Geometry: Angles & Figures

7.F.1Angle pairs01/03

Vertical, adjacent, complementary, supplementary angles.

Step01of 03

Four key angle relationships.

Complementary	→sum to 90°
Supplementary	→sum to 180°
Vertical	→opposite angles when two lines cross — always congruent
Adjacent (linear pair)	→side by side, share a ray; if non-shared sides form a line, they're supplementary

7.F.1 — Vertical, adjacent, complementary, supplementary angles.

Step 01 of 03

Four key angle relationships.

Complementary	→sum to 90°
Supplementary	→sum to 180°
Vertical	→opposite angles when two lines cross — always congruent
Adjacent (linear pair)	→side by side, share a ray; if non-shared sides form a line, they're supplementary

Step 02 of 03

If $∠ A = 35°$ and is COMPLEMENTARY to $∠ B$ : $∠ B = 90 - 35 = 55°$ . SUPPLEMENTARY: $180 - 35 = 145°$ .

Step 03 of 03

Vertical angles trick: if two lines cross with one angle $70°$ , the OPPOSITE angle is also $70°$ ; the two adjacent angles are $110°$ each.

Key insight

C → Corner (90°). S → Straight (180°). Vertical = opposite, always equal. Linear pair = adjacent + supplementary.

7.F.2 — Triangle angle relationships.

Step 01 of 03

Interior angles of any triangle sum to $180°$ .

∠ A + ∠ B + ∠ C = 180°

Step 02 of 03

If two angles are $50°$ and $70°$ , the third is $180 - 50 - 70 = 60°$ .

Step 03 of 03

Exterior angle = sum of the two NON-adjacent interior angles.

Key insight

Triangle interior angles sum to 180°. Equilateral → all 60°. In a right triangle, the two non-right angles sum to 90° (complementary).

7.F.3 — Scale drawings.

Step 01 of 03

A scale drawing represents a real-world object with proportional dimensions. The scale factor = drawing length ÷ actual length.

Step 02 of 03

If a 1 cm drawing = 5 m actual, the scale is 1:500 (cm matches cm). A 4 cm length on the drawing = 20 m actual.

Step 03 of 03

Set up a proportion: $\frac{drawing 1}{actual 1} = \frac{drawing 2}{actual 2}$ . Cross-multiply.

Key insight

Scale = drawing/actual. All measurements scale by the same factor — set up a proportion to convert between drawing and actual.

7.F.4 — Cross-sections of 3D figures.

Step 01 of 03

A cross-section is the 2D shape revealed when you slice a 3D figure with a plane. The shape depends on the original solid AND the slice angle.

Step 02 of 03

Common results.

Cube — slice parallel to face	→square
Cube — diagonal slice	→rectangle, triangle, or hexagon
Cylinder — horizontal	→circle
Cylinder — vertical through axis	→rectangle
Cone — horizontal	→circle
Cone — vertical through apex	→triangle

Step 03 of 03

Slicing a sphere is always a circle (size depends on how close to the center).

Key insight

Cross-section depends on solid AND angle. Cube can become rectangle, triangle, or hexagon. Cylinder gives circle (horizontal) or rectangle (vertical). Sphere is always a circle.

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7.F.1 – 7.F.4 · CCSS 7.G.A / 7.G.B · Full curriculum →