Cambridge Lower Secondary Stage 7 — Unit 5. When two parallel lines are cut by a transversal, special angle pairs are formed. Learn to spot them, name them, and use them!
Angles at Parallel Lines — Watch the Pattern!
The Three Angle Relationships
Z-Angles (Alternate Angles)
When a transversal crosses parallel lines, alternate angles are between the parallel lines, on opposite sides of the transversal. They form a Z-shape. They are equal.
Alternate angles are EQUAL
F-Angles (Corresponding Angles)
Corresponding angles are on the same side of the transversal and the same side of each parallel line. They form an F-shape. They are equal.
Corresponding angles are EQUAL
C-Angles (Co-interior Angles)
Co-interior angles (also called same-side interior or consecutive angles) are between the parallel lines, on the same side of the transversal. They form a C or U shape. They add up to 180°.
Co-interior angles add up to 180°
Geometric Reasoning
In maths, a reason must be stated precisely. Use these exact phrases:
"Alternate angles are equal" — for Z-angles
"Corresponding angles are equal" — for F-angles
"Co-interior angles add up to 180°" — for C-angles
"Angles on a straight line add up to 180°"
"Angles at a point add up to 360°"
"Vertically opposite angles are equal"
Constructions with Compass
Bisecting a Line Segment
To find the exact midpoint of line AB using a compass:
Open your compass to more than half the length of AB.
Place the compass on point A and draw two arcs (one above, one below the line).
Without changing the compass width, place on point B and draw two more arcs crossing the first pair.
Draw a straight line through the two crossing points. This is the perpendicular bisector.
Bisecting an Angle
To cut angle ABC exactly in half:
Place the compass on vertex B and draw an arc that crosses both arms of the angle at points D and E.
Place the compass on D and draw an arc inside the angle.
Without changing compass width, place on E and draw another arc crossing the first.
Draw a line from B through the crossing point. This is the angle bisector.
Constructing Triangles
SSS (Side-Side-Side)
Given three side lengths: draw the base, then use arcs from each end to locate the third vertex.
SAS (Side-Angle-Side)
Given two sides and the angle between them: draw one side, use a protractor to mark the angle, then measure the second side.
ASA (Angle-Side-Angle)
Given a side and the angles at each end: draw the side, then use a protractor at each end to draw the two angles until they meet.
Loci
Equidistant from a Point
All points the same distance from a fixed point form a circle.
Equidistant from a Line
All points the same distance from a fixed line form two parallel lines (one on each side).
Equidistant from Two Lines
All points equidistant from two intersecting lines lie on the angle bisectors of the angles between the lines.
Worked Examples
Study each example carefully before practising!
Example 1 — Alternate Angles (Z-Angles)
Question: Find the value of x. Give a reason.
Solution:
x and 65° are alternate angles (they are between the parallel lines on opposite sides of the transversal — Z-shape).
Alternate angles are equal.
x = 65°
Example 2 — Corresponding Angles (F-Angles)
Question: Find y.
Solution:
Both angles are in the upper-right position at each intersection — they are corresponding angles (F-shape, same side of transversal, same relative position).
Corresponding angles are equal.
y = 118°
Example 3 — Co-interior Angles (C-Angles)
Question: Find z.
Solution:
The two angles are between the parallel lines, on the same side of the transversal — they are co-interior angles (C-shape).
Co-interior angles add up to 180°.
z + 72° = 180°
z = 108°
Example 4 — Multi-step Problem
Question: Find angles a and b.
Solution — Step 1: Find a
The 130° angle and a are co-interior angles (same side of transversal 1, between the parallel lines).
130° + a = 180° ⇒ a = 50°
Step 2: Find b
The 130° angle and b are corresponding angles at transversals 1 and 2 (parallel transversals... here we use the fact that the two transversals are not parallel, so we use angles on a straight line at the top parallel line).
Angle on a straight line at top with T1: 180° − 130° = 50° (angles on straight line). Then b and 50° are corresponding angles.
b = 50°
Interactive Angle Finder
Click on an angle label, type its value, then press Calculate to fill in all 8 angles automatically!
Relationships reminder:
p & r are vertically opposite (equal) | q & s are vertically opposite (equal)
p & q are supplementary (180°) | p & u are co-interior (180°) | p & t are corresponding (equal) | p & v are corresponding (equal) | r & t are alternate (equal) | s & u are alternate (equal)
Ex 1 — Angle Type Identifier
Look at each diagram. Click the correct angle relationship!
Score: 0 / 10
Ex 2 — Calculate the Angle
Find the missing angle. Type your answer (whole number, degrees) and press Check.
Score: 0 / 10
Ex 3 — Missing Angles Chain
Multi-step problems. Find all the missing angles in order.
Score: 0 / 6
Ex 4 — Construction Steps Sorter
Click the steps in the CORRECT ORDER (1st, 2nd, 3rd...) for each construction.
Part A — Bisecting a Line Segment
Click the steps in the correct order:
Part B — Bisecting an Angle
Click the steps in the correct order:
Ex 5 — True or False?
Click TRUE or FALSE for each statement. Immediate feedback given!
Score: 0 / 12
Ex 6 — Angle Reasoning
For each question, choose the correct reason AND enter the angle value.
Score: 0 / 12
Practice Questions
Work these out on paper. All involve parallel lines unless stated otherwise.
Two parallel lines are cut by a transversal. One angle formed is 55°. Find the alternate angle and state the reason.
A transversal crosses two parallel lines forming a corresponding angle of 112°. Find the value of the corresponding angle on the second line.
Two co-interior angles are formed where a transversal meets parallel lines. One angle is 74°. Find the other.
Find the value of x. A transversal crosses two parallel lines; the upper angle is (2x + 10)° and the alternate angle at the lower line is 70°.
Three angles are at a point on a straight line: 40°, x, and 85°. Find x.
Angles at a point are 90°, 130°, x, and 45°. Find x.
Two parallel lines are cut by a transversal. A co-interior angle pair is (3x − 5)° and (2x + 15)°. Find x and both angles.
In the diagram, AB is parallel to CD. Angle ABE = 63° and angle CDE = 47°. Find angle BED. (Hint: draw a line through E parallel to both.)
Explain why alternate angles are equal using the idea of a Z-shape and parallel lines.
A triangle has angles 50°, 70°, and x. Find x. (Angles in a triangle = 180°.)
The exterior angle of a triangle is 115°. One of the non-adjacent interior angles is 48°. Find the other non-adjacent interior angle.
Describe in full the steps to bisect a line segment of length 8 cm using only a compass and ruler.
A triangle has sides 5 cm, 7 cm, and 9 cm. Name the method used to construct it and describe the key steps.
A triangle has two sides of 6 cm and 8 cm with an included angle of 50°. What construction method is used? Describe the steps.
Describe the locus of all points exactly 4 cm from a fixed point P.
Describe the locus of all points equidistant from two parallel lines 6 cm apart.
A co-interior angle pair sums to 180°. One angle is (4x + 10)°. The other is (2x + 50)°. Find x.
In a diagram, angle p = 128°. Write the values of the three angles vertically opposite and on a straight line from p at the same intersection.
Two corresponding angles are (5x − 20)° and (3x + 40)°. Find x and the angle size.
Angle ABF = 38° where AB ∥ CD and EF is a transversal. State the value of angle CDG (corresponding to ABF) and angle EFD (co-interior with ABF).
Alternate angle = 55°. Reason: Alternate angles are equal.
112°. Corresponding angles are equal.
180° − 74° = 106°. Co-interior angles add up to 180°.
Alternate angles are equal: 2x + 10 = 70 → 2x = 60 → x = 30°.
Draw line through E parallel to AB and CD. Upper alt. angle at E = 63°; lower alt. angle at E = 47°. Angle BED = 63° + 47° = 110°.
Because the parallel lines never meet, the transversal cuts them at the same angle. The Z-shape shows the angles on opposite sides are a perfect mirror — so they must be equal.
x = 180° − 50° − 70° = 60°.
Exterior angle = sum of two non-adjacent interior angles. Other angle = 115° − 48° = 67°.
Open compass to more than half AB. Arc from A above and below. Same width, arc from B crossing first pair. Line through intersections = perpendicular bisector (midpoint).
SSS construction. Draw 9 cm base. Arc of radius 5 cm from one end; arc of radius 7 cm from other end. Their intersection is the third vertex. Join up.
SAS. Draw 6 cm line. Use protractor at one end to mark 50°. Measure 8 cm along that ray. Join end points.
A circle of radius 4 cm centred on P.
A line exactly halfway between the two parallel lines (3 cm from each), running parallel to both.
Angle CDG = 38° (corresponding). Angle EFD = 180° − 38° = 142° (co-interior, or angles on a straight line at F).
The Hard Challenge
Multi-step and algebraic problems. Show all working on paper!
In a diagram, AB ∥ CD. A transversal meets AB at P and CD at Q. Angle APQ = (3x + 15)° and angle PQD = (5x − 25)°. These are co-interior angles. Find x, and find each angle. Then find angle QPD (on a straight line at P).
Two parallel lines are cut by two transversals that meet at a point X between the lines. The angle at the upper line (left transversal) is 55°. The angle at the lower line (right transversal) is 40°. Find the angle at X between the two transversals inside the parallel lines. (Use alternate angles and angles in a triangle.)
In triangle ABC, AB ∥ DE where D is on AC and E is on BC. Angle ABC = 72° and angle BAC = 48°. Find angle ADE giving full geometric reasons.
A transversal crosses three parallel lines at points A, B, and C respectively. The angle at A (upper-left) is (4x − 10)°. The angle at B (lower-right, between lines 1 and 2) is (2x + 30)°. Lines 1 and 2 are parallel. Find x. Then find angle at C (corresponding to angle at B, between lines 2 and 3).
ABCD is a trapezium where AB ∥ CD. Angle DAB = 112° and angle ABC = 68°. Find angles ADC and BCD. Explain why angles DAB + ADC = 180° using a geometric reason.
Two lines PQ and RS intersect at T. Angle PTR = (6x − 12)° and angle QTS = (4x + 28)°. Find x. What geometric fact do you use? Find all four angles at T.
In a diagram, a transversal crosses parallel lines creating angle a = (2x + 3y)° at the top and angle b = (3x − y)° at the bottom. Angle a and b are corresponding angles. A separate equation is a + b = 250°. Find x and y, and find angle a.
Point P is equidistant from two intersecting lines. Describe the complete locus of P and explain why it has two parts. What construction would you use to find points on this locus?
A triangle has angles (x + 10)°, (2x − 5)°, and (x + 15)°. Find x and all three angles. Is the triangle acute, right, or obtuse? Then find the exterior angle at the largest interior angle.
In the diagram, PQ ∥ RS. A transversal meets PQ at A and RS at B. Point C lies between the lines with angle CAB = 32° and angle CBR = 47°. Find angle ACB. (Draw a line through C parallel to PQ and RS, then use alternate and co-interior angle relationships.)
Let left transversal make angle 55° with upper parallel line (upper-left). By alternate angles, interior angle at upper line (lower-right) = 180°−55° = 125° (on straight line); actually alternate with X: the angle inside the triangle at the upper parallel line = 180°−55° = 125°... Better: angle at upper line below = 180°−55°=125°, or the alternate angle below line = 55°. By alternate angles at lower line, interior angle = 180°−40°=140°... Using triangle: angle at upper line (alt.) = 55°; angle at lower line (alt.) = 40°. Triangle angle sum: angle at X = 180°−55°−40° = 85°.
Since AB∥DE, angle ADE = angle DAB = 48° (corresponding angles, AB∥DE). But wait — angle ADE is at D on line AC. Corresponding to angle BAC: angle ADE = 48°. Reason: Corresponding angles are equal (AB∥DE).
Angles at A and B are co-interior (between parallel lines 1 and 2, same side): (4x−10)+(2x+30)=180 → 6x+20=180 → 6x=160 → x=26.67°. Angle at B = 2(26.67)+30 = 83.33°. Angle at C = 83.33° (corresponding, lines 2 and 3 are also parallel).
Corresponding: 2x+3y = 3x−y → 4y = x → x = 4y. Sum: a+b = 250° and a=b (corresponding), so 2a=250 → a=125°. So 2x+3y=125 and x=4y → 8y+3y=125 → 11y=125 → y≈11.36°, x≈45.45°.
The locus has TWO parts — both angle bisectors of the two pairs of vertically opposite angles formed by the intersecting lines. This is because a point equidistant from both lines could be in any of the four angle regions; the bisector of each angle pair gives equal distances to both lines. Construction: bisect each of the four angles at the intersection using a compass.
Sum=180: (x+10)+(2x−5)+(x+15)=180 → 4x+20=180 → 4x=160 → x=40. Angles: 50°, 75°, 55°. All less than 90° → acute triangle. Exterior angle at largest (75°) = 180°−75° = 105°.
Draw line XY through C, XY∥PQ∥RS. Angle XCA = 32° (alternate angles, XY∥PQ). Angle YCB = 47° (alternate angles, XY∥RS). Angle ACB = 180°−32°−47° = 101° (angles on a straight line at C).