangle has vertices at (0,0), (4,0), and (2,3). After translation by (−1, 2), what are the new vertices? — (−1,2), (3,2), (1,5).Add (−1, 2) to each vertex: (0−1, 0+2)=(−1,2); (4−1, 0+2)=(3,2); (2−1, 3+2)=(1,5).
Which transformation preserves orientation (does not flip the figure)? — Translation.Translations and rotations preserve orientation (sense of handedness). Reflections reverse orientation.
After a rotation of 360° about any point, what is the result? — The figure returns to its original position.A 360° rotation returns every point to its original location — it is the identity transformation.
What is the line of reflection if point A(3, 4) maps to A′(3, −4)? — x-axis.The y-coordinate is negated while x stays the same: (x, y) → (x, −y) is the rule for reflection across the x-axis.
A glide reflection is a composition of which two transformations? — Translation and reflection.A glide reflection is a translation followed by a reflection across a line parallel to the direction of translation (or vice versa).
In △PQR and △XYZ, ∠P = ∠X, PQ = XY, and ∠Q = ∠Y. Which criterion proves them congruent? — ASA.∠P = ∠X, PQ = XY (included side between the angles), ∠Q = ∠Y. The side PQ is between ∠P and ∠Q, matching ASA.
Two right triangles each have a hypotenuse of 13 and a leg of 5. Which criterion proves them congruent? — HL.HL (Hypotenuse-Leg) applies to right triangles. Equal hypotenuses (13) and equal legs (5) satisfy HL.
If △ABC ≅ △DEF by SAS, with AB = DE = 7, ∠A = ∠D = 50°, and AC = DF = 9, what else can you conclude? — All remaining sides and angles are equal (CPCTC).Once △ABC ≅ △DEF is established, CPCTC guarantees all remaining corresponding parts are equal: BC = EF, ∠B = ∠E, ∠C = ∠F.
Why can't AAA (Angle-Angle-Angle) prove triangle congruence? — Equal angles only guarantee similarity, not congruence.AAA guarantees the triangles have the same shape (similar), but they can be different sizes. Two triangles with all equal angles need not have equal sides.
In △JKL and △MNP, ∠K = ∠N = 90°, KL = NP, and JL = MP. Which criterion applies? — HL.Both triangles are right triangles (90° angle at K and N). JL and MP are hypotenuses (opposite the right angle), and KL = NP are legs. This is HL.
If △ABC ≅ △DEF and you need to prove AC = DF, at what step can you use CPCTC? — After proving the triangles are congruent.CPCTC is a consequence of triangle congruence — it can only be used AFTER you have proven the triangles congruent in the proof.
A diagonal of a parallelogram divides it into two triangles. Which congruence criterion most directly proves these triangles congruent? — SSS.The diagonal is a shared side. Opposite sides of a parallelogram are equal, giving two more pairs of equal sides. So all three pairs of sides are equal — SSS applies.
Given: △ABC with AB = CB and M is the midpoint of AC. Prove △ABM ≅ △CBM. Which criterion applies? — SSS.AB = CB (given), AM = CM (M is midpoint), BM = BM (reflexive). All three pairs of sides are equal — SSS.
In △RST and △UVW, RS = UV, ST = VW, and ∠S = ∠V. The equal angle is included between the equal sides. Which criterion applies? — SAS.∠S is between RS and ST; ∠V is between UV and VW. Two sides and the included angle matches SAS.
Which reason justifies: "∠1 and ∠2 are vertical angles, therefore ∠1 = ∠2"? — Vertical Angles Theorem.The Vertical Angles Theorem states that vertical angles (formed by two intersecting lines) are congruent.
In a proof, "∠ABD = ∠CBD because BD bisects ∠ABC." What justification is used? — Definition of angle bisector.An angle bisector divides an angle into two equal parts. This is the definition of angle bisector.
What is the Segment Addition Postulate? — AB + BC = AC when B is between A and C.The Segment Addition Postulate states: if B is between A and C, then AB + BC = AC.
A proof shows △AMC ≅ △BMD and then states AC = BD. What reason justifies this step? — CPCTC.AC and BD are corresponding sides of the congruent triangles. Once congruence is proven, CPCTC justifies AC = BD.
In a proof, you write "∠1 + ∠2 = 180° because they form a straight line." What postulate is this? — Linear Pair Postulate.The Linear Pair Postulate states adjacent angles that form a straight line are supplementary (sum to 180°).
Which is a valid reason to justify "AB = CD" in a proof? — Given or a proven theorem/calculation.Every statement in a proof needs a logical justification: it is either Given, follows from a definition, or from a previously proved theorem or postulate.
In the proof: "1. AB = DE (Given); 2. BC = EF (Given); 3. ∠B = ∠E (Given); 4. △ABC ≅ △DEF (?)" — what is the missing reason? — SAS.AB = DE (side), ∠B = ∠E (included angle between the sides), BC = EF (side). The angle is between the two sides — SAS.
What does it mean for a proof to be "valid"? — Each statement follows logically from previous statements with a valid reason.A valid proof requires each statement to follow logically from prior statements using accepted definitions, postulates, or theorems as reasons.
A proof uses "∠ABC ≅ ∠ABC." What property justifies this? — Reflexive Property.Any angle is congruent to itself. This is the Reflexive Property applied to angles.
In a proof involving a parallelogram, you use "AB ∥ CD." What reason could justify this? — Definition of a parallelogram.By definition, a parallelogram has two pairs of parallel sides. "Definition of parallelogram" justifies AB ∥ CD.
Two lines are parallel if the alternate interior angles formed by a transversal are: — Congruent.The Converse of the Alternate Interior Angles Theorem: if alternate interior angles are congruent, the lines are parallel.
In △ABC, ∠A = 45° and ∠B = 75°. What is ∠C? — 60°.∠A + ∠B + ∠C = 180° → 45° + 75° + ∠C = 180° → ∠C = 60°.
An exterior angle at vertex C of △ABC measures 130°. If ∠A = 50°, what is ∠B? — 80°.Exterior angle = ∠A + ∠B → 130° = 50° + ∠B → ∠B = 80°.
A rectangle is a special parallelogram. What additional property does a rectangle have? — All angles are 90°.A rectangle is a parallelogram with all four interior angles equal to 90°. Its diagonals are equal in length but not necessarily perpendicular.
A rhombus has all sides equal. What additional property do its diagonals have? — They are perpendicular bisectors of each other.In a rhombus, the diagonals are perpendicular to each other and each diagonal bisects the other (midpoint intersection).
If corresponding angles formed by two lines and a transversal are supplementary, the two lines are: — Perpendicular.If corresponding angles are supplementary (sum to 180°) and each pair also equals 90°, the transversal meets each line at 90°, meaning the lines are perpendicular. Actually, supplementary corresponding angles (each 90°) means both lines are perpendicular to the transversal, hence parallel to each other only if both are 90°. In general, if corresponding angles are supplementary but not 90°, the lines are NOT parallel.
In a parallelogram ABCD, ∠A = 70°. What is ∠B? — 110°.Consecutive angles of a parallelogram are supplementary: ∠A + ∠B = 180° → ∠B = 180° − 70° = 110°.
A transversal crosses two lines. The alternate exterior angles measure 5x° and (3x + 20)°. If the lines are parallel, find x. — 10.Alternate exterior angles are equal when lines are parallel: 5x = 3x + 20 → 2x = 20 → x = 10.
A square is both a rectangle and a rhombus. How many lines of symmetry does a square have? — 4.A square has 4 lines of symmetry: 2 through opposite vertices (diagonals) and 2 through midpoints of opposite sides.
In parallelogram PQRS, diagonal PR divides it into two triangles. What congruence criterion proves △PQR ≅ △RSP? — SSS.PQ = RS, QR = SP (opposite sides of parallelogram), PR = PR (shared diagonal). All three pairs of sides are equal — SSS.