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of corresponding angles has equal measure."},{"text":"Corresponding sides and angles are proportional.","correct":false,"explanation":"Proportional measurements are associated with similar triangles and do not necessarily mean the triangles are the same size. For congruent triangles, corresponding sides and angles are equal."},{"text":"Corresponding sides and angles are parallel.","correct":false,"explanation":"Corresponding parts occupy matching positions but do not have to be parallel. Congruence allows you to conclude that corresponding sides have equal lengths and corresponding angles have equal measures."},{"text":"Corresponding sides and angles are perpendicular.","correct":false,"explanation":"Perpendicular lines meet at $90^\\circ$, which is unrelated to the general consequence of congruence. Corresponding sides and angles in congruent triangles are equal."}]},{"id":"252581e0-5798-81b9-bb4d-4d1966c7509e","cardType":"cloze","points":null,"front":"In $\\triangle ABC \\cong \\triangle DEF$, the order shows B matches [ ].","back":"In $\\triangle ABC \\cong \\triangle DEF$, the order shows B matches E.","mcqOptions":[{"text":"E","correct":true,"explanation":"Congruence notation lists corresponding vertices in the same order. Since B is the second vertex of $ABC$, it matches E, the second vertex of $DEF$."},{"text":"D","correct":false,"explanation":"Matching B with D incorrectly pairs the second vertex of the first triangle with the first vertex of the second. The order shows that B matches E."},{"text":"F","correct":false,"explanation":"Matching B with F incorrectly pairs the second vertex with the third vertex. Reading both triangle names in order gives $A\\leftrightarrow D$, $B\\leftrightarrow E$, and $C\\leftrightarrow F$."},{"text":"A","correct":false,"explanation":"A belongs to the first triangle and cannot be B’s corresponding vertex in the second triangle. The notation pairs B, the second vertex, with E."}]},{"id":"f419dcea-7c02-3f5d-fff2-67a818b6783b","cardType":"basic","points":1,"front":"If $\\triangle PQR \\cong \\triangle XZY$, which angle matches $\\angle QPR$?","back":"$$\\angle QPR$ matches $\\angle ZXY$.","mcqOptions":[{"text":"$$\\angle QPR$ matches $\\angle ZXY$.","correct":true,"explanation":"The order gives $P\\leftrightarrow X$, $Q\\leftrightarrow Z$, and $R\\leftrightarrow Y$. Therefore $Q-P-R$ corresponds to $Z-X-Y$, so the matching angle is $\\angle ZXY$."},{"text":"$$\\angle QPR$ matches $\\angle XZY$.","correct":false,"explanation":"This places Z at the angle’s vertex, so it corresponds to the angle at Q rather than P. Since $P\\leftrightarrow X$, the correct angle is $\\angle ZXY$."},{"text":"$$\\angle QPR$ matches $\\angle XYZ$.","correct":false,"explanation":"This angle has Y as its vertex, corresponding to R rather than P. Using $Q\\leftrightarrow Z$, $P\\leftrightarrow X$, and $R\\leftrightarrow Y$ gives $\\angle ZXY$."},{"text":"$$\\angle QPR$ matches $\\angle YZX$.","correct":false,"explanation":"This choice reverses the correspondence and places Z at the vertex. The middle letter P must match X, so the corresponding angle is $\\angle ZXY$."}]},{"id":"4d76e6aa-bc8b-ed70-1e7d-deea545657c3","cardType":"basic","points":1,"front":"If $\\triangle PQR \\cong \\triangle XZY$, which side matches $QR$?","back":"$$QR$ matches $ZY$.","mcqOptions":[{"text":"$$QR$ matches $ZY$.","correct":true,"explanation":"The congruence order pairs Q with Z and R with Y. A side’s endpoints must map to their corresponding endpoints, so $QR$ matches $ZY$."},{"text":"$$QR$ matches $XZ$.","correct":false,"explanation":"The side $XZ$ joins the vertices corresponding to P and Q, so it matches $PQ$, not $QR$. Since $Q\\leftrightarrow Z$ and $R\\leftrightarrow Y$, $QR$ matches $ZY$."},{"text":"$$QR$ matches $XY$.","correct":false,"explanation":"The side $XY$ joins the vertices corresponding to P and R, so it matches $PR$. Mapping Q to Z and R to Y shows that $QR$ matches $ZY$."},{"text":"$$QR$ matches $ZX$.","correct":false,"explanation":"Reversing the letters of a side does not change its endpoints: $ZX$ is the same side as $XZ$, which corresponds to $PQ$. The match for $QR$ is $ZY$."}]},{"id":"d34577bb-c5ae-9160-2033-57e046c86925","cardType":"cloze","points":null,"front":"The SSS congruence test needs [ ] equal.","back":"The SSS congruence test needs three corresponding sides equal.","mcqOptions":[{"text":"three corresponding sides","correct":true,"explanation":"SSS stands for side-side-side. The test proves congruence when all three sides of one triangle are equal to the three corresponding sides of the other triangle."},{"text":"two corresponding sides","correct":false,"explanation":"Two equal side pairs alone do not fix a triangle’s shape, so different triangles may satisfy this information. SSS requires all three pairs of corresponding sides to be equal."},{"text":"three corresponding angles","correct":false,"explanation":"Three equal angle pairs establish similarity, not congruence, because the triangles may have different sizes. SSS specifically requires three corresponding sides to be equal."},{"text":"two corresponding angles","correct":false,"explanation":"Two corresponding angles determine the triangle’s shape but not its scale, so this information alone cannot prove congruence. SSS requires three corresponding sides to be equal."}]},{"id":"977715f4-d80c-0ad8-98c6-d93cbf3030f9","cardType":"basic","points":1,"front":"For SAS congruence, where must the equal angle be?","back":"The included angle between the two equal sides.","mcqOptions":[{"text":"The included angle between the two equal sides.","correct":true,"explanation":"For SAS, the angle must be enclosed by the two sides known to be equal. This fixes the triangle uniquely and allows congruence to be established."},{"text":"The non-included angle beside one equal side.","correct":false,"explanation":"Using a non-included angle gives SSA information, which can produce more than one possible triangle. SAS requires the included angle between the two equal sides."},{"text":"The opposite angle across from both equal sides.","correct":false,"explanation":"An angle opposite one of the known sides is not enclosed by both sides and gives ambiguous SSA information. The required angle is the included angle between the equal sides."},{"text":"The largest angle inside each of the triangles.","correct":false,"explanation":"The size ranking of the angle is irrelevant to SAS. It must occupy the specific position between the two corresponding equal sides, making it the included angle."}]},{"id":"882e8246-b31b-95ba-9eae-cadc6d400dda","cardType":"cloze","points":null,"front":"The ASA/AAS test uses two corresponding [ ] and one corresponding side.","back":"The ASA/AAS test uses two corresponding angles and one corresponding side.","mcqOptions":[{"text":"angles","correct":true,"explanation":"ASA and AAS each use two pairs of equal corresponding angles and one pair of equal corresponding sides. The side’s position distinguishes ASA from AAS."},{"text":"vertices","correct":false,"explanation":"Vertices are points and are not measured as equal evidence for congruence. ASA and AAS require two corresponding angles and one corresponding side."},{"text":"lengths","correct":false,"explanation":"Treating the two measurements as lengths would give three side measurements rather than ASA or AAS. These tests use two corresponding angles and one corresponding side."},{"text":"diagonals","correct":false,"explanation":"A triangle has no diagonals because each pair of non-adjacent vertices is already joined by a side. ASA and AAS use two corresponding angles and one corresponding side."}]},{"id":"19ec4d1e-ac85-91b7-3b70-7eeb08367df9","cardType":"basic","points":1,"front":"For RHS congruence, what angle fact must be true?","back":"Both triangles are right-angled.","mcqOptions":[{"text":"Both triangles are right-angled.","correct":true,"explanation":"RHS stands for right angle-hypotenuse-side, so each triangle must contain a $90^\\circ$ angle. Equal hypotenuses and one other equal side can then establish congruence."},{"text":"Both triangles are acute-angled.","correct":false,"explanation":"Acute triangles contain only angles smaller than $90^\\circ$, so the RHS test cannot apply. RHS requires both triangles to be right-angled."},{"text":"Both triangles are obtuse-angled.","correct":false,"explanation":"An obtuse triangle has an angle greater than $90^\\circ$ and cannot satisfy the right-angle condition in RHS. Both triangles must instead be right-angled."},{"text":"Both triangles are equiangular.","correct":false,"explanation":"Equiangular triangles have three equal angles and are similar, but they are not necessarily congruent or right-angled. RHS specifically requires both triangles to be right-angled."}]},{"id":"af855511-26c7-743a-6b26-807af059fd89","cardType":"basic","points":1,"front":"For RHS congruence, which side must be equal as the longest side?","back":"The hypotenuses are equal.","mcqOptions":[{"text":"The hypotenuses are equal.","correct":true,"explanation":"In right-angled triangles, the hypotenuse is always the longest side. RHS congruence requires the two hypotenuses to have equal lengths."},{"text":"The corresponding bases are equal.","correct":false,"explanation":"Treating a chosen base as the longest side ignores the defining feature of a right-angled triangle. The hypotenuses must be equal because each is opposite the right angle."},{"text":"The right-angle sides are equal.","correct":false,"explanation":"The sides forming the right angle are shorter legs, not the longest sides. RHS requires equal hypotenuses, together with one equal pair of corresponding legs."},{"text":"The perpendicular sides are equal.","correct":false,"explanation":"The perpendicular sides form the right angle and are not hypotenuses. The required longest sides are the hypotenuses, which lie opposite the right angles."}]},{"id":"9fd4f39e-9af9-7af9-5cb9-814a2309c69d","cardType":"basic","points":1,"front":"For RHS congruence, what side fact is needed besides equal hypotenuses?","back":"One other corresponding side is equal.","mcqOptions":[{"text":"One other corresponding side is equal.","correct":true,"explanation":"RHS uses a right angle, equal hypotenuses and one other pair of equal corresponding sides. These three facts fix the triangles’ size and shape."},{"text":"One other non-corresponding side is equal.","correct":false,"explanation":"Comparing sides that do not occupy matching positions cannot establish RHS congruence. One other pair of corresponding sides must be equal alongside the equal hypotenuses."},{"text":"One other corresponding side is parallel.","correct":false,"explanation":"Parallel sides describe direction rather than length, so this does not meet the RHS condition. One other pair of corresponding sides must have equal lengths."},{"text":"One other corresponding side is twice as long.","correct":false,"explanation":"A doubled side length would make the triangles different sizes rather than congruent. RHS requires one other corresponding side to be equal in length."}]},{"id":"055cd480-96f4-9a15-2001-5b47bb253b08","cardType":"basic","points":2,"front":"Why do three equal angles not prove congruent triangles?","back":"AAA gives the same shape, not necessarily the same size.","mcqOptions":[{"text":"AAA gives the same shape, not necessarily the same size.","correct":true,"explanation":"Equal corresponding angles guarantee similarity, so the triangles have the same shape. Their corresponding sides may be scaled, meaning the triangles need not be congruent."},{"text":"AAA gives the same size, but not necessarily the same shape.","correct":false,"explanation":"This reverses the similarity result. Three equal corresponding angles fix the shape, but side lengths may differ, so the triangles are not necessarily the same size."},{"text":"AAA proves both triangles have equal corresponding side lengths.","correct":false,"explanation":"Equal angles do not determine absolute side lengths because one triangle may be an enlargement of the other. AAA establishes the same shape, not necessarily the same size."},{"text":"AAA only applies when all three angles are right angles.","correct":false,"explanation":"A triangle cannot contain three right angles because its angles total 180°. AAA can show similarity for any three matching angles, but it does not prove congruence."}]},{"id":"55a7d644-f951-6ca4-df88-edc15560a148","cardType":"cloze","points":null,"front":"A common side is [ ] in both triangles.","back":"A common side is equal to itself in both triangles.","mcqOptions":[{"text":"equal to itself","correct":true,"explanation":"A common side is the same line segment in both triangles, so its length is equal to itself. This shared length can be used in a congruence proof."},{"text":"different in length","correct":false,"explanation":"A shared side cannot have two different lengths because it is the same line segment in both triangles. The correct fact is that it is equal to itself."},{"text":"twice as long","correct":false,"explanation":"This treats the shared segment as though it changes length between the triangles. A common side is identical in both, so it is equal to itself."},{"text":"perpendicular to itself","correct":false,"explanation":"A side does not form a right angle with itself. The useful congruence fact is about length: the common side is equal to itself in both triangles."}]},{"id":"aeb60d8d-ec7e-ae35-68ac-c81c4b8d09d9","cardType":"cloze","points":null,"front":"In a parallelogram, opposite sides are [ ].","back":"In a parallelogram, opposite sides are equal.","mcqOptions":[{"text":"equal","correct":true,"explanation":"A defining property of every parallelogram is that each pair of opposite sides has equal length. This fact can provide corresponding equal sides in a congruence proof."},{"text":"perpendicular","correct":false,"explanation":"Perpendicular sides meet at 90°, which is not generally true of opposite sides in a parallelogram. Opposite sides are equal in length and parallel."},{"text":"adjacent","correct":false,"explanation":"Opposite sides lie across from one another rather than sharing a vertex, so they are not adjacent. The required length property is that opposite sides are equal."},{"text":"unequal","correct":false,"explanation":"This contradicts a defining property of parallelograms. Both pairs of opposite sides have equal lengths, which is often used when proving triangles congruent."}]},{"id":"a8546204-a619-c1ca-0d33-88e12559b800","cardType":"basic","points":1,"front":"What do diagonals of a parallelogram do to each other?","back":"They bisect each other - cut each other in half.","mcqOptions":[{"text":"They bisect each other - cut each other in half.","correct":true,"explanation":"The diagonals of every parallelogram meet at their shared midpoint. Therefore, each diagonal divides the other into two segments of equal length."},{"text":"They bisect the angles - cut each corner angle in half.","correct":false,"explanation":"Confusing parallelograms with rhombuses makes this tempting, but parallelogram diagonals do not generally bisect the corner angles. They bisect each other at their intersection."},{"text":"They meet at right angles - cross at 90°.","correct":false,"explanation":"Perpendicular diagonals occur in special parallelograms such as rhombuses, but not in every parallelogram. The universal property is that the diagonals bisect each other."},{"text":"They are equal in length - have the same measurement.","correct":false,"explanation":"Equal diagonals are a property of rectangles, not all parallelograms. In every parallelogram, the diagonals bisect each other, although their total lengths may differ."}]},{"id":"fdec480f-050c-20b6-630f-b12ce4e14121","cardType":"basic","points":1,"front":"When two straight lines cross, what angle equality can you use?","back":"Vertically opposite angles are equal.","mcqOptions":[{"text":"Vertically opposite angles are equal.","correct":true,"explanation":"When two straight lines intersect, the angles directly opposite one another are vertically opposite angles. Each opposite pair always has equal angle measures."},{"text":"Adjacent angles are equal.","correct":false,"explanation":"Adjacent angles at an intersection usually form a straight line and sum to 180°, rather than being equal. Vertically opposite angles are the equal pairs."},{"text":"Corresponding angles are equal.","correct":false,"explanation":"Corresponding angles require a transversal crossing parallel lines, not merely two intersecting lines. At a simple intersection, vertically opposite angles are equal."},{"text":"Angles on a straight line are equal.","correct":false,"explanation":"Angles on a straight line add to 180° but are not generally equal. The equality created whenever two lines cross is between vertically opposite angles."}]},{"id":"ee213ba8-107b-451b-0677-2732bff9da34","cardType":"basic","points":1,"front":"On parallel lines, which angle equality can help prove triangles congruent?","back":"Alternate angles are equal.","mcqOptions":[{"text":"Alternate angles are equal.","correct":true,"explanation":"When a transversal crosses parallel lines, alternate angles lie between the lines on opposite sides of the transversal. These angles are equal and can support congruence proofs."},{"text":"Co-interior angles are equal.","correct":false,"explanation":"Co-interior angles between parallel lines add to 180° rather than being equal in general. Alternate angles form an equal pair that can be used in congruence proofs."},{"text":"Adjacent angles are equal.","correct":false,"explanation":"Adjacent angles share a vertex and side but are not automatically equal. With parallel lines, the relevant guaranteed equality here is between alternate angles."},{"text":"Angles on a straight line are equal.","correct":false,"explanation":"Angles on a straight line sum to 180° and need not have equal measures. Alternate angles formed by a transversal across parallel lines are equal."}]},{"id":"85bbb3ad-912f-c89e-b354-239d5a07604e","cardType":"cloze","points":null,"front":"A radius to the point where a tangent touches a circle is [ ] to the tangent.","back":"A radius to the point where a tangent touches a circle is perpendicular to the tangent.","mcqOptions":[{"text":"perpendicular","correct":true,"explanation":"A tangent meets the radius at the point of contact at a right angle, so the radius is perpendicular to the tangent."},{"text":"parallel","correct":false,"explanation":"A tangent cannot be parallel to the radius at their point of contact. The radius and tangent meet at a right angle, so the correct term is perpendicular."},{"text":"oblique","correct":false,"explanation":"An oblique angle is not a right angle. The circle theorem states that a radius to the point of contact is perpendicular to the tangent."},{"text":"inclined at $45^\\circ$","correct":false,"explanation":"The angle is not specifically $45^\\circ$; it is always $90^\\circ$. Therefore, the radius is perpendicular to the tangent at the point of contact."}]},{"id":"401ab2d1-de17-81f5-7ce7-dd3737dc6423","cardType":"basic","points":1,"front":"In a circle with centre O, why is $OP=OQ$ for points P and Q on the circle?","back":"$$OP$ and $OQ$ are radii of the same circle.","mcqOptions":[{"text":"$$OP$ and $OQ$ are radii of the same circle.","correct":true,"explanation":"Both segments run from the centre O to points on the circumference. All radii of the same circle have equal length, so $OP=OQ$."},{"text":"$$OP$ and $OQ$ are equal chords of the same circle.","correct":false,"explanation":"Chords have both endpoints on the circumference, whereas these segments start at the centre. They are radii of the same circle, so their lengths are equal."},{"text":"$$OP$ and $OQ$ are diameters of the same circle.","correct":false,"explanation":"A diameter passes through the centre and has two endpoints on the circumference. Each segment has only one radius length, so both are radii of the same circle."},{"text":"$$OP$ and $OQ$ are tangents from the same point.","correct":false,"explanation":"A tangent touches a circle at one point and does not run from the centre to the circumference. These segments are equal because they are radii of the same circle."}]},{"id":"63093f58-07fd-f38a-af17-b99a51eb624b","cardType":"basic","points":1,"front":"In the tangent proof with triangles XPO and XQO, which side is the shared hypotenuse?","back":"$XO$.","mcqOptions":[{"text":"$XO$.","correct":true,"explanation":"Both right-angled triangles contain $XO$, and it lies opposite their right angles at P and Q. It is therefore their shared hypotenuse."},{"text":"$XP$.","correct":false,"explanation":"$$XP$ is a leg of triangle XPO and does not belong to triangle XQO. The side shared by both triangles, and their hypotenuse, is $XO$."},{"text":"$XQ$.","correct":false,"explanation":"$$XQ$ is a leg of triangle XQO but is not a side of triangle XPO. The common hypotenuse in both triangles is $XO$."},{"text":"$OP$.","correct":false,"explanation":"$$OP$ is a radius and a perpendicular leg of triangle XPO, not a shared side. Both triangles instead contain the hypotenuse $XO$."}]},{"id":"71a660f1-0fb6-594e-ac17-7fa540dd3594","cardType":"cloze","points":null,"front":"A midpoint cuts a line segment into [ ].","back":"A midpoint cuts a line segment into two equal parts.","mcqOptions":[{"text":"two equal parts","correct":true,"explanation":"By definition, a midpoint lies halfway along a line segment and divides it into exactly two sections of equal length."},{"text":"two unequal parts","correct":false,"explanation":"A point that creates unequal sections is not the midpoint. A midpoint is halfway along the segment and therefore cuts it into two equal parts."},{"text":"three equal parts","correct":false,"explanation":"A single midpoint makes exactly two sections, not three. Dividing a segment into three equal parts requires two trisection points rather than one midpoint."},{"text":"two parts in a $1:2$ ratio","correct":false,"explanation":"A $1:2$ division places the point away from the centre of the segment. A midpoint divides the segment in a $1:1$ ratio, producing two equal parts."}]},{"id":"cdd93403-b7b8-52b9-f60f-f0d1dd36313c","cardType":"basic","points":2,"front":"In the midpoint-parallelogram proof, how do you get $XM=NP$?","back":"

$XM=MZ$ from midpoint M.
$MZ=NP$ in parallelogram ZMNP.

","mcqOptions":[{"text":"

$XM=MZ$ from midpoint M.
$MZ=NP$ in parallelogram ZMNP.

","correct":true,"explanation":"M is the midpoint of $XZ$, giving $XM=MZ$. Opposite sides of parallelogram ZMNP are equal, giving $MZ=NP$, so $XM=NP$ by transitivity."},{"text":"

$XM=MZ$ from midpoint M.
$MZ=MN$ in parallelogram ZMNP.

","correct":false,"explanation":"The midpoint equality is valid, but adjacent sides of a parallelogram are not necessarily equal. The required second equality is $MZ=NP$ because these are opposite sides."},{"text":"

$XM=XZ$ from midpoint M.
$MZ=NP$ in parallelogram ZMNP.

","correct":false,"explanation":"A midpoint makes each half equal, not one half equal to the entire segment. The correct chain is $XM=MZ$ and $MZ=NP$, hence $XM=NP$."},{"text":"

$XM=MN$ from midpoint M.
$MN=NP$ in parallelogram ZMNP.

","correct":false,"explanation":"The midpoint of $XZ$ relates $XM$ to $MZ$, not to $MN$, while adjacent parallelogram sides need not be equal. Use $XM=MZ=NP$."}]},{"id":"e02db502-5ac9-3a47-80cb-9c2fed84de48","cardType":"basic","points":1,"front":"In quadrilateral WXYZ with $WX=WZ$ and $YX=YZ$, which common side completes SSS for triangles WXY and WZY?","back":"$WY$ is common to both triangles.","mcqOptions":[{"text":"$WY$ is common to both triangles.","correct":true,"explanation":"Both triangles include the diagonal $WY$, so $WY=WY$. Together with the two given pairs of equal sides, this establishes SSS congruence."},{"text":"$WX$ is common to both triangles.","correct":false,"explanation":"$$WX$ belongs to triangle WXY but not triangle WZY, so it cannot be the common side. Both triangles contain the diagonal $WY$."},{"text":"$WZ$ is common to both triangles.","correct":false,"explanation":"$$WZ$ is a side of triangle WZY only, rather than a shared side. The diagonal $WY$ appears in both triangles and completes SSS."},{"text":"$XY$ is common to both triangles.","correct":false,"explanation":"$$XY$ belongs only to triangle WXY; triangle WZY contains $YZ$ instead. The side present in both triangles is $WY$."}]},{"id":"d13d0fff-b064-6c66-acff-c08c4b555bb7","cardType":"basic","points":1,"front":"Once $\\triangle WXY \\cong \\triangle WZY$, why is $\\angle WXY=\\angle WZY$?","back":"They are corresponding angles in congruent triangles.","mcqOptions":[{"text":"They are corresponding angles in congruent triangles.","correct":true,"explanation":"The congruence order pairs vertex X in the first triangle with vertex Z in the second. Therefore, the angles at those corresponding vertices are equal."},{"text":"They are vertically opposite angles in congruent triangles.","correct":false,"explanation":"Vertically opposite angles arise where two straight lines intersect, which is not the reason here. The angles are equal because they correspond in congruent triangles."},{"text":"They are alternate interior angles in congruent triangles.","correct":false,"explanation":"Alternate interior angles require parallel lines and a transversal, neither of which establishes this equality. Congruence makes the angles equal because they are corresponding angles."},{"text":"They are base angles in congruent triangles.","correct":false,"explanation":"No single isosceles triangle contains these as its equal base angles. They are equal because vertex X corresponds to vertex Z in the congruent triangles."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Component 1"},"heading":"Congruent Triangles","title":"Congruent Triangles Flashcards","topbar":{"backHref":"/gcse/maths/eduqas/flashcards"},"topicSidebar":"$L36"}]}],"topicId":"49cba4ee-16e0-447a-b979-d61944d6a031","walled":"$L37"}],"$L38","$L39"]