Construction means drawing figures with only an unmarked straightedge and a pair of compasses. The straightedge joins two points; the compasses draw circles and carry lengths from one place to another.
With just these two tools you can cut a segment exactly in half and draw the line through its midpoint at right angles to it: the perpendicular bisector.
Two figures with the same shape and size, which fit exactly when placed on top of each other, are congruent. For triangles, it is enough for one of these to match: three sides (SSS), two sides and the included angle (SAS), or one side and the two angles at its ends (ASA).
On this page you follow the perpendicular bisector construction in four steps, and change two sides and the angle between them to see the third side fixed automatically.
Construction with straightedge and compasses
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Four steps to construct the perpendicular bisector of segment AB with a straightedge and compasses
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👀 The rules of construction
①Constructions use only an unmarked straightedge and compasses
②The straightedge draws or extends a segment through two points; the compasses draw circles and measure and carry lengths
③For the perpendicular bisector, both arcs need the same radius, and it must be more than half of AB or the arcs will not meet
④The crossing points P and Q are each one radius away from both A and B (PA = PB). That is why line PQ passes through the midpoint M of AB and meets AB at right angles
Two triangles with the same two sides and included angle
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Triangles ABC and A'B'C' with the same sides a, b and included angle C
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🔍 What congruent means
①Two figures with the same shape and size, which fit exactly when one is placed on the other, are congruent
②The vertices, sides and angles that land on each other are corresponding vertices, sides and angles. In the picture, sides with the same tick marks correspond, and so do the marked angles
③The right-hand triangle is the left one flipped over. It still fits exactly when placed on top, so the two are congruent
④Once the sliders fix two sides and the included angle, the length of the third side and the other two angles leave no room for choice. That is what SAS congruence means
Congruence conditions for triangles
SSS
All three pairs of sides are equal
a = a', b = b', c = c'
SAS
Two pairs of sides and the angles between them are equal
a = a', b = b', ∠C = ∠C'
ASA
One pair of sides and the angles at both ends are equal
c = c', ∠A = ∠A', ∠B = ∠B'
💡 Choosing a condition
①SSS is side-side-side, SAS is side-angle-side and ASA is angle-side-angle
②In SAS the angle must be the one between the two sides. With two sides and an angle that is not between them (SSA), two different triangles can sometimes be drawn, so it does not prove congruence
③If only the three angles match (AAA), the triangles have the same shape but may differ in size. They are similar, not congruent
④If two angles match, the third does too, so one side and two angles can always be turned into ASA, with the angles at the ends of that side
Check with examples
Example 1
△ABC ≅ △DEF and BC = 7. Find the length of side EF.
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Following the order of the letters, A ↔ D, B ↔ E and C ↔ F, so the side corresponding to BC is EF.
BC ↔ EF (corresponding sides)
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Corresponding sides of congruent figures are equal.
EF = BC = 7
▸ EF = 7
Find corresponding sides from the order of the vertices in the congruence statement, not from how the letters look.
Example 2
△ABC ≅ △DEF, ∠A = 50° and ∠B = 60°. Find ∠F.
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The angles of a triangle add up to 180°, so find ∠C first.
∠C = 180° − 50° − 60° = 70°
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C and F are both third in their triangle's name, so they correspond and ∠F = ∠C.
∠F = ∠C = 70°
▸ ∠F = 70°
∠F is not given directly, but finding its corresponding angle and using the angle sum gets you there.
Exam checklist
Congruence notation
△ABC ≅ △DEF
Write corresponding vertices in the same order: A ↔ D, B ↔ E, C ↔ F
Grade 7 school test style
Which condition always makes two triangles congruent?
①All three pairs of angles are equal
②Two pairs of sides are equal
③All three pairs of sides are equal
④The areas are equal
⑤The perimeters are equal
▸ ③ All three pairs of sides are equal
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The congruence conditions for triangles are SSS, SAS and ASA.
SSS, SAS, ASA
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Three pairs of equal sides is SSS. With only equal angles the sizes can differ (①), two sides alone do not fix the angle between them (②), and equal areas or perimeters still allow different shapes (④ ⑤).
three pairs of equal sides ⇒ SSS
🎯 Exam checklist
①The congruence conditions for triangles are SSS, SAS and ASA
②The angle in SAS must be the included angle; SSA and AAA are not congruence conditions
③Congruent figures have equal corresponding sides and equal corresponding angles
④Write congruent triangles with corresponding vertices in the same order
⑤When describing a construction, say in order where the compass point goes and what radius it uses