Construction & Congruence

Construction means drawing figures with only an unmarked ruler and a compass, one of geometry's oldest rules. The ruler joins two points, the compass carries a set distance, and the two can split a segment exactly in half. Figures with the same shape and size that overlap perfectly are congruent, and SSS, SAS, or ASA is enough to prove it. Step through the construction and reshape the triangle to see congruence for yourself.

What is Construction?

Construction step0

👀 Construction Rules

①Use only an unmarked straightedge and compass

②Straightedge: draws lines/segments between two points

③Compass: draws circles (arcs)

④Use to construct triangles, angle bisectors, perpendicular bisectors

Congruent Triangles

Side a6

Side b5

Included angle C50

🔍 What is Congruence?

①Figures with exactly the same shape and size

②They overlap perfectly when superimposed

③Corresponding sides have equal lengths

④Corresponding angles have equal measures

Triangle Congruence Conditions

SSS

All three sides are equal

a = a', b = b', c = c'

SAS

Two sides and the included angle are equal

a = a', b = b', ∠C = ∠C'

ASA

One side and the two adjacent angles are equal

∠A = ∠A', a = a', ∠B = ∠B'

💡 Memorize Congruence Conditions

①SSS: side-side-side

②SAS: side-angle-side (included angle!)

③ASA: angle-side-angle

④Caution: SSA (side-side-angle) is NOT a valid condition!

Work It Out

Example 1

If △ABC ≡ △DEF and BC = 7, find the length of side EF.

In congruent figures, corresponding sides are equal. BC and EF correspond.

BC ↔ EF (corresponding sides)

Equal corresponding sides give the length.

EF = BC = 7

▸ EF = 7

Congruent ⇒ corresponding sides and angles are each equal.

Example 2

If △ABC ≡ △DEF with ∠A = 50° and ∠B = 60°, find ∠F.

The angles of a triangle sum to 180°, so find ∠C.

∠C = 180° − 50° − 60° = 70°

By congruence, ∠F equals its corresponding angle ∠C.

∠F = ∠C = 70°

▸ ∠F = 70°

Find corresponding parts from the vertex order (A↔D, B↔E, C↔F).

Exam Key Points

Congruence Notation

△ABC ≅ △DEF

List vertices in corresponding order (order matters!)

Grade-7 school exam type

Which condition always makes two triangles congruent?

①three angles are equal

②two sides are equal

③three sides are equal, each to each

④equal areas

⑤equal perimeters

▸ ③ three sides are equal, each to each

The congruence conditions are SSS, SAS, ASA.

SSS, SAS, ASA

Three equal sides give SSS congruence (equal angles alone is only similarity).

three equal sides ⇒ SSS congruence

🎯 Exam Key Points

①Three conditions: SSS, SAS, ASA

②Use corresponding-vertex order in notation

③If congruent → corresponding sides/angles are equal

④Describe construction steps precisely

⑤In SAS, verify it is the INCLUDED angle (AAa is insufficient)

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Positional Relations

Polygons & Circles

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