Angles between rectangles and squares

Find the whole corners first

Every corner of a rectangle or square is 90°. When two shapes meet, follow each corner’s two sides from the shared point. Decide whether those right corners occupy separate regions or share an overlapping part before adding their angles.

The sides of two different squares need not have the same length. Rectangles need not have equal neighbouring sides. Their right corners, rather than apparent lengths or positions, supply the angle facts here.

Separate corners and the remaining gaps

Squares ABCD and AEFG meet at A. The corner BAD uses 90°, and EAG uses another 90°. Those regions do not overlap in this figure. A complete turn around A is 360°, so the two remaining gaps together occupy 360°-90°-90° = 180°.

Follow the two right corners around A

BAD and EAG are the squares' right corners. Angle a is DAE, and b is GAB.

View figure data

Given shapes, lines and angles

Given	Information
ABCD	square.
AEFG	square.
Already drawn lines	AE, EF, FG, GA. Use the boundary and these lines; construct no extra line.
Angle a	DAE = 30°.
Angle b	GAB 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.
AEFG	square.
Already drawn lines	AE, EF, FG, GA. Use the boundary and these lines; construct no extra line.
Angle a	DAE = 30°.
Angle b	GAB 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.

Check your understanding 1 Use all the regions exactly once

Find b. State the whole-turn equation and explain why b is not 90°-30°.

Your angle, equation and corner distinction

Show answer

The four separate regions total 360°: 90° + 30° + 90° + b = 360°. Thus b = 150°. Subtracting 30° from 90° would assume that a and b share one right corner. They are different gaps between the two square corners.

Overlapping corners need a different partition

The next two shapes are rectangles. At A, BAD is one right corner, and EAG is the other. This time they overlap: angle b, EAD, belongs to both right corners. Adding 90° and 90° would count b twice.

Identify the shared part before adding

Angle a is BAE; b is EAD; c is DAG. The whole smaller angle BAG is requested by its three point names.

View figure data

Given shapes, lines and angles

Given	Information
ABCD	rectangle.
AEFG	rectangle.
Already drawn lines	AE, EF, FG, GA. Use the boundary and these lines; construct no extra line.
Angle a	BAE = 45°.
Angle b	EAD is requested.
Angle c	DAG is requested.
Angle BAG	BAG is requested; its middle letter names the vertex.
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.
AEFG	rectangle.
Already drawn lines	AE, EF, FG, GA. Use the boundary and these lines; construct no extra line.
Angle a	BAE = 45°.
Angle b	EAD is requested.
Angle c	DAG is requested.
Angle BAG	BAG is requested; its middle letter names the vertex.
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.

Check your understanding 2 Connect the shared part to both rectangles

Find b, c and the whole smaller angle BAG. Use the two whole right corners, then combine the three separate regions inside BAG. The middle letter A names its vertex. Explain why this whole angle is not 180°.

Your three angles and shared-part explanation

Show answer

At A in rectangle ABCD, a + b = 90°, so b = 45°. In rectangle AEFG, b + c = 90°, so c = 45°. The whole BAG contains a, b and c once each, giving ∠ BAG = 45° + 45° + 45° = 135°. Equivalently, subtract the shared 45° from 90° + 90°. The two right corners overlap; they do not form two separate adjacent 90° regions.

Check your understanding 3 Separate a known total from two known values

Return to the two squares with separate corners. Suppose neither gap angle is supplied. Can you conclude that both gaps are 90°? State what is known and what extra information would let you find them individually.

Your decision, known total and missing information

Show answer

No. The two right corners leave 180° in total, but the individual gaps need not be equal. One gap value, or a justified relationship between them, is needed. The shapes being squares does not make the two gaps equal. A balanced-looking sketch supplies no extra equality.