MathematicsChapter 10: Practical Geometry
Sas Triangles
To copy a triangle you need the right three measurements.
Learning objectives
- Explain sas triangles in clear mathematical language.
- Use the main rule, formula, test, or construction correctly.
- Work through a simple example and show each important step.
- Identify common mistakes and write an exam-ready conclusion.
Definition
To copy a triangle you need the right three measurements. SAS means two sides and the included angle (the angle between those two sides). With SAS the triangle is unique: any two people who copy the same two sides and the same included angle get triangles that match. If the equal angle is not the one between the two given sides, you do not have SAS, and uniqueness can fail.
Where / symbols
- two sides
- SAS means two sides and the included angle (the angle between those two sides).
- included angle
- SAS means two sides and the included angle (the angle between those two sides).
- unique
- With SAS the triangle is unique: any two people who copy the same two sides and the same included angle get triangles that match.
Overview
Draw the included angle first in the middle, then lock the two arms with a compass:
Why Sas Is Unique
Fix side AB, then fix the angle at A, then fix side AC. Point C can sit in only one place on that ray once the compass opening is AC. The third side BC is then forced. That is why SAS builds one triangle, not a family of triangles.
Examples Sides 5 cm, 7 cm with included angle 50 deg: one triangle · Two different included angles with the same sides make different triangles · Angle not between the two sides is not SAS
How To Draw Sas
Draw one side as a base. At one endpoint copy the included angle with a compass or protractor so a ray appears. On that ray mark the second given side with the compass. Join the new point to the other end of the base. Label the vertices in the order that matches the given angle.
Examples Base AB = 6 cm, included angle A = 60 deg, then AC = 4 cm · Copy the angle by arcs if the question forbids a protractor · Join C to B last; do not draw BC first
WHAT — Core idea
To copy a triangle you need the right three measurements. SAS means two sides and the included angle (the angle between those two sides). With SAS the triangle is unique: any two people who copy the same two sides and the same included angle get triangles that match. If the equal angle is not the one between the two given sides, you do not have SAS, and uniqueness can fail.
HOW — Rules, method, and examples
1. WHY SAS IS UNIQUE: Fix side AB, then fix the angle at A, then fix side AC. Point C can sit in only one place on that ray once the compass opening is AC. The third side BC is then forced. That is why SAS builds one triangle, not a family of triangles. 2. HOW TO DRAW SAS: Draw one side as a base. At one endpoint copy the included angle with a compass or protractor so a ray appears. On that ray mark the second given side with the compass. Join the new point to the other end of the base. Label the vertices in the order that matches the given angle.
WHY — Meaning and connections
Included means 'in between'. SAS is a construction recipe and a uniqueness fact. In similarity (Chapter 9) SAS used side ratios; here SAS uses actual lengths. Keep those two meanings of SAS on the correct chapter.
HOW — Answer in an exam
SAS needs the angle between the two sides. Draw one side, copy the included angle, then compass the second side. The triangle is unique. If the angle is opposite a side, that is SSA, not SAS.
Explanation
Included means 'in between'. SAS is a construction recipe and a uniqueness fact. In similarity (Chapter 9) SAS used side ratios; here SAS uses actual lengths. Keep those two meanings of SAS on the correct chapter.
In simple words
SAS needs the angle between the two sides. Draw one side, copy the included angle, then compass the second side. The triangle is unique. If the angle is opposite a side, that is SSA, not SAS.
Key points
- two sides
- included angle
- unique
- only one place
- forced
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Sas Triangles
Sticky Notes · Quick Revision
WHAT
- To copy a triangle you need the right three measurements.
Key Points
- To copy a triangle you need the right three measurements.
- Fix side AB, then fix the angle at A, then fix side AC.
- Draw one side as a base.
- Included means 'in between'.
HOW
- 1.
- Fix side AB, then fix the angle at A, then fix side AC.
- Draw one side as a base.
Examples
- Sides 5 cm, 7 cm with included angle 50 deg: one triangle
- Two different included angles with the same sides make different triangles
- Angle not between the two sides is not SAS
- Base AB = 6 cm, included angle A = 60 deg, then AC = 4 cm
- Copy the angle by arcs if the question forbids a protractor
Common Mistakes
- Correction: SAS needs the angle between the two sides.
- Correction: Draw one side, copy the included angle, then compass the second side.
- The triangle is unique.
Exam Tip
- SAS needs the angle between the two sides. Draw one side, copy the included angle, then compass the second side. The triangle is unique. If the angle is opposite a side, that is SSA, not SAS.
60-second revision
- To copy a triangle you need the right three measurements.
- Fix side AB, then fix the angle at A, then fix side AC.
- Draw one side as a base.
- Included means 'in between'.
- SAS needs the angle between the two sides.
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