الرياضيات (بالإنجليزية) الصف الثاني الإعدادي، الأسبوع 8: Congruent triangles: ASA, AAS and RHS, and using them to find lengths and prove equalities

Congruence

الأسبوع 8 · التطابق: تطابق المضلعات وحالات تطابق المثلثين الخمس · 3 دروس

Congruent triangles: ASA, AAS and RHS, and using them to find lengths and prove equalities

Mathematics Prep 2 Term 1, Unit 3, Lesson 1

GoalUse ASA, AAS and RHS, know why AAA and SSA do not work, and use congruence to find lengths and to prove equalities.

New words

ASA · two angles and the side between themAAS · two angles and a side that is not between themRHS · right-angled triangles: hypotenuse and one legvertically opposite angles · the equal angles formed when two lines cross

Explanation

Two more cases prove congruence. ASA: two angles and the side between them. AAS: two angles and a side not between them; since the third angle is always known (the angles add to 180°), AAS is really ASA in disguise. RHS: two right-angled triangles with equal hypotenuses and one equal leg; here the right angle is not between the equal sides. Two things do NOT work. AAA: a triangle drawn with the same angles can be bigger or smaller, for example two equilateral triangles of sides 3 and 5. SSA: two sides and an angle that is not between them can fit two different triangles. Facts hidden in a figure help: a common side, vertically opposite angles, a midpoint, a perpendicular. After proving congruence, corresponding parts are equal; this gives us lengths and angles.

Rule	What must be equal	Works?
SSS	3 sides	✓
SAS	2 sides and the angle between	✓
ASA / AAS	2 angles and a side	✓
RHS	hypotenuse and a leg	✓
AAA	3 angles	✗
SSA	2 sides and a non-included angle	✗

Worked examples

Two triangles: ∠B = ∠E = 40°, ∠C = ∠F = 60° and BC = EF = 8 cm. The side BC lies between ∠B and ∠C, and EF between ∠E and ∠F, so Δ ABC ≅ Δ DEF by ASA. Two right-angled triangles with hypotenuse 13 cm and one leg 5 cm each are congruent by RHS.

Lines AD and BC cross at O, and O is the midpoint of both. Prove AB = DC. In Δ AOB and Δ DOC: AO = OD (midpoint), BO = OC (midpoint), ∠AOB = ∠DOC (vertically opposite). So Δ AOB ≅ Δ DOC by SAS, and AB = DC.

Δ ABC ≅ Δ DEF with AC = 2x + 3 and DF = 3x − 1. Corresponding sides are equal, so 2x + 3 = 3x − 1 and x = 4. The side is 2 × 4 + 3 = 11. Check: 3 × 4 − 1 = 11.

Class activity

Show AAA fails. Draw a triangle with angles 60°, 60°, 60° and sides 4 cm. Draw another with the same angles and sides 6 cm. Do they match? Then draw an angle of 30°, a side of 5 cm and another side of 3 cm opposite the 30° angle. How many different triangles can you draw?

Practice

∠B = ∠E = 35°, ∠C = ∠F = 75° and BC = EF = 9. Which case?

Two right-angled triangles have hypotenuse 10 cm and a leg of 6 cm each. Are they congruent?

Two triangles both have angles 50°, 60° and 70°. Are they congruent?

Δ ABC ≅ Δ DEF with AB = 5x − 2 and DE = 2x + 7. Find x.

ABCD is a square with diagonal AC. Prove Δ ABC ≅ Δ ADC.