Angles with parallel lines

{"questions":[{"content":"One of the alternate interior angles is $48^\circ$. What is the measure of the other alternate interior angle?[[choice-1]]","widgets":{"choice-1":{"type":"choice","options":["$48^\circ$","$132^\circ$","$96^\circ$","$312^\circ$"],"explanations":["Alternate interior angles with parallel lines are equal, so the second angle is also $48^\circ$.","They are not adjacent: for alternate interior angles, equality is used, not the sum of $180^\circ$.","Doubling the angle is not applied here.","An angle greater than $180^\circ$ is not considered in this problem."],"answer":[0]}},"hints":["Pay attention to how the alternate interior angles are positioned relative to the transversal.","Alternate interior angles with parallel lines are equal."]},{"content":"One of the consecutive interior angles is $65^\circ$. What is the measure of the other consecutive interior angle?[[choice-10]]","widgets":{"choice-10":{"type":"choice","options":["$65^\circ$","$115^\circ$","$125^\circ$","$245^\circ$"],"explanations":["Consecutive interior angles do not equal each other but complement each other to $180^\circ$.","The sum of consecutive interior angles is $180^\circ$, so $180^\circ — 65^\circ = 115^\circ$.","Check: $65^\circ + 125^\circ = 190^\circ$, not $180^\circ$.","An angle greater than $180^\circ$ is not suitable."],"answer":[1]}},"hints":["Consecutive interior angles are on the same side of the transversal.","The sum of consecutive interior angles with parallel lines is $180^\circ$."]},{"content":"One of the corresponding angles is $102^\circ$. What is the measure of the corresponding angle?[[choice-28]]","widgets":{"choice-28":{"type":"choice","options":["$78^\circ$","$102^\circ$","$180^\circ$","$204^\circ$"],"explanations":["The difference with $180^\circ$ is used for consecutive angles, not corresponding.","Corresponding angles with parallel lines are equal, so the second angle is $102^\circ$.","Corresponding angles do not necessarily sum to $180^\circ$.","An angle greater than $180^\circ$ is not considered."],"answer":[1]}},"hints":["Such angles occupy the same position relative to the transversal.","Corresponding angles with parallel lines are equal."]},{"content":"[[image-56]][[matcher-65]]","widgets":{"image-56":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/1-01-19.svg"},"matcher-65":{"type":"matcher","labels":["Alternate interior","Consecutive interior","Corresponding"],"items":["$3$ and $5$","$5$ and $4$","$5$ and $2$","$5$ and $8$"]}},"hints":["Alternate interior angles lie across from each other relative to the transversal.","Consecutive interior angles are on the same side relative to the transversal."]},{"content":"[[image-141]][[matcher-154]]","widgets":{"image-141":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/1-01-19.svg"},"matcher-154":{"type":"matcher","labels":["Consecutive interior","Alternate interior","Corresponding"],"items":["$6$ and $3$","$4$ and $6$","$3$ and $7$","$3$ and $4$"]}},"hints":["Alternate interior angles lie across from each other relative to the transversal.","Consecutive interior angles are on the same side relative to the transversal."]},{"content":"[[image-624]][[matcher-647]]","widgets":{"image-624":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/2-01-19.svg"},"matcher-647":{"type":"matcher","labels":["Corresponding","Alternate interior","Consecutive interior"],"items":["$3$ and $7$","$4$ and $6$","$4$ and $5$","$2$ and $3$"]}},"hints":["Consecutive angles are on the same side relative to the transversal and inside parallel lines.","One of the corresponding angles lies inside, and the other outside the parallel lines. Such angles are equal."]},{"content":"[[image-835]][[matcher-862]]","widgets":{"image-835":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/2-01-19.svg"},"matcher-862":{"type":"matcher","labels":["Consecutive interior","Alternate interior","Corresponding"],"items":["$6$ and $3$","$3$ and $5$","$2$ and $6$","$1$ and $2$"]}},"hints":["The sum of consecutive interior angles is $180^\circ$.","Corresponding, like alternate interior angles, are equal."]},{"content":"[[image-1429]][[matcher-1464]]","widgets":{"image-1429":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/3-01-19.svg"},"matcher-1464":{"type":"matcher","labels":["Corresponding","Alternate interior","Consecutive interior"],"items":["$1$ and $2$","$3$ and $4$","$5$ and $6$","No correct answer"]}},"hints":["Consecutive interior angles lie inside the parallel lines and are on the same side relative to the transversal.","Alternate interior angles lie inside the parallel lines and are across from each other relative to the transversal."]},{"content":"When two parallel lines are intersected by a transversal, one of the corresponding angles is $77^\circ$. What is the measure of the second corresponding angle?[[choice-2005]]","widgets":{"choice-2005":{"type":"choice","options":["$77^\circ$","$103^\circ$","$154^\circ$","$206^\circ$"],"explanations":["Corresponding angles are equal, so the second angle is $77^\circ$.","The property of sum $180^\circ$ applies to consecutive interior angles.","Doubling is not used here.","An angle greater than $180^\circ$ is not suitable."],"answer":[0]}},"hints":["Corresponding angles are in the same position relative to the transversal.","With parallel lines, they are equal."]},{"content":"One of the alternate interior angles is $56^\circ$. What is the measure of the other alternate interior angle?[[choice-3001]]","widgets":{"choice-3001":{"type":"choice","options":["$56^\circ$","$124^\circ$","$66^\circ$","$134^\circ$"],"explanations":["Alternate interior angles with parallel lines are equal, so the second angle is also $56^\circ$.","Sum $180^\circ$ applies to consecutive interior angles, not alternate interior.","Such a result does not follow from the properties of alternate interior angles.","Such a result does not follow from the properties of alternate interior angles."],"answer":[0]}},"hints":["Consider alternate interior angles: they are inside the parallel lines on opposite sides of the transversal.","Alternate interior angles with parallel lines are equal."]},{"content":"One of the consecutive interior angles is $118^\circ$. What is the measure of the second consecutive interior angle?[[choice-3002]]","widgets":{"choice-3002":{"type":"choice","options":["$118^\circ$","$62^\circ$","$72^\circ$","$152^\circ$"],"explanations":["Consecutive interior angles do not equal each other: they complement each other to $180^\circ$.","The sum of consecutive interior angles is $180^\circ$, so $180^\circ — 118^\circ = 62^\circ$.","Check: $118^\circ + 72^\circ = 190^\circ$, not $180^\circ$.","Check: $118^\circ + 152^\circ = 270^\circ$, not $180^\circ$."],"answer":[1]}},"hints":["Consecutive interior angles lie inside the parallel lines and on the same side of the transversal.","The sum of consecutive interior angles with parallel lines is $180^\circ$."]},{"content":"One of the corresponding angles is $143^\circ$. What is the measure of the corresponding angle?[[choice-3003]]","widgets":{"choice-3003":{"type":"choice","options":["$37^\circ$","$143^\circ$","$147^\circ$","$180^\circ$"],"explanations":["The difference with $180^\circ$ applies to consecutive angles, not corresponding.","Corresponding angles with parallel lines are equal, so the second angle is $143^\circ$.","Such a result does not follow from the property of corresponding angles.","Corresponding angles are not required to sum to $180^\circ$."],"answer":[1]}},"hints":["Corresponding angles occupy the same position relative to the transversal and parallel lines.","Corresponding angles with parallel lines are equal."]},{"content":"One of the alternate interior angles is $129^\circ$. What is the measure of the other alternate interior angle?[[choice-3004]]","widgets":{"choice-3004":{"type":"choice","options":["$51^\circ$","$129^\circ$","$121^\circ$","$39^\circ$"],"explanations":["The difference with $180^\circ$ applies to consecutive angles, not alternate interior.","Alternate interior angles are equal, so the second angle is $129^\circ$.","Such a result does not follow from the property of alternate interior angles.","Such a result does not follow from the property of alternate interior angles."],"answer":[1]}},"hints":["Alternate interior angles are inside the parallel lines and on opposite sides relative to the transversal.","Alternate interior angles with parallel lines are equal."]},{"content":"One of the consecutive interior angles is $74^\circ$. What is the measure of the other consecutive interior angle?[[choice-3005]]","widgets":{"choice-3005":{"type":"choice","options":["$74^\circ$","$96^\circ$","$106^\circ$","$116^\circ$"],"explanations":["Consecutive interior angles are not equal: they sum to $180^\circ$.","Check: $74^\circ + 96^\circ = 170^\circ$, not $180^\circ$.","The sum of consecutive interior angles is $180^\circ$, so $180^\circ — 74^\circ = 106^\circ$.","Check: $74^\circ + 116^\circ = 190^\circ$, not $180^\circ$."],"answer":[2]}},"hints":["Find the pair of consecutive interior angles: they are positioned inside the parallel lines on the same side relative to the transversal.","Consecutive interior angles with parallel lines sum to $180^\circ$."]},{"content":"One of the corresponding angles is $88^\circ$. What is the measure of the corresponding angle?[[choice-3006]]","widgets":{"choice-3006":{"type":"choice","options":["$92^\circ$","$88^\circ$","$102^\circ$","$180^\circ$"],"explanations":["Such a result does not follow from the property of corresponding angles.","Corresponding angles with parallel lines are equal, so the second angle is $88^\circ$.","Such a result does not follow from the property of corresponding angles.","Corresponding angles are not required to sum to $180^\circ$."],"answer":[1]}},"hints":["Corresponding angles are on the same side relative to the transversal and occupy the same position at two parallel lines.","Corresponding angles with parallel lines are equal."]}]}

{"questions":[{"content":"Lines $a$ and $b$ are parallel, $c$ is a transversal. One of the angles is $48^\circ$. What is the measure of the corresponding angle? [[choice-1]]","widgets":{"choice-1":{"type":"choice","options":["$48^\circ$","$132^\circ$","$96^\circ$","$42^\circ$"],"explanations":["Corresponding angles with parallel lines are equal.","This value would be an additional angle to $180^\circ$, but adjacent angles are not needed here.","Invalid operation: multiplication by 2 is not related to the definition of corresponding angles.","The number does not follow from the properties of angles with parallel lines."],"answer":[0]}},"step":1,"hints":["One of the corresponding angles is inside the parallel lines, the other outside.","With parallel lines, corresponding angles are equal, so the second angle is $48^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. One of the alternate interior angles is $67^\circ$. What is the measure of the other alternate interior angle? [[choice-2]]","widgets":{"choice-2":{"type":"choice","options":["$67^\circ$","$113^\circ$","$134^\circ$","$33^\circ$"],"explanations":["Alternate interior angles with parallel lines are equal.","This is an additional number to $180^\circ$, but for alternate interior angles, equality is used.","Doubling does not apply to the properties of alternate interior angles.","The value does not follow from the data and the definition of alternate interior angles."],"answer":[0]}},"step":1,"hints":["Remember the property of alternate interior angles with parallel lines.","Alternate interior angles are equal, so the sought angle is $67^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. A consecutive interior angle is $125^\circ$. What is the measure of the other consecutive interior angle? [[choice-3]]","widgets":{"choice-3":{"type":"choice","options":["$55^\circ$","$125^\circ$","$65^\circ$","$145^\circ$"],"explanations":["Consecutive interior angles sum to $180^\circ$: $180^\circ-125^\circ=55^\circ$.","Equality is not true here: consecutive interior angles are not equal but complement each other to $180^\circ$.","Error in subtraction: $180^\circ-125^\circ$ is not $65^\circ$.","Such a number does not come from the sum of $180^\circ$."],"answer":[0]}},"step":1,"hints":["Identify the pair of consecutive interior angles.","They complement each other to $180^\circ$, so $180^\circ-125^\circ=55^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. One of the consecutive interior angles is $88^\circ$. Find the other consecutive interior angle. [[choice-4]]","widgets":{"choice-4":{"type":"choice","options":["$92^\circ$","$88^\circ$","$44^\circ$","$102^\circ$"],"explanations":["Consecutive interior angles give $180^\circ$: $180^\circ-88^\circ=92^\circ$.","Here there is no equality, but a sum of $180^\circ$.","Half an angle is not related to this property.","Such a value does not come from subtracting from $180^\circ$."],"answer":[0]}},"step":1,"hints":["Remember that the sum of consecutive interior angles is $180^\circ$.","Subtract from $180^\circ$: $180^\circ-88^\circ=92^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. One of the consecutive interior angles is $140^\circ$. What is the measure of the other interior angle? [[choice-5]]","widgets":{"choice-5":{"type":"choice","options":["$40^\circ$","$140^\circ$","$70^\circ$","$20^\circ$"],"explanations":["Exterior consecutive angles (on the same side of the transversal, outside) sum to $180^\circ$: $180^\circ-140^\circ=40^\circ$.","For consecutive angles, the sum of $180^\circ$ applies, not equality.","Halving does not relate to this type of angles.","Invalid subtraction: $180^\circ-140^\circ$ is not $20^\circ$."],"answer":[0]}},"step":1,"hints":["Consecutive interior angles with parallel lines complement each other to $180^\circ$.","Find: $180^\circ-140^\circ=40^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. One of the angles formed by the transversal with line $a$ is $36^\circ$. What is the measure of the corresponding angle? [[choice-6]]","widgets":{"choice-6":{"type":"choice","options":["$36^\circ$","$144^\circ$","$72^\circ$","$54^\circ$"],"explanations":["Corresponding angles with $a \parallel b$ are equal.","This is an additional value to $180^\circ$, but not for corresponding angles.","Doubling is not used in the definition of corresponding angles.","Such a value does not follow from the property of corresponding angles."],"answer":[0]}},"step":1,"hints":["One of the corresponding angles is inside the parallel lines, the other outside.","Corresponding angles are equal, so the sought angle is $36^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. One of the alternate interior angles is $101^\circ$. Find the second alternate interior angle. [[choice-7]]","widgets":{"choice-7":{"type":"choice","options":["$101^\circ$","$79^\circ$","$180^\circ$","$99^\circ$"],"explanations":["Alternate interior angles with parallel lines are equal.","This number results from $180^\circ-101^\circ$, but equality is used for alternate interior angles.","Angle $180^\circ$ cannot be alternate interior in a regular configuration of a transversal intersecting parallel lines.","The value does not follow from the property of alternate interior angles."],"answer":[0]}},"step":1,"hints":["Remember: alternate interior angles with parallel lines are equal.","Therefore, the other alternate interior angle is also $101^\circ$."]},{"content":"Lines $a \parallel b$. A consecutive interior angle is $39^\circ$. Find the second consecutive interior angle. [[choice-8]]","widgets":{"choice-8":{"type":"choice","options":["$141^\circ$","$39^\circ$","$51^\circ$","$149^\circ$"],"explanations":["Consecutive interior angles give $180^\circ$: $180^\circ-39^\circ=141^\circ$.","They are not equal, but complement each other to $180^\circ$.","Invalid subtraction: $180^\circ-39^\circ$ is not $51^\circ$.","Invalid subtraction: $180^\circ-39^\circ$ is not $149^\circ$."],"answer":[0]}},"step":1,"hints":["Use the property of consecutive interior angles: their sum is $180^\circ$.","Calculate: $180^\circ-39^\circ=141^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. One of the alternate interior angles is $62^\circ$. What is the measure of the second alternate interior angle? [[choice-9]]","widgets":{"choice-9":{"type":"choice","options":["$62^\circ$","$118^\circ$","$124^\circ$","$28^\circ$"],"explanations":["Exterior alternate interior angles with parallel lines are equal.","This is an additional value to $180^\circ$, but for alternate interior angles, equality is used.","Doubling does not relate to this property.","Such a value does not follow from the condition."],"answer":[0]}},"step":1,"hints":["Alternate interior angles lie 'cross-wise' relative to the transversal.","Alternate interior angles are equal, so the second angle is also $62^\circ$."]},{"content":"Lines $a \parallel b$, $c$ is a transversal. One of the consecutive interior angles is $95^\circ$. Find the second consecutive interior angle. [[choice-10]]","widgets":{"choice-10":{"type":"choice","options":["$85^\circ$","$95^\circ$","$75^\circ$","$105^\circ$"],"explanations":["Exterior consecutive angles give $180^\circ$: $180^\circ-95^\circ=85^\circ$.","These angles are not equal, but complement each other to $180^\circ$.","Invalid subtraction: $180^\circ-95^\circ$ is not $75^\circ$.","Invalid subtraction: $180^\circ-95^\circ$ is not $105^\circ$."],"answer":[0]}},"step":1,"hints":["For consecutive interior angles, use the sum of $180^\circ$.","Subtract: $180^\circ-95^\circ=85^\circ$."]},{"content":"[[image-381]][[matcher-404]]","widgets":{"image-381":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/2-01-19.svg"},"matcher-404":{"type":"matcher","labels":["Corresponding","Consecutive interior","Alternate interior"],"items":["$2$ and $6$","$6$ and $3$","$6$ and $4$","$4$ and $2$"]}},"hints":["Alternate interior angles are arranged 'cross-wise' relative to the transversal.","Consecutive interior angles are on the same side of the transversal."]},{"content":"[[image-592]][[matcher-619]]","widgets":{"image-592":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/1-01-19.svg"},"matcher-619":{"type":"matcher","labels":["Corresponding","Consecutive interior","Alternate interior"],"items":["$7$ and $3$","$3$ and $6$","$4$ and $6$","$4$ and $1$"]}},"hints":["One of the corresponding angles is inside, and the other outside the parallel lines, and they are equal.","Consecutive angles have different magnitudes and sum to $180^\circ$."]},{"content":"[[image-825]][[matcher-856]]","widgets":{"image-825":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/5-01-13.svg"},"matcher-856":{"type":"matcher","labels":["Corresponding","Consecutive interior","Alternate interior"],"items":["$a$ and $e$","$d$ and $f$","$c$ and $f$","$a$ and $f$"]}},"hints":["Corresponding angles are positioned relative to the parallel lines.","The sum of consecutive interior angles is $180^\circ$."]},{"content":"[[image-1011]][[matcher-1046]]","widgets":{"image-1011":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/5-01-13.svg"},"matcher-1046":{"type":"matcher","labels":["Corresponding","Consecutive interior","Alternate interior"],"items":["$d$ and $n$","$e$ and $c$","$e$ and $d$","$n$ and $e$"]}},"hints":["Pay attention to the word 'interior'. It is not present in the names of all pairs of angles.","Corresponding angles, like alternate interior angles, are equal."]},{"content":"[[image-1455]][[matcher-1494]]","widgets":{"image-1455":{"type":"image","url":"https://obrazavr.ru/wp-content/uploads/2026/02/4-01-18.svg"},"matcher-1494":{"type":"matcher","labels":["Corresponding","Consecutive interior","Alternate interior"],"items":["$1$ and $2$","$3$ and $4$","$5$ and $6$","No correct answer"]}},"hints":["One of the corresponding angles is inside, and the other outside the parallel lines, and they are equal.","Alternate interior angles are arranged 'cross-wise' relative to the transversal."]}]}