A Tale of Three Intersecting Lines · Lesson 5 of 10
Two Sides and the Included Angle
“Construct a triangle by fixing the angle between two given sides.”
• Identify the included angle between two specified sides. • Construct a triangle from two sides and their included angle. • Use a protractor to place a ray with the correct interior angle. • Explain why the construction fixes the third vertex. • Recognise the valid range of an included interior angle.
Two Sides and the Included Angle
Three side lengths are one way to specify a triangle. Another is to give two side lengths and the angle between them. Imagine two rods hinged at one end: the lengths of the rods tell us their sizes, and the angle at the hinge tells us how wide they open. Joining their free endpoints closes the triangle.
The angle formed by the two given sides at their common vertex. For sides AB and AC, the included angle is ∠BAC, or ∠A.
The word “included” is essential. If the two sides are AB and AC, an angle at B is not the included angle because AC does not meet AB there. Before drawing, make a rough sketch and mark the two given lengths and the angle where they meet. This prevents using correct numbers in the wrong positions.
A Construction with Two Sides and an Angle
To construct AB = 5 cm, AC = 4 cm, and ∠A = 45°, we first draw AB. The required angle tells us the direction of AC, but not where C lies along that direction. The length AC then fixes C on the angle’s second arm. Joining C to B completes the triangle.
- Draw AB = 5 cm using a marked ruler.
- Place the protractor’s centre at A and align its baseline with AB. Read the scale that begins at 0° along AB; locate 45°. Draw a ray from A through that direction.
- Mark C on this ray at a distance of 4 cm from A, using a ruler or a 4 cm compass opening.
- Join B and C. Check AB, AC, and the angle at A.
Here a ray means a straight path that begins at A and continues in one direction. Once C is placed on the ray 4 cm from A, AC has both the required direction and the required length. BC does not need to be guessed or specified separately: the two endpoints already determine the segment.
Problem
Construct a triangle with AB = 5 cm, AC = 4 cm, and ∠A = 45°.
- 1.Draw the 5 cm base AB.
- 2.Draw a ray from A making a 45° angle with AB, taking care to read the correct protractor scale.
- 3.Locate C 4 cm from A on that ray and join BC.
- 4.The construction gives the two required side lengths and their included angle directly.
Different Included Angles
The construction does not require an acute angle. An obtuse angle also works, provided it is less than a straight angle. The third vertex may then lie beyond one end of the base when viewed across the page. That appearance is perfectly consistent with the given angle.
Problem
Construct a triangle with sides 3 cm and 7 cm and included angle 75°.
- 1.Draw PQ = 7 cm. The other given side will be PR = 3 cm, so the included angle is at P.
- 2.Draw a 75° ray from P and mark R 3 cm along it.
- 3.Join RQ. Check the 75° angle between PQ and PR, not an angle at Q.
Problem
Construct a triangle with sides 3 cm and 8 cm and included angle 120°.
- 1.Draw AB = 8 cm. Draw a ray from A making 120° with AB.
- 2.Mark C on that ray with AC = 3 cm. C will lie to the left of A in the usual drawing with AB pointing right.
- 3.Join BC. The triangle is valid: the two positive sides meet at an angle between 0° and 180°.
When Does This Information Form a Triangle?
With two positive side lengths, any included angle strictly between 0° and 180° produces a triangle. The two arms point in different, non-opposite directions, so their chosen endpoints and their shared endpoint do not lie on one straight line. Joining the free endpoints therefore encloses a region.
At 0°, the two sides point along the same ray. At 180°, they point along opposite rays on the same line. Both cases are collinear and fail to make a triangle. An angle greater than 180° is a reflex angle, not the interior angle being specified in this construction.
Do not use an angle merely because it appears somewhere in the triangle. It must be between the two given sides. Also check which protractor scale starts at 0° on the base ray; reading the other scale can turn a required 25° angle into 155°.
Construct a triangle with sides 6 cm and 3 cm and included angle 25°. Then repeat with the same sides and included angle 90°. The two lengths stay fixed, but the opening and the third side change. Explain which part of the method controls each change.
Quiz
For given sides AB and AC, which angle is included?
After drawing the angle ray at A, how is C placed if AC = 4 cm?
Which included angle with two positive sides gives a triangle?
Why does a 180° included angle fail?
For sides 3 cm and 7 cm with included angle 75°, which construction is correct?
Practice Problems
- Construct triangles from the measurements 3 cm, 75°, 7 cm; 6 cm, 25°, 3 cm; and 3 cm, 120°, 8 cm, taking each angle between its two given sides.
- Sketch ΔPQR with PQ = 6 cm and PR = 4 cm. Name the included angle and explain why ∠Q is not that angle.
- Write the construction steps for AB = 4 cm, AC = 2 cm, and ∠A = 90°.
- Explain why two positive side lengths with an included angle 0° or 180° do not make a triangle.
- A learner needs an included angle of 25° but reads 155° on the protractor. Explain which measurement has changed and how to avoid the error.
- Use the same two sides with included angles 40° and 100°. Compare the drawings and explain why both are valid.
Key Takeaways
• The included angle is at the common endpoint of the two given sides. • Draw one side, construct the included angle, locate the second side’s endpoint, and join the remaining endpoints. • The direction and length together fix the third vertex. • An obtuse included angle can produce a valid triangle. • The two lengths must be positive and the included interior angle must lie strictly between 0° and 180°.