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Constructing
Geometric Shapes

Learn to draw triangles from given sides and angles, and discover when conditions create a unique triangle, many triangles, or none at all.

7th Grade Math

What Does Construct Mean?

In geometry, constructing a shape means drawing it so that it matches a set of specific conditions: given side lengths, angle measures, or both. You can draw freehand, use a ruler and protractor, or use technology.

Three Possible Outcomes

Unique Triangle
Exactly one triangle can be drawn. The conditions fully determine the shape and size.
More Than One
Multiple different triangles satisfy the given conditions. The shape is not fully determined.
No Triangle
The conditions are impossible to satisfy. No valid triangle can exist with those measurements.

The Building Blocks: Sides and Angles

Side a Side b Side c A B C A B C

The Triangle Inequality Theorem

3 + 4 > 5 ✓ 4 5 3 Valid ✓ 2 3 Side = 5 2 + 3 = 5, not > 5 ✕ Invalid ✕

The Angle Sum Rule

50° 60° 70° 50° + 60° + 70° = 180° ✓

Unique vs. Non-Unique Conditions

Condition Unique? Why?
SSS (3 sides) Yes Fully determines shape and size
SAS (2 sides + included angle) Yes The angle locks the shape
ASA (2 angles + included side) Yes Third angle is forced by \(180^\circ\) rule
AAA (3 angles only) No: many Shape is fixed, but size can vary
Sides violate inequality No: none Sides cannot close into a triangle

How to Determine the Outcome

1
Identify the given info. List all sides and angles you are given. Count how many of each.
2
Check for "no triangle" first. Do the sides satisfy the triangle inequality? Do the angles add to \(180^\circ\)?
3
Classify the condition type. Is it SSS, SAS, ASA, AAS, or AAA? Match it to the uniqueness chart.
4
Construct or explain. If unique, draw it. If many, show two different examples. If none, explain why.

Example 1: SSS Condition

Draw a triangle with sides \(5\) cm, \(7\) cm, and \(9\) cm.

9 cm 5 cm 7 cm A B C

Example 2: AAA Condition

Draw a triangle with angles \(40^\circ\), \(60^\circ\), and \(80^\circ\).

Large Small 40° 60° 80° 40° 60° 80° Same angles, different sizes

Example 3: No Triangle Possible

Can you draw a triangle with sides \(2\) cm, \(3\) cm, and \(10\) cm?

10 cm 2 cm 3 cm Gap ✕

Example 4: SAS Gives a Unique Triangle

Draw a triangle with sides \(6\) cm and \(8\) cm and an included angle of \(50^\circ\).

50° 6 cm 8 cm B C A

Example 5: ASA Also Unique

Draw a triangle where \(\angle B = 45^\circ\), side \(BC = 10\) cm, and \(\angle C = 60^\circ\).

45° 60° 10 cm B C A

Condition Types Summary

SSS
3 sides given. Always unique (if inequality holds).
SAS
2 sides + included angle. Always unique.
ASA
2 angles + included side. Always unique.
AAA
3 angles only. Many triangles (similar shapes, different sizes).
Fails Inequality
Side sums don't work. No triangle possible.

Common Mistakes to Avoid

✕ Forgetting the Inequality Check
Always verify that the sum of the two shorter sides is greater than the longest side before trying to draw.
✕ Thinking AAA is Unique
Three angles fix the shape but not the size. You can always scale the triangle larger or smaller.
✕ Confusing "Included" and "Non-Included"
In SAS, the angle must be between the two sides. A non-included angle (SSA) can sometimes give two triangles.

Question 1 of 5

A triangle has sides of length 3 cm, 4 cm, and 8 cm. What is the result?

Question 2 of 5

You are given three angles: \(50^\circ\), \(60^\circ\), and \(70^\circ\). How many triangles can you draw?

Question 3 of 5

Which condition always produces a unique triangle?

Question 4 of 5

A triangle has sides 5 cm, 12 cm, and 13 cm. Which statement is true?

Question 5 of 5

You know \(\angle A = 40^\circ\), \(\angle B = 90^\circ\), and side \(AB = 6\) cm. This is an example of: