Drawing Triangles: Conditions and Construction

Lesson 1 of 2

In this lesson:

• Use tools to draw shapes from given conditions
• Construct triangles from three side lengths (SSS)
• Apply the triangle inequality to test validity

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Draw geometric shapes using freehand sketches, ruler and protractor, and dynamic geometry tools
2. Construct a triangle given three side lengths (SSS) and determine whether the given lengths can form a triangle using the triangle inequality
3. Explain why two different triangles with the same three side lengths must be congruent

One Condition vs. Two Conditions

One condition: many different rectangles. Two conditions: exactly one rectangle.

Three Outcomes When Drawing with Conditions

Given conditions for a triangle, exactly one of these is true:

• Unique — only one possible triangle (up to rotation and reflection)
• Multiple — infinitely many triangles satisfy the conditions
• Impossible — no triangle can be drawn

The central question: which condition sets give which outcome?

Three Tools for Drawing Shapes

• Freehand sketch — quick exploration; imprecise but useful for reasoning
• Ruler and protractor — precise construction; primary tool today
• Dynamic geometry software (e.g., GeoGebra) — rapid exploration; ideal for the SSA ambiguous case in Lesson 2

Warm-Up: Same Angles, Different Triangles?

Draw a triangle with angles 50°, 60°, and 70°.

Compare your triangle with a neighbor's.

What do you notice? Are the two triangles the same?

Five Condition Sets for Triangles

SSS Construction: Sides 5, 7, and 4 cm

• Draw cm. Arc from : radius cm. Arc from : radius cm
• Label the intersection . Draw and

SSS Gives a Unique Triangle

Could you draw a different triangle with sides 5, 7, and 4 cm?

• Any such triangle can be flipped or rotated to match yours exactly
• A flip (reflection) is still congruent — same shape and size
• Conclusion: SSS uniquely determines one triangle (up to congruence)

When SSS Fails: Impossible Triangle

Draw cm. Arc from : radius cm. Arc from : radius cm.

The arcs do not intersect — they fall short.

Why? The two shorter sides () cannot span the base of cm. No triangle is possible.

The Triangle Inequality Rule Explained

A triangle with sides , , exists when all three hold:

Shortcut: Only test the two shorter sides vs. the longest.

Check-In: Valid Triangle or Not?

Test each set of side lengths:

1. Sides , , cm — valid?
2. Sides , , cm — valid?

Check the two shorter sides against the longest, then confirm with a construction.

The Degenerate Case: Exactly Equal

Sides 3, 4, 7: — equals the longest side.

• The two arcs just barely touch on the baseline
• The result is a straight line, not a real triangle
• Degenerate triangle: zero interior area

Worked Example: Testing the Borderline

Sides 3, 4, 7

• Longest side:
• Sum of shorter sides:
• Is ? No → no triangle (degenerate)

Sides 3, 4, 6

• ✓ → triangle exists

Practice: Classify These Side Sets

For each set, classify as valid (unique triangle), degenerate (straight line), or impossible (arcs miss):

1. , ,
2. , ,
3. , ,
4. , ,
5. , ,
6. , ,

Apply the inequality test, then construct the two valid triangles.

Answers: Check Your Triangle Inequality Work

Flipped or Rotated: Still Congruent

• Translation: different position — same triangle
• Rotation: different orientation — same triangle
• Reflection: mirror image — still congruent

Two triangles are congruent if one can be moved onto the other by translation, rotation, or reflection.

Key Takeaways from Lesson 1

✓ SSS gives a unique triangle when the triangle inequality holds

✓ Two shorter sides must sum strictly more than the longest

✓ Equal → degenerate. Less than → impossible.

Reflected triangle = congruent — not a different triangle

Shortcut: only check the two shorter sides vs. longest

Coming Up in Lesson 2

• SAS: Two sides + included angle → unique
• ASA: Two angles + included side → unique
• AAA: Three angles only → infinitely many similar triangles
• SSA: The ambiguous case → 0, 1, or 2 triangles