Included angle — Geometry, what it means and where it is taught
The angle between two named sides. The A in SAS must be included.
Taught in Geometry.
Where it is taught
What uses it later
This idea is needed for:
- SAS similarity — Two pairs of sides in ratio with the included angles congruent.
- Ambiguous case — Two sides and a non-included angle, which can describe two different triangles.
Practise this idea
7 questions in the course bank use Included angle, in Triangle Congruence Proofs, Similarity & Proportionality, The Law of Sines & Cosines.
Problem 1
AB ≅ DE, ∠ B ≅ ∠ E, BC ≅ EF. Name the shortcut.
Worked answer
SAS
∠ B is formed by AB and BC.
Included angle, so SAS.
Problem 2
Given AB ≅ CB and ∠ ABD ≅ ∠ CBD, with BD shared. Name the shortcut proving △ ABD ≅ △ CBD.
Worked answer
SAS
Side AB, included angle ∠ ABD, side BD (reflexive).
That is SAS.
Problem 3
Two pairs of sides in ratio with the included angles congruent proves
- SAS similarity
- SSS similarity
- AA similarity
- congruence
Worked answer
SAS similarity
Two sides in ratio plus the angle between them.
That is SAS similarity.
Problem 4
A triangle has sides 5 and 6 with an included angle of 60°. Find the third side to one decimal place.
Worked answer
5.6
25+36-2(5)(6)(0.5)
= 61-30 = 31
c = √31 ≈ 5.6
Problem 5
A triangle has sides 2 and 3 with an included angle of 120°. Find the third side to one decimal place.
Worked answer
4.4
4+9-2(2)(3)(-0.5)
= 13+6 = 19
c = √19 ≈ 4.4
Problem 6
Find the area of a triangle with sides 6 and 8 and an included angle of 30°.
Worked answer
12
12(6)(8)(0.5)
= 12
Problem 7
A triangle has sides 7 and 9 with an included angle of 90°. Find the third side to one decimal place.
Worked answer
11.4
49+81 = 130
c = √130 ≈ 11.4
Exactly Pythagoras