Angles in composite figures

Read what is given before calculating

For a composite figure, identify each named triangle, rectangle, square, parallelogram, rhombus or trapezium. Mark only properties justified by those names or by given equality and parallel marks. A sketch that looks like a square does not establish four equal sides or right angles.

Consider a rectangle ABCD with a triangle ABP inside it. Equal sides AP and BP are marked, and angle APB is 40°. In triangle ABP, the base angles are (180°-40°)/2 = 70°. The rectangle angle DAB is 90°, so angle DAP is 90°-70° = 20°.

We used two known shapes and the existing ray AP. We did not assume an angle from the drawing or add a construction line.

State the dependency of each step

A parallelogram has consecutive vertices A, B, C, D and angle DAB = 65°. Angle ABC is 115° because adjacent angles in a parallelogram total 180°. If a drawn ray BP lies inside angle ABC with angle ABP = 42°, then angle PBC is 115°-42° = 73°.

If P lay outside the angle, that subtraction would not describe the same regions. Position and incidence matter as much as numbers.

Check your understanding 1 An equal-sided triangle shares a square’s side

Square WXYZ and equilateral triangle WYP share the square’s diagonal WY as a side. Point P lies on the same side of WY as X, inside angle XWY. Find angle PWX if the square’s diagonal makes angle XWY = 45°. Is the stated position possible?

Your answer or drawing on paper

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An equilateral triangle requires angle PWY = 60°. A ray inside the 45° angle XWY cannot make 60° with WY. The given position is impossible; subtracting 60° from 45° is a signal to check the conditions, not to report a negative geometric angle.

Solve a consistent figure

Check your understanding 2 The first result is not the final corner part

Rectangle ABCD contains isosceles triangle ABE, with AE = BE. The vertex angle AEB, labelled a, is 40°. AE and BE are already drawn. Find b, EAB; c, ABE; and d, DAE. State which shape you use at each step.

Connect the equal sides to a rectangle corner

The equal sides meet at E. At A, b and d together fill the rectangle's right angle.

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Given shapes, lines and angles

Given	Information
ABCD	rectangle.
ABE	isosceles triangle. The equal sides meet at E.
Already drawn lines	AE, BE. Use the boundary and these lines; construct no extra line.
Angle a	AEB = 40°.
Angle b	BAE is requested.
Angle c	ABE is requested.
Angle d	EAD 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.

Given shapes, lines and angles

Given	Information
ABCD	rectangle.
ABE	isosceles triangle. The equal sides meet at E.
Already drawn lines	AE, BE. Use the boundary and these lines; construct no extra line.
Angle a	AEB = 40°.
Angle b	BAE is requested.
Angle c	ABE is requested.
Angle d	EAD 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.

Your three angles and geometric reasons

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In triangle ABE, b and c are opposite equal sides. They share 180°-40° = 140° equally, so b = c = 70°. At A, the rectangle corner DAB is 90°, so d = 90°-70° = 20°. Check the triangle’s total and the whole corner separately. The drawing places AE inside DAB, which makes this subtraction valid.

Decide whether an equal split is justified

A drawn diagonal creates triangles, but the word diagonal alone does not make their sides or angles equal. Use the original shape’s side facts.

Check your understanding 3 Derive the split from equal sides

ABCD is a square and AC is already drawn. Find b, CAD, and c, BCA. Explain why these values follow without an angle measurement, and why a general rectangle would need more information.

Use a right corner and two equal sides

The square's equal sides make both triangle ABC and triangle ACD isosceles.

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Given shapes, lines and angles

Given	Information
ABCD	square.
Already drawn lines	AC. Use the boundary and these lines; construct no extra line.
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.

Given shapes, lines and angles

Given	Information
ABCD	square.
Already drawn lines	AC. Use the boundary and these lines; construct no extra line.
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.

Your two angles and side-property argument

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In triangle ABC, AB = BC and ABC = 90°, so its other two angles share 90° equally. Thus c = 45°. Triangle ACD likewise has AD = CD and ADC = 90°, giving b = 45°. A rectangle supplies the right corners, but it need not supply AB = BC or AD = CD. Its diagonal need not make 45° with a side.

Check your understanding 4 Keep the whole opposite corner

In parallelogram ABCD, DAB is 63°. AC is already drawn and a, BAC, is 25°. Find b, c and d. Which value is a triangle part, and which total belongs to the whole corner?

An equal opposite corner can have unequal parts

At A, a and b fill DAB. At C, c and d fill BCD.

View figure data

Given geometry and angles

Given	Information
Geometry	ABCD is a parallelogram. AC is already drawn.
Whole angle DAB	63°
Angle a (BAC)	25°

Given geometry and angles

Given	Information
Geometry	ABCD is a parallelogram. AC is already drawn.
Whole angle DAB	63°
Angle a (BAC)	25°

Your three angles and the whole-corner checks

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At A, b = 38°. The neighbouring corner ABC is 117°. Triangle ABC then gives c = 180°-117°-25° = 38°. The whole opposite corner BCD is 63°, so d = 25°. Check a + b = c + d = 63°. The corner totals are equal, but their two parts need not be equal to each other.

Check your understanding 5 Find what the extra side equality changes

The next shape is a rhombus, with DAB = 112° and AC already drawn. Find b and c. Identify the side equality that the general parallelogram did not guarantee.

Four equal sides constrain the triangle parts

AB = BC makes triangle ABC isosceles; AD = CD does the same for ACD.

View figure data

Given shapes, lines and angles

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.

Given shapes, lines and angles

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.

Your two angles and the decisive equality

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ABC is 180°-112° = 68°. Since AB = BC, triangle ABC’s other angles are both 56°, so c = 56°. The remaining part at A is also 112°-56° = 56°, giving b = 56°. Equal neighbouring side lengths justify the equal base angles. In a general parallelogram, only opposite side lengths are guaranteed equal.

Check your understanding 6 Follow the non-parallel side to the whole corner

ABCD is a trapezium, with AB parallel to CD. DAB is 75°, ABC is 65°, and a, BAC, is 46°. AC is already drawn. Find b, c and d. Explain why c is neither the whole angle BCD nor the supplement of DAB.

Connect a triangle total to the correct side pair

At A, a and b fill DAB. At C, c and d fill BCD; BC joins corners ABC and BCD.

View figure data

Given shapes, lines and angles

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.

Given shapes, lines and angles

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.

Your three angles, whole corner and reasons

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At A, b = 29°. Triangle ABC gives c = 180°-65°-46° = 69°. Along BC, the whole angle BCD is 180°-65° = 115°, so d = 46°. Thus c + d = 115°. Corner DAB lies on the other leg, AD; its 75° does not supply the supplement for BCD.

Check the chain, not just the final number

Name each triangle or whole corner before using its total. Record the condition that justifies an equality or a supplement. Check every requested angle, and ask whether the stated position is possible. Without an equal-side condition or another given angle, a remaining total can leave two parts individually unknown. Reading a balanced-looking sketch does not resolve that uncertainty.