Parallel quadrilaterals and their angles
Use marked parallel sides and named quadrilateral properties to justify unknown angles.
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Start with the stated side properties
| Shape | Side properties | Useful angle relationships |
|---|---|---|
| Parallelogram | Both pairs of opposite sides are parallel and equal in length | Opposite angles are equal; neighbouring angles total 180° |
| Rhombus | All four sides are equal; both opposite pairs are parallel | The parallelogram angle relationships still apply |
| Trapezium here | One pair of opposite sides is parallel | Angles along each non-parallel side total 180° |
A rectangle is a parallelogram with four right angles. A square has four right angles and four equal sides, so it is also a rhombus. A general rhombus need not have right angles. Equal side lengths, equal angles and parallel directions are different properties: do not exchange one for another without a reason.
One parallelogram angle determines the other three
Distinguish opposite and neighbouring corners
ABCD is a parallelogram. Angles a and c are opposite; b and d are the other opposite pair.
View figure data
| Given | Information |
|---|---|
| Geometry | ABCD is a parallelogram. AB is parallel to CD; BC is parallel to DA. |
| Angle a | DAB |
| Angle b | ABC |
| Angle c | BCD |
| Angle d | CDA |
| Angle a | 63° |
The angle opposite a is c = 63°. The angles next to a are b = d = 180°-63° = 117°. Opposite angles are equal; it is the neighbouring pair that totals 180°. Check both relationships rather than assuming every corner is the same size.
Check your understanding 1 List what the class guarantees
For parallelogram ABCD above, name both equal-side pairs, both parallel-side pairs and both equal-corner pairs. Then write all four neighbouring-corner sums. Use = for equality, || or ∥ for parallel sides, and capital letters for corners. Give each pair once.
Show answer
Equal sides: AB = CD; BC = DA. Parallel sides: AB || CD; BC || DA. Equal corners: A = C; B = D. Neighbouring sums: A + B = 180; B + C = 180; C + D = 180; D + A = 180, in degrees. Reversing the order of a pair states the same relation. AB = BC is an extra equality that a general parallelogram does not guarantee.
Check your understanding 2 Equal sides do not make every angle equal
ABCD is a rhombus and a is 112°. Find b, c and d. State why four equal sides do not make this a square.
Use the rhombus's parallel sides
All four sides are equal, and opposite side pairs are parallel. The marked corner a is obtuse.
View figure data
| Given | Information |
|---|---|
| Geometry | ABCD is a rhombus. AB is parallel to CD; BC is parallel to DA. All four sides are equal. |
| Angle a | DAB |
| Angle b | ABC |
| Angle c | BCD |
| Angle d | CDA |
| Angle a | 112° |
Show answer
Opposite angles are equal, so c = 112°. The neighbouring angles are b = d = 180°-112° = 68°. A square needs four right angles; these are 112° and 68°. Four equal side lengths alone do not require equal angles.
Check your understanding 3 Decide which property changes
PQRS is a rhombus. Give a chain naming all its equal sides. Then list its parallel-side pairs, equal-corner pairs and neighbouring-corner sums. Which part differs from the general parallelogram’s guarantee?
Show answer
PQ = QR = RS = SP. The parallel pairs are PQ || RS and QR || SP. Opposite corners satisfy P = R and Q = S. The sums are P + Q = 180; Q + R = 180; R + S = 180; S + P = 180. All four sides, rather than only opposite pairs, are guaranteed equal. The angle relationships remain those of a parallelogram; P = Q = R = S is not guaranteed without right-angle information.
A trapezium has two separate angle pairs
Use the given parallel pair. With AB parallel to CD, the angles at A and D lie along one non-parallel side, while those at B and C lie along the other. Thus a + d = 180° and b + c = 180°. The angles along AB need not total 180°, and opposite angles need not be equal.
Check your understanding 4 Follow the tags when the figure turns
The two sides tagged p are parallel. Name that pair using || or ∥, then write the angle sums along both non-parallel sides. Do not choose pairs just because their corners are across the page from each other.
The parallel pair is vertical
Matching p tags identify the parallel pair in trapezium ABCD. No equal lengths or equal angles are given.
View figure data
| Given | Information |
|---|---|
| Shape | ABCD is a trapezium. The two sides tagged p are parallel. |
| Use of the drawing | Positions identify sides and corners. Use only the stated class and marks; do not infer extra equal lengths or right angles from appearance. |
Show answer
AB || CD. Side AD connects A and D, so A + D = 180. Side BC connects B and C, so B + C = 180. The totals are in degrees. Neither opposite-corner pair is guaranteed equal or supplementary by the trapezium’s stated properties.
Check your understanding 5 Match each given angle to its own side
AB is parallel to CD. Angle a is 75° and angle b is 65°. Find c and d, and identify the pair you use for each answer.
Follow the non-parallel sides
AD joins corners a and d; BC joins corners b and c. AB and CD are the parallel pair.
View figure data
| Given | Information |
|---|---|
| Geometry | ABCD is a trapezium with AB parallel to CD. |
| Angle a | DAB |
| Angle b | ABC |
| Angle c | BCD |
| Angle d | CDA |
| Angle a | 75° |
| Angle b | 65° |
Show answer
Along BC, b + c = 180°, so c = 180°-65° = 115°. Along AD, a + d = 180°, so d = 180°-75° = 105°. Using a to find c would pair the wrong corners.
Check your understanding 6 Do not infer a second angle from the sketch
AB is parallel to CD, and only a = 105° is supplied. Which other marked angle can you determine? Are b and c individually known?
One leg has no angle value
AB and CD are parallel. No equality of legs or base angles is stated.
View figure data
| Given | Information |
|---|---|
| Geometry | ABCD is a trapezium. AB is parallel to CD. |
| Angle a | 105° |
Show answer
Angle d = 180°-105° = 75°. The other pair satisfies b + c = 180°, but neither b nor c is individually determined. Equal legs or another angle value would be additional information; it cannot be read into the sketch.
Combine a whole corner with a triangle
Check your understanding 7 Use the diagonal already shown
ABCD is a parallelogram. The whole angle DAB is 63°, and AC is already drawn with a = 25°. Find b, c and d. Keep the whole angle at A separate from its two parts.
The given diagonal creates two triangles
At A, a and b fill DAB. At C, c and d fill BCD. In triangle ABC, the marked angles are a and c.
View figure data
| Given | Information |
|---|---|
| Geometry | ABCD is a parallelogram. AC is already drawn. |
| Whole angle DAB | 63° |
| Angle a (BAC) | 25° |
Show answer
At A, b = 63°-25° = 38°. The neighbouring whole angle ABC is 180°-63° = 117°. Triangle ABC then gives c = 180°-117°-25° = 38°. The whole opposite angle BCD is 63°, so d = 63°-38° = 25°. Check a + b = 63° and c + d = 63°. A general parallelogram’s diagonal does not have to halve its corner angles.
Check your understanding 8 When equal sides justify equal parts
ABCD is a rhombus and angle DAB is 112°. AC is already drawn. Find b, CAD, and c, BCA. Use the triangles’ sides to justify the split; do not carry the same rule to every parallelogram.
A rhombus diagonal creates equal-sided triangles
Triangle ABC has AB = BC; triangle ACD has AD = CD.
View figure data
| Given | Information |
|---|---|
| ABCD | rhombus. |
| Already drawn lines | AC. Use the boundary and these lines; construct no extra line. |
| Angle DAB | 112° |
| Angle b | CAD is requested. |
| Angle c | BCA is requested. |
| Use of the drawing | The stated classes, alignments and angle values are exact. Use them rather than measuring the picture; appearance supplies no extra equalities or right angles. |
Show answer
The neighbouring whole angle ABC is 180°-112° = 68°. In triangle ABC, AB = BC, so BAC and BCA share 180°-68° = 112° equally: each is 56°. Thus c = 56°, and b is the remaining 112°-56° = 56° at A. The equal sides supply a reason for these equal parts; a general parallelogram does not supply them.
Check your understanding 9 Keep a whole corner separate from a triangle part
ABCD is a trapezium with AB parallel to CD. The whole corners DAB and ABC are 75° and 65°. AC is already drawn, and a, BAC, is 46°. Find b, CAD; c, BCA; and d, ACD. Explain which total determines each result.
Use the given diagonal and the correct leg
At A, a and b fill DAB. At C, c and d fill BCD.
View figure data
| Given | Information |
|---|---|
| ABCD | trapezium. AB and CD are its parallel sides. |
| Already drawn lines | AC. Use the boundary and these lines; construct no extra line. |
| Angle DAB | 75° |
| Angle ABC | 65° |
| Angle a | BAC = 46°. |
| Angle b | CAD is requested. |
| Angle c | BCA is requested. |
| Angle d | ACD is requested. |
| Use of the drawing | The stated classes, alignments and angle values are exact. Use them rather than measuring the picture; appearance supplies no extra equalities or right angles. |
Show answer
At A, b = 75°-46° = 29°. Triangle ABC gives c = 180°-65°-46° = 69°. Along BC, the whole BCD is 180°-65° = 115°, so d = 115°-69° = 46°. The triangle angle c is only part of that 115° corner. Using 75° to find the whole BCD would join the wrong leg.
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