Geometry & Proof

Q: Which statement about alternate angles on parallel lines is correct?

They are equal — Alternate angles on parallel lines are equal — they are the pair that forms a Z shape.

Q: What do the three angles inside a triangle add up to?

180° — The angles inside a triangle always add up to 180°.

Q: What does the congruence condition "SAS" mean?

Two sides and the angle between them are equal — SAS stands for Side-Angle-Side: two sides plus the angle between them.

Q: In a right-angled isosceles triangle, how big are the two acute angles?

45° and 45° — Take the right angle of 90° away from the total of 180°, which leaves 90°. Split that evenly and you get 45° twice.

Q: Triangles ABC and DEF are similar, with BC = 6 cm, EF = 9 cm, and AB = 4 cm. How long is DE?

6 cm — The scale factor is EF/BC = 9/6 = 3/2, so DE = AB × 3/2 = 4 × 1.5 = 6 cm.

Q: Exterior angle theorem: an exterior angle of a triangle is equal to what, in terms of the two interior angles that are not next to it?

Their sum — An exterior angle of a triangle equals the sum of the two opposite interior angles.

Q: In triangle ABC, AB = AC (so it is isosceles). How are angle B and angle C related?

∠B = ∠C — The base angles of an isosceles triangle are equal.

Where this is used

The triangular trusses in a bridge, the graphics on your phone, the way GPS pins down your position — all of them rest on being able to say "this has to be true" about something you cannot measure directly. Congruence, similarity, and the behaviour of parallel lines are the reasoning that real designs are built on.

Key points

Parallel lines and angles

When two lines are parallel: corresponding angles are equal, alternate angles are equal, and the two interior angles on the same side add up to 180°.

When two triangles are congruent

(1) all three sides are equal (SSS); (2) two sides and the angle between them are equal (SAS); (3) two angles and the side between them are equal (ASA).

When two triangles are similar

(1) all three pairs of sides are in the same ratio; (2) two pairs of sides are in the same ratio and the angle between them is equal; (3) two pairs of angles are equal.

How to write a proof

(1) List what you are given; (2) name the definition or theorem that justifies each step; (3) reach the conclusion. The flow is "because such-and-such is true (∵), therefore this follows (∴)".

Why triangles are so strong

Fix the three side lengths of a triangle and only one shape is possible (that is SSS congruence). A four-sided frame, by contrast, can be pushed out of shape. That is why bridges and cranes are built from triangles.

Check yourself 10questions

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Q1 Which statement about alternate angles on parallel lines is correct?

Q2 What do the three angles inside a triangle add up to?

Q3 What does the congruence condition "SAS" mean?

Q4 In a right-angled isosceles triangle, how big are the two acute angles?

Q5 Triangles ABC and DEF are similar, with BC = 6 cm, EF = 9 cm, and AB = 4 cm. How long is DE?

Q6 Exterior angle theorem: an exterior angle of a triangle is equal to what, in terms of the two interior angles that are not next to it?

Q7 What is one interior angle of an equilateral triangle?

Q8 In triangle ABC, AB = AC (so it is isosceles). How are angle B and angle C related?

Q9 Two similar figures have areas in the ratio 4:9. What is the ratio of their sides?

Q10 When proving two triangles are congruent, what reason is used most often?

Jobs that use this

Architectural designer $85k

3D graphics engineer $95k

Land surveyor $60k

Architectural designer$85k

Uses the rigidity of triangles and the properties of similar shapes when designing a building's structure. The same proof-style reasoning goes into earthquake-resistance calculations.

Maths used: Geometry and proofThe Pythagorean theoremVectorsDifferential equations

3D graphics engineer$95k

Represents 3D models as polygons (triangles), then renders them and runs physics simulations on them.

Maths used: Geometry and proofVectorsMatricesDifferential geometry

Land surveyor$60k

Measures the area and position of land precisely using triangulation. Similar shapes and trigonometric ratios are the main tools of the job.

Maths used: Geometry and proofTrigonometric ratiosCoordinatesError analysis

Where this is used

Key points

Check yourself 10questions

Jobs that use this

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