Table of Contents
Last modified on August 3rd, 2023
ASA means ‘Angle-Side-Angle’. ASA triangles are triangles where two angles and their common side are known. Shown below is an ASA triangle, △ABC, with given angles, ∠B and ∠C with their common side ‘a’ between them.
It involves two steps:
Step1: Use the angle sum rule of a triangle to find the missing angle
Step 2: Use The Law of Sines to find each of the other two sides
Let us take some examples to understand the concept better.
Find the missing sides and the angle in the given ASA triangle.
In the triangle, the given angles and side is:
∠A = 70°
∠B = 30°
Side c = 12
Step 1:
Using the angle sum theorem, we will find the missing angle, ∠C
∠A + ∠B + ∠C = 180°, here ∠A = 70°, ∠B = 30°
70° + 30° + ∠C = 180°
∠C = 180° – (70° + 30°)
∠C = 80°
Step 2:
Now, we will find side a using the Law of Sines
a/sin A = c/sin C, here ∠A = 70°, c = 12, ∠C= 80°
a/sin (70°) = c/sin (80°)
a = sin (70°) x 12/sin (80°)
a = 0.93 x 12/0.98
a = 11.38
Similarly, we will find side b using the Law of Sines
b/sin B = c/sin C, here ∠B= 30°, c = 12, ∠C = 80°
b/sin 30° = 12/sin 80°
b = 5.50 x sin 30°/sin 80°
b = 12 x 0.5/0.98
b = 6.12
Find the missing sides and the angle in the given triangle.
The given triangle is also an ASA triangle.
Here,
∠A = 68°
∠C = 46°
Side b = 16.2
Step 1:
Using the angle sum theorem, we will find the missing angle, ∠B
∠A + ∠B + ∠C = 180°, here ∠A = 68°, ∠C = 46°
∠68° + ∠B + 46° = 180°
∠B = 180° – (68° + 46°)
∠B = 180° – 114°
∠B = 66°
Step 2:
Now, we will find side a using the Law of Sines
a/sin A = b/sin B, here ∠A = 68°, b = 16.2, ∠B= 66°
a/sin 68° = 16.2/sin 66°
a = sin 68° x 16.2/sin 66°
a = 0.92 x 16.2/0.91
a = 16.37
Similarly, we will find side c using the Law of Sines
b/sin B = c/sin C, here b = 16.2, ∠B= 66°, ∠C = 46°
c = b x sin C/sin B
c = 16.2 x sin 46°/sin 66°
c = 16.2 x 0.71/0.91
c = 12.63
Let us prove the above theorem.
To prove: ΔABC ≅ ΔDCB
Proof:
| Steps | Statements | Reasons |
|---|---|---|
| 1. | AB || CD ∠ACB = ∠DBC | Given |
| 2. | ∠ABC ≅∠DCB | Alternate interior angles |
| 3. | CB ≅ CB | Reflexive property of congruence |
| 4. | ΔABC ≅ ΔDCB | AAS postulate (Hence proved) |
Identify which pair of given triangles illustrates an angle-side-angle (ASA) relationship.
a) △EFG ≅ △CBA, b) △PSQ ≅ △PSR, c) △OAB ≅ △OCD
Name two of the given triangles that are congruent by ASA.
△MLN and △XZY are congruent by ASA as ∠MLN ≅ ∠XZY, LN ≅ ZY, and ∠LNM ≅∠ZYX
Given the following information about an ASA triangle. ∠A = 40°, ∠B = 70°, side c = 6 cm. Find its missing sides and the angle.
Using the angle sum theorem, we will find the missing angle, ∠C
∠A + ∠B + ∠C = 180°, here ∠A = 40°, ∠B = 70°
40° + 70° + ∠C = 180°
∠C = 180° – (40° + 70°)
∠C = 70°
Now, we will find side a using the Law of Sines
a/sin A = c/sin C, here ∠A = 40°, c = 6 cm, ∠C= 70°
a = sin A x c/sin C
a = sin 40° x 6/sin 70
a = 0.64 x 6/0.93
a = 4.12 cm
Similarly, we will find side b using the Law of Sines
b/sin B = c/sin C, here ∠B = 70°, c = 6 cm, ∠C= 70°
b = sin B x c/sin C
b = sin 70° x 6/sin 70°
b = 6 cm
Thus the missing sides are a = 4.12 cm, b = 6 cm, and missing angle, ∠C = 70°
Last modified on August 3rd, 2023
