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GCSE · Geometry and Measures

4.10D Angles in Parallel Lines: Fluency and Exam Drill

EDEXCEL 1MA1 · Calculators not allowed · about 70 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Lines l and m are parallel and are crossed by a straight transversal. Angle y is in the corresponding position to the 118 degree angle marked above line l.

Diagram NOT accurately drawn

(Total for Question 1 is 2 marks)

Lines l and m are parallel and are crossed by a straight transversal. Angle y is in the alternate position to the 47 degree angle marked between l and m.

Diagram NOT accurately drawn

(Total for Question 2 is 2 marks)

Lines l and m are parallel and are crossed by a straight transversal. Angle y is co-interior (allied) with the 108 degree angle marked between l and m.

Diagram NOT accurately drawn

(Total for Question 3 is 2 marks)

Lines AB and CD are parallel and are both crossed by a straight transversal EF, meeting AB at G and CD at H. Angle x, below CD on the right of the transversal, is 63 degrees. Angle z lies above CD, on the same (right) side of the transversal. Angle y lies above AB, on the right of the transversal, in the same relative position as angle z.

Diagram NOT accurately drawn

(Total for Question 4 is 3 marks)

Work out the value of x.

Diagram NOT accurately drawn

(Total for Question 5 is 2 marks)

Work out the value of x, then find the size of the smaller of the two angles.

Diagram NOT accurately drawn

(Total for Question 6 is 3 marks)

Work out the value of x, then find the size of each angle.

Diagram NOT accurately drawn

(Total for Question 7 is 3 marks)

Lines AB and CD are parallel and are crossed by a straight transversal. Angle t is marked between AB and CD, above CD and to the left of the transversal. Three further angles are also marked: angle p (above AB, to the left of the transversal), angle s (between AB and CD, below AB, to the right of the transversal) and angle r (between AB and CD, below AB, to the left of the transversal). For each pair below, state whether the angles are corresponding, alternate or co-interior (allied).

Diagram NOT accurately drawn

(a)State the relationship between angle p and angle t.(1)

(b)State the relationship between angle s and angle t.(1)

(c)State the relationship between angle r and angle t.(1)

(Total for Question 8 is 3 marks)

The diagram shows a Z-shape formed between two parallel lines: angle x and a 74 degree angle are alternate angles. Write down the size of angle x.

Diagram NOT accurately drawn

(Total for Question 9 is 1 mark)

The diagram shows an F-shape formed by a transversal crossing two parallel lines: angle x and a 39 degree angle are corresponding angles. Write down the size of angle x.

Diagram NOT accurately drawn

(Total for Question 10 is 1 mark)

The diagram shows a C-shape (co-interior angles) formed between two parallel lines. One marked angle is 101 degrees. Write down the size of the other marked angle, x.

Diagram NOT accurately drawn

(Total for Question 11 is 1 mark)

Three parallel lines, l, m and n, are all crossed by the same straight transversal. The angle between the transversal and l is 58 degrees. Angle y is in the corresponding position where the transversal crosses n. Work out y, giving a reason.

Diagram NOT accurately drawn

(Total for Question 12 is 2 marks)

Triangle ABC has a straight line DAE through vertex A, parallel to side BC (D on the same side as B, E on the same side as C). The angle between DAE and AB is 52 degrees; the angle between DAE and AC is 65 degrees. Work out the size of angle BAC. Show your reasoning.

Diagram NOT accurately drawn

(Total for Question 13 is 3 marks)

Two parallel lines are cut by a transversal. Angle x and angle y both lie between the two parallel lines, on opposite sides of the transversal. Explain why angle x = angle y.

Diagram NOT accurately drawn

(Total for Question 14 is 2 marks)

The diagram shows two parallel lines crossed by a straight transversal. The angle between the two lines, on the lower-right side of the upper intersection, is 108 degrees. Angle y is between the two lines, on the same (right) side at the lower intersection. Angle z is between the two lines, on the other side at the lower intersection. Work out the values of y and z. Give a reason for each answer.

Diagram NOT accurately drawn

(Total for Question 15 is 4 marks)

Two parallel lines are cut by a transversal. Two co-interior angles between the lines are (4x + 10) degrees and (2x + 20) degrees. Work out the value of x, then find the size of each angle.

Diagram NOT accurately drawn

(Total for Question 16 is 4 marks)

A straight transversal crosses line AB at point G and line CD at point H. The angle between the transversal and AB at G, measured on the upper-right side, is 121 degrees. The angle between the transversal and CD at H, in the corresponding position (upper-right side), is 118 degrees. Ama claims that AB is parallel to CD. Determine, with full reasons, whether Ama is correct.

Diagram NOT accurately drawn

(Total for Question 17 is 5 marks)

Triangle ABC is isosceles with AB = AC. A straight line DAE passes through vertex A, parallel to BC (D on the same side as B, E on the same side as C). Angle DAB = 47 degrees. Work out the size of angle BAC. You must show your reasoning.

Diagram NOT accurately drawn

(Total for Question 18 is 4 marks)

Two parallel lines AB and CD are joined by a straight support strut EF, crossing AB at G and CD at H. Angle EGB = (2x + 15) degrees and angle GHD = (3x - 25) degrees; these two angles are alternate angles. Work out the value of x, then find the size of angle EGB.

Diagram NOT accurately drawn

(Total for Question 19 is 4 marks)

Lines AB and CD are parallel and are crossed by a straight transversal EF at G (on AB) and H (on CD). Angle BGH = (3y + 10) degrees. Angle GHD is co-interior with angle BGH. Angle CHF is vertically opposite angle GHD, and angle CHF = (y + 50) degrees. Work out the value of y, and hence find the size of angle GHD.

Diagram NOT accurately drawn

(Total for Question 20 is 4 marks)

Triangle ABC has angle BAC = 70 degrees. A straight line DAE passes through vertex A, parallel to BC (D on the same side as B, E on the same side as C). Angle DAB = (2x + y) degrees and angle EAC = (3x - y) degrees. Given that angle ABC is 18 degrees greater than angle ACB, work out the values of x and y.

Diagram NOT accurately drawn

(Total for Question 21 is 6 marks)

Lines AB and CD are crossed by a straight transversal EF at G (on AB) and H (on CD). Angle EGB = (5x - 12) degrees and angle GHD = (3x + 8) degrees; these two angles are co-interior.

Diagram NOT accurately drawn

(a)Find the value of x for which AB would be parallel to CD.(3)

(b)Priya says that, in fact, x = 11, and that AB is parallel to CD. Determine, with full reasons, whether Priya is correct.(3)

(Total for Question 22 is 6 marks)

Mark scheme · 4.10D Angles in Parallel Lines: Fluency and Exam Drill

Question 1

B1 y = 118 cao
B1 correct reason: corresponding angles are equal (l parallel to m)
Answer: y = 118 degrees

Question 2

B1 y = 47 cao
B1 correct reason: alternate angles are equal (l parallel to m)
Answer: y = 47 degrees

Question 3

M1 180 - 108 oe, using co-interior angles sum to 180
A1 y = 72 cao
Answer: y = 72 degrees

Question 4

M1 z = 180 - 63 = 117 (angles on a straight line EF at H)
M1 y = z (corresponding angles are equal)
A1 y = 117 cao
Answer: y = 117 degrees

Question 5

M1 3x + 5 = 65 oe (corresponding angles are equal)
A1 x = 20 cao
Answer: x = 20

Question 6

M1 (2x + 10) + (3x - 20) = 180 oe, using co-interior angles sum to 180
A1 x = 38 cao
A1 smaller angle = 86 degrees, ft from their x
Answer: x = 38; smaller angle = 86 degrees

Question 7

M1 4x - 8 = 2x + 30 oe (alternate angles are equal)
A1 x = 19 cao
A1 each angle = 68 degrees, ft from their x
Answer: x = 19; each angle = 68 degrees

Question 8

(a) B1 corresponding
(a) Answer: Corresponding angles
(b) B1 alternate
(b) Answer: Alternate angles
(c) B1 co-interior (allied)
(c) Answer: Co-interior (allied) angles

Question 9

B1 x = 74 cao
Answer: x = 74 degrees

Question 10

B1 x = 39 cao
Answer: x = 39 degrees

Question 11

B1 x = 79 cao
Answer: x = 79 degrees

Question 12

B1 y = 58 cao
B1 correct reason: corresponding angles are equal, since l, m and n are all parallel
Answer: y = 58 degrees

Question 13

M1 angle ABC = 52 (alternate angles, DAE parallel to BC)
M1 angle ACB = 65 (alternate angles, DAE parallel to BC)
A1 angle BAC = 180 - 52 - 65 = 63 cao (angle sum of a triangle)
Answer: angle BAC = 63 degrees

Question 14

B1 correctly identifies x and y as alternate angles
B1 states that alternate angles on parallel lines are equal
Answer: x = y because they are alternate angles, and alternate angles between parallel lines are equal

Question 15

M1 y = 180 - 108, using co-interior (allied) angles sum to 180
A1 y = 72
M1 z = 180 - y (angles on a straight line) or z = 108 (corresponding angles are equal)
A1 z = 108
Answer: y = 72 degrees; z = 108 degrees

Question 16

M1 (4x + 10) + (2x + 20) = 180 oe, using co-interior angles sum to 180
M1 6x + 30 = 180 (oe simplification)
A1 x = 25 cao
A1 angles = 110 degrees and 70 degrees, ft from their x
Answer: x = 25; angles = 110 degrees and 70 degrees

Question 17

B1 identifies the two marked angles as being in corresponding positions
M1 states the correct test: two lines are parallel if and only if a pair of corresponding angles is equal
A1 121 is not equal to 118 (correct comparison of the two given values)
A1 correct conclusion that AB is not parallel to CD
B1 full explanation given, e.g. if AB were parallel to CD the corresponding angles at G and H would both be equal, but 121 and 118 differ by 3 degrees, so the lines are not parallel
Answer: Ama is incorrect; AB is not parallel to CD, since the corresponding angles (121 degrees and 118 degrees) are not equal

Question 18

M1 angle ABC = 47 (alternate angles, DAE parallel to BC)
M1 angle ACB = angle ABC = 47 (base angles of an isosceles triangle with AB = AC are equal)
M1 angle BAC = 180 - 47 - 47 (angles in a triangle sum to 180)
A1 angle BAC = 86 cao
Answer: angle BAC = 86 degrees

Question 19

M1 2x + 15 = 3x - 25 oe (alternate angles are equal)
A1 x = 40 cao
M1 substitutes x = 40 into 2x + 15
A1 angle EGB = 95 degrees, ft from their x
Answer: x = 40; angle EGB = 95 degrees

Question 20

M1 angle GHD = angle CHF = y + 50 (vertically opposite angles are equal)
M1 (3y + 10) + (y + 50) = 180 oe, using co-interior angles sum to 180
M1 correctly simplifies to 4y + 60 = 180
A1 y = 30 and angle GHD = 80 degrees
Answer: y = 30; angle GHD = 80 degrees

Question 21

M1 angle ABC = angle DAB and angle ACB = angle EAC (alternate angles, DAE parallel to BC)
M1 angle DAB + angle BAC + angle EAC = 180 (angles on a straight line DAE)
M1 (2x + y) + 70 + (3x - y) = 180 simplifies to 5x + 70 = 180 (the y-terms cancel)
A1 x = 22
M1 forms a second equation from the given angle difference: (2x + y) - (3x - y) = 18 oe
A1 y = 20
Answer: x = 22, y = 20 (angle ABC = 64 degrees, angle ACB = 46 degrees)

Question 22

(a) M1 (5x - 12) + (3x + 8) = 180 oe, using co-interior angles sum to 180 for parallel lines
(a) M1 8x - 4 = 180 (oe simplification)
(a) A1 x = 23 cao
(a) Answer: x = 23
(b) M1 substitutes x = 11 into both expressions: 5(11) - 12 = 43 and 3(11) + 8 = 41
(b) M1 43 + 41 = 84, compared with 180
(b) A1 correct conclusion: since 84 is not equal to 180, AB is not parallel to CD, so Priya is incorrect
(b) Answer: Priya is incorrect; when x = 11 the co-interior angles sum to 84 degrees, not 180 degrees, so AB is not parallel to CD

Angles in Parallel Lines: Fluency and Exam Drill - Worksheets, Questions and Revision

4.10D Angles in Parallel Lines: Fluency and Exam Drill

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