Choose perpendicular sections and calculate spatial angles

The common edge determines the section

Two distinct intersecting planes meet along a line ℓ. At a point O on ℓ, choose a line in each plane perpendicular to ℓ. The angle between those lines measures a dihedral angle. For the smaller angle between the planes, take an angle greater than zero and at most 90 degrees. Arbitrary edges of the planes need not measure this angle

Which pair correctly defines a plane–plane angle?

A perpendicular cross-section has horizontal run 4 m and vertical rise 3 m. Find tan θ for the plane angle

The section need not be right-angled

If the perpendiculars OA and OB and the join AB form an oblique triangle, apply the sine or cosine rule within that triangle. Let the lengths of OA, OB and AB be u, v and w respectively. Apply the cosine rule to the angle at O. The rays can enclose an obtuse angle even though the smaller angle between the planes is acute

A perpendicular section has OA=3, OB=5, AB=7. Find cos∠AOB

A perpendicular section has OA=3, OB=5, AB=7. Find ∠AOB in degrees

Two planes have a perpendicular section with OA=3, OB=5 and AB=7. Find the smaller angle between the planes, in degrees

Apply the supplement only when requested

A specified interior dihedral angle can be obtuse. Do not replace it by its supplement when that interior angle is the requested quantity. Here the smaller plane angle is explicitly requested, so its cosine is |cos∠AOB| and its value is at most 90°

An oblique base followed by a vertical section

In horizontal triangle ABC, use the two given sides and their included angle to find AC by the cosine rule. Let D be 24 units directly above C. The vertical section ACD then gives AD by Pythagoras. The line–plane angle of AD uses AC as its projection

Horizontal triangle ABC

\(\displaystyle {AB=5},\allowbreak\quad {BC=8},\allowbreak\quad{\angle ABC=60^\circ}\)

D directly above C

\(\displaystyle {CD=24},\allowbreak\quad{CD\perp\text{plane }ABC}\)

For that spatial example, find AD

Horizontal triangle ABC

\(\displaystyle {AB=5},\allowbreak\quad {BC=8},\allowbreak\quad{\angle ABC=60^\circ}\)

D directly above C

\(\displaystyle {CD=24},\allowbreak\quad{CD\perp\text{plane }ABC}\)

Find the exact sine of the angle AD makes with the horizontal plane

Horizontal triangle ABC

\(\displaystyle {AB=5},\allowbreak\quad {BC=8},\allowbreak\quad{\angle ABC=60^\circ}\)

D directly above C

\(\displaystyle {CD=24},\allowbreak\quad{CD\perp\text{plane }ABC}\)

Line–plane angle

\(\displaystyle \angle(AD,\text{plane }ABC)\)

Find the exact cosine of that line–plane angle

Justify a plane-angle construction, ungraded

Draw two planes meeting along ℓ. Construct a cross-section perpendicular to ℓ, identify the two traces and explain when a supplement is required. In the perpendicular section, let OA=3 and OB=5 be the two segments from O, with AB=7 joining their endpoints. Use these lengths to justify both the 120° section angle and the 60° smaller plane angle. This construction and proof are ungraded

Model: distinguish rays from planes

The section intersects each plane in a line through O perpendicular to ℓ. With OA, OB and AB measuring 3, 5 and 7 respectively, the cosine rule gives the angle between the selected rays as 120°. Each trace is a whole line and also has the opposite ray; the other included angle is 60°. The smaller plane angle is therefore 60°. A three-dimensional sketch alone cannot certify the needed perpendicularity

Model a solid with two sections, ungraded

Create and label a solid with horizontal triangle ABC, where AB is 5, BC is 8 and angle ABC is 60°. Place D directly above C, with CD perpendicular to the base plane and 24 units long. Derive the length AD and its angle with the base plane, naming the law and triangle at each step. Explain why the two plane-angle perpendiculars are a different construction. Independent modelling is ungraded

Model: keep the intermediate geometry

In horizontal triangle ABC, AB is 5, BC is 8 and angle ABC is 60°. D is directly above C, with CD perpendicular to the base plane and 24 units long. Use by the cosine rule. The perpendicular height CD then gives AD by Pythagoras. Use the resulting right triangle to obtain the three trigonometric ratios. This uses a line and its projection. A plane–plane angle instead uses a common-edge perpendicular in each plane; that extra plane is not supplied merely by having a diagonal