Grade 7 Constructing Triangles Lesson: Rules

Contents

Key takeaway: This grade 7 constructing triangles lesson explains when a set of measurements can actually be drawn as a triangle. Three side lengths work only when the two shorter sides together beat the longest one, a rule called the triangle inequality. Seventh graders also learn which conditions give exactly one triangle, many triangles, or none at all.

First, you test side lengths with the inequality. Next, you use the angle sum of 180 degrees. Finally, you decide whether a protractor and compass drawing is unique, ambiguous, or impossible.

A triangle can be drawn from given measurements only when the conditions allow it. Three side lengths work when the two shorter sides add up to more than the longest side. Two sides with the angle between them, or two angles with a side, give exactly one triangle. Three angles alone give many triangles.

Grade 7Common Core: 7.G.A.2

Grade 7 Constructing Triangles Lesson: key facts

Triangle inequality: the sum of any two side lengths must be greater than the third side length.
The three angles of a triangle always add up to \(180^{\circ}\).
Three sides, two sides with the angle between them, or two angles with a side give exactly one triangle when a triangle is possible.
Three angles alone give infinitely many triangles that have the same shape but different sizes.
Two sides and an angle that is not between them can give no triangle, one triangle, or two triangles.

How do you draw a triangle from given measurements?

To draw a figure with given conditions, use a ruler to measure lengths and a protractor to measure angles. Start with one side, add each angle or side in order, and close the figure.

Worked method: two sides and the angle between them

Draw triangle ABC with \(AB=6\text{ cm}\), \(AC=4\text{ cm}\) and \(\angle A=50^{\circ}\).

Use the ruler to draw segment AB, 6 cm long.
Place the center of the protractor on A and line up its 0 line with AB. Mark a dot at \(50^{\circ}\).
Draw a ray from A through the dot. Measure 4 cm along it and label the point C.
Join B to C. The side BC measures about 4.6 cm.

Figure 1. The triangle built in the steps above: 6 cm, \(50^{\circ}\), 4 cm.

When can three lengths make a triangle?

Triangle inequality

Three lengths can be the sides of a triangle only if the sum of any two of them is greater than the third. It is enough to check the two shorter sides against the longest side.

Sticks of 3, 4, and 8 cannot form a triangle because \(3+4=7\) is less than 8. Sticks of 3, 4, and 6 can, because \(3+4=7\) is greater than 6. If the sum equals the longest side, the sticks lie flat on a line and make no triangle.

Figure 2. Sides 3, 4, 8 fail the triangle inequality. Sides 3, 4, 6 pass it.

Which conditions give exactly one triangle?

The angle measures of every triangle add up to \(180^{\circ}\), so two angles are enough to find the third. Different sets of measurements lead to different counts of triangles.

Given	Triangles possible
Three sides that pass the triangle inequality	Exactly one
Two sides and the angle between them	Exactly one
Two angles and a side (angle sum less than \(180^{\circ}\))	Exactly one
Three angles that add up to \(180^{\circ}\)	Infinitely many
Two sides and an angle not between them	None, one, or two
Sides that fail the inequality, or angles that do not add up to \(180^{\circ}\)	None

Figure 3. Two angles and the side between them fix one triangle.

With three angles only, the shape is fixed but not the size. The triangles in Figure 4 both have angles of \(50^{\circ}\), \(60^{\circ}\), and \(70^{\circ}\), yet one is larger.

Figure 4. Same three angles, different sizes: the angles alone do not decide the triangle.

With two sides and a non-included angle, the compass arc can cross the other ray twice. In Figure 5, \(\angle A=30^{\circ}\), \(AC=8\text{ cm}\), and \(BC=5\text{ cm}\) fit two different triangles.

Figure 5. Angle A, side AC, and side BC can fit two triangles: ABC with B at \(B_1\) or at \(B_2\).

Drawing Geometric Figures and Constructing Triangles: frequently asked questions

What is the triangle inequality?

The triangle inequality says that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For lengths 5, 6, and 12, the sum \(5+6=11\) is less than 12, so no triangle exists.

Can three angles form a unique triangle?

No. Three angles that add up to \(180^{\circ}\) fix the shape but not the size, so infinitely many triangles fit. A side length is needed to choose one size.

How many triangles can be made with two sides and a non-included angle?

There can be none, one, or two triangles. The compass arc drawn from the end of the known side may miss the ray, touch it once, or cross it twice. This is why side-side-angle does not guarantee one triangle.

How do you use a protractor to draw an angle?

Put the center mark on the vertex and line up one side of the angle with the 0 line. Read the scale that starts at 0 on that side, mark the angle, and draw a ray from the vertex through the mark.

How do you find the third angle of a triangle?

Add the two known angles and subtract the sum from \(180^{\circ}\). For \(40^{\circ}\) and \(60^{\circ}\), the third angle is \(180-100=80\), so it measures \(80^{\circ}\).

When can three lengths form a triangle?

Add the two shorter lengths and compare the total with the longest side. Sides of 3, 4, and 5 inches work because 3 plus 4 is 7, which beats 5. Sides of 5, 5, and 10 fail because 5 plus 5 only equals 10, so the sticks lie flat in a line. In other words, the inequality must be strict.

How does a grade 7 constructing triangles lesson use angle sums?

The three angles of any triangle total 180 degrees. If two angles are 48 and 67, the third must be 65. A set such as 50, 60, and 80 fails, because the total is 190. Therefore a drawing with those angles is impossible.

Which measurements lock in a single triangle?

Three sides, two sides with the angle between them, and two angles with a side each fix one shape. Draw a 7-centimeter side, add a 40-degree angle, and mark a 5-centimeter side on the other ray. Everyone who follows those steps gets the same triangle. Three angles alone fix only the shape, so the size can vary.

What goes wrong with two sides and a different angle?

When the given angle is not between the two sides, an arc can cross the ray twice. An angle of 30 degrees with sides of 12 and 7 centimeters gives two possible triangles. This is why the order of the measurements matters.

Draw before you move on

Cut three straws and test whether they close into a triangle, then tape the result onto paper. Measure each angle with a protractor and see whether the total comes close to 180. After that, try the worksheet and start with the stick and campsite problems.