In plain English: 7.G.A.2 is the Common Core grade 7 math standard that asks students to draw shapes that meet given conditions, freehand, with a ruler and protractor, and with technology. The focus is on triangles built from three side lengths or angle measures, and on noticing whether those measures give exactly one triangle, more than one triangle, or no triangle at all.
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them. Also written as 7.G.2 · Official standard
Students draw shapes that must meet given conditions, such as "a triangle with a 40° angle and two equal sides." They work three ways: freehand (a quick sketch without tools, to plan), with a ruler and protractor (a protractor measures and draws angles in degrees), and with technology (a geometry app where points can be dragged). A compass, a tool that draws circles and parts of circles called arcs, helps place a side of an exact length.
The main question is how many triangles fit three given measures. Sometimes the measures give exactly one triangle: every triangle anyone draws with them is the same shape and the same size, so the copies would match if you cut them out and stacked them. Sometimes they give more than one triangle, and sometimes no triangle. Students test side lengths with the triangle inequality (the two shorter sides together must be longer than the longest side) and angles with the 180° angle sum. Proving why some measures always give one triangle comes later, in high school geometry (HSG.CO.B.8).
Learning Objectives
By the end of this lesson, students will be able to:
Sketch a shape freehand to plan it, then draw it accurately with a ruler, protractor and compass
Construct a triangle from three side lengths, from two sides and the angle between them, and from two angles and the side between them
Use the triangle inequality to decide whether three lengths make a triangle
Explain why three angles that add to 180° give more than one triangle, and angles with any other sum give none
Use a geometry app to test when two sides and an angle that is not between them give no triangle, one triangle or two triangles
Prior Knowledge Required
Students should already be comfortable with:
Measuring and drawing angles in whole degrees with a protractor 4.MD.C.6
Naming triangles by their angles (acute: all angles less than 90°; right: one 90° angle; obtuse: one angle more than 90°) and by their sides (two equal sides, or none) 5.G.B.4
Knowing that the three angles of a triangle add up to 180° (the warm-up checks this with paper corners)
Measuring lengths to the nearest millimeter with a ruler
Give every student a sheet of plain paper. Ask them to sketch freehand, without a ruler, the three triangles in the prompt below. They have two minutes and do not need to be neat.
Warm-Up Prompt
"Sketch a triangle with one obtuse angle (more than 90°). Sketch a triangle with two equal sides. Now try to sketch a triangle with two obtuse angles. What happens?"
The first two are easy. The third one fails: two angles that are each more than 90° already add to more than 180°, and the sides never close. Confirm the 180° rule quickly: each student cuts out any paper triangle, tears off the three corners, and lines them up point to point. The corners always form a straight line, which is 180°. Tell students that today's question is: when you are given three measures, can you always draw the triangle, and is there only one?
Direct Instruction20 minutes
Explain the words first. A segment is the straight path between two points, and a ray starts at a point and goes on forever in one direction; the two arms of an angle are rays. A side is one of the three segments of a triangle, and a vertex is a corner where two sides meet. The included angle of two sides is the angle between them, at the vertex where they meet. The included side of two angles is the side that joins their vertices. Model each construction on the board with a large ruler, protractor and compass while students copy it on paper.
Sketch first: draw a quick freehand triangle and label the given measures on it, so you know which measure goes where.
Draw one side with the ruler. It is usually the longest given side, or the side between two given angles.
Add the other measures: swing compass arcs for side lengths, or draw angles with the protractor at the ends of the side.
Decide how many: the pieces meet in one place (one triangle), in more than one way (more than one triangle), or not at all (no triangle).
Three sides: exactly one triangle
Construct a triangle with sides 5 cm, 6 cm and 9 cm.
Equation: Check first: 5 + 6 = 11, which is more than 9, so a triangle exists. Draw the 9 cm side. Set the compass to 5 cm and swing an arc from the left end; set it to 6 cm and swing an arc from the right end. The arcs cross at the third vertex. Every student who does this gets the same triangle, so there is exactly one.
Three sides: no triangle
Try to construct a triangle with sides 3 cm, 4 cm and 8 cm.
Equation: 3 + 4 = 7, which is less than 8. Draw the 8 cm side and swing the 3 cm and 4 cm arcs: they stop 1 cm apart and never meet. No triangle is possible. If the two short sides added to exactly 8 cm, they would lie flat on the long side, which is still not a triangle.
Three angles: more than one triangle, or none
Can you draw a triangle with angles 50°, 60° and 70°? What about 50°, 60° and 80°?
Equation: 50° + 60° + 70° = 180°, so yes. Draw any side, put 50° at one end and 60° at the other, and the third angle is 70° by itself. A longer first side gives a bigger triangle with the same angles, so there are many different triangles: scaled copies, as in scale drawings (7.G.A.1). 50° + 60° + 80° = 190°, so the second set gives no triangle.
Two sides and the included angle, or two angles and the included side: exactly one
Construct (a) a triangle with sides 7 cm and 5 cm and a 40° angle between them, and (b) a triangle with angles 45° and 65° and a 7 cm side between them.
Equation: (a) Draw the 7 cm side, draw a 40° angle at one end, and mark 5 cm along the new ray. Join the two ends: only one third side is possible. (b) Draw the 7 cm side, draw 45° at one end and 65° at the other. The rays cross in one place, and the third angle is 180° - 45° - 65° = 70°. Both give exactly one triangle.
Two sides and an angle that is not between them: two triangles
Angle A is 30°, side AB is 8 cm, and side BC, across from angle A, is 5 cm.
Equation: Draw a 30° angle at A and mark B 8 cm up one ray. Swing a 5 cm arc from B. It crosses the other ray twice, at C₁ and at C₂ (about 3.9 cm and 9.9 cm from A). Both triangles have a 30° angle, an 8 cm side and a 5 cm side, but they are different shapes. So these measures give two triangles (Diagram 2).
Show Diagram 1 for the first two examples. Then state the triangle inequality: three lengths make a triangle only when the two shorter lengths add up to more than the longest one. Students only need to test the two shorter sides, because any sum that includes the longest side is automatically bigger than the third side.
Show Diagram 2 for the last example. The dashed line from B is the shortest distance from B to the other ray, 4 cm. A side BC shorter than that cannot reach the ray, and a side of exactly 4 cm just touches it, making one right triangle. A side between 4 cm and 8 cm reaches it twice. A side of 8 cm or more crosses the ray only once (its other crossing lands behind A, off the ray), so there is one triangle. Students discover this by drawing, not by memorizing: they swing arcs of different lengths and count the crossings.
Summary: what three measures give
Measures given
Number of triangles
Three sides
One if the two shorter sides add to more than the longest; none otherwise
Two sides and the angle between them
One (if the angle is less than 180°)
Two angles and the side between them
One if the two angles add to less than 180°; none otherwise
Three angles
More than one (many sizes) if they add to 180°; none otherwise
Two sides and an angle not between them
None, one or two: draw it and count the arc crossings
Guided Practice15 minutes
Pairs work through the table. For each row, one partner sketches it freehand and predicts the answer, and the other checks with ruler, protractor and compass. Then they switch roles.
Guided practice conditions with answers
Conditions
Answer
Sides 4 cm, 7 cm and 10 cm
Exactly one triangle: 4 + 7 = 11, more than 10
Sides 2 cm, 5 cm and 8 cm
No triangle: 2 + 5 = 7, less than 8
Angles 25°, 65° and 90°
More than one triangle: the sum is 180°, and any size works
Angles of 70° and 120° with a 3 cm side between them
No triangle: 70° + 120° = 190°, so the rays spread apart and never meet
Two 6 cm sides with a 60° angle between them
Exactly one triangle; the third side also measures 6 cm, so all three sides are equal
A four-sided shape with every side 4 cm
More than one shape: a square, or any leaning shape with four 4 cm sides (a rhombus). Four sides alone do not fix the angles
Listen for these errors: adding the longest side to a short one when testing lengths, putting a given angle at the wrong vertex, and thinking that equal angles mean equal sizes. Ask pairs to hold up two of their 25°, 65°, 90° triangles next to each other: the angles match, but one is bigger.
Independent Practice15 minutes
Students construct each triangle on their own, label every measure, and write "one," "more than one" or "none" next to it.
Independent practice with answers
Construct
Answer
Sides 6 cm, 8 cm and 10 cm, with a compass. Measure the largest angle.
One triangle; the largest angle measures 90°
Angles 40° and 85° with a 5 cm side between them
One triangle; the third angle is 55°
Sides 7 cm and 4 cm with a 90° angle between them
One right triangle
A freehand sketch of a right triangle with two equal sides, then its angles
One shape for each size; the angles are 90°, 45° and 45°
Angles 30°, 30° and 30°
No triangle: the sum is only 90°
Closure5 minutes
Exit ticket: (1) Can 5 cm, 5 cm and 11 cm make a triangle? Explain in one sentence. (Answer: no, because 5 + 5 = 10, which is less than 11.) (2) Write three angle measures that give more than one triangle, and explain why there is more than one. (3) Which kind of three measures gave two different triangles today?
Differentiation Strategies
For Struggling Students
Start with the straws from Activity 1 before any compass work, so students feel why short sides cannot close
Give a checklist card: "Sketch, label, draw the longest side, add the other two measures, count"
Pre-draw the first side and the given angle, and let the student finish the construction
For Advanced Students
Challenge: find all whole-number lengths for BC that give two triangles when angle A is 30° and AB is 14 cm, and explain the pattern
Ask whether four side lengths ever fix one four-sided shape, and test with straws or the app
Ask students to write their own set of three measures for each result: one triangle, more than one, and none
Assessment Guidance
What to Look For
Check that students sketch and label before they construct, that constructed sides are within 1 mm and angles within 1° of the given measures, and that arcs are visible rather than erased. When students decide how many triangles, listen for a reason: the sum of the two shorter sides, the angle sum, or the number of arc crossings. "It looked like it worked" is not yet a reason.
02
Classroom Activities
3 Activities
1
Straw Triangles
15 minPairs
Each pair gets six straw pieces: 3, 4, 5, 6, 8 and 10 cm long. There are 20 different ways to choose three pieces. Pairs try every set of three, lay the pieces on the desk end to end, and record whether they close into a triangle.
Procedure
Make a table with columns: three lengths, sum of the two shorter, longest, triangle (yes or no)
Try each of the 20 sets and fill in a row for each
Circle the sets where the two shorter pieces add up to exactly the longest one, and look closely at what happens
Write a rule that predicts "yes" or "no" without using the straws
Answer Key
13 of the 20 sets make a triangle. Two sets lie flat, because the two shorter pieces add up to exactly the longest: 3, 5, 8 and 4, 6, 10. The other five sets fall short: 3, 4, 8 and 3, 4, 10 and 3, 5, 10 and 3, 6, 10 and 4, 5, 10.
Discussion Questions
Every set that falls short or lies flat uses the 8 cm or the 10 cm piece. Why do the long pieces cause the trouble?
Straws have thickness, so the flat sets can seem to almost work. Why does the math say no triangle?
Did any set of three pieces close into two different triangles?
Modification for Distance Learning
Students cut strips of paper to the six lengths at home, or drag segments of those lengths in a geometry app, and record the same table.
2
One, Many or None? Card Sort
20 minGroups of 3
Each group gets 8 condition cards and three sorting mats labeled "exactly one," "more than one" and "none." Groups predict first, then check at least one card from each pile with ruler, protractor and compass.
Condition Cards
C1: Sides 5 cm, 5 cm and 8 cm
C2: Angles 30°, 70° and 80°
C3: Sides 3 cm, 6 cm and 10 cm
C4: Angles 45° and 95° with a 6 cm side between them
C5: Angles 60°, 60° and 70°
C6: Sides 7 cm and 9 cm with a 35° angle between them
C7: A 30° angle at A, AB = 10 cm, and BC = 7 cm across from angle A
C8: Angles 95° and 90° with a 4 cm side between them
Answer Key
Exactly one: C1 (5 + 5 = 10, more than 8), C4 (third angle 40°), C6 (the angle is between the sides)
More than one: C2 (the angles add to 180°, any size), C7 (the 7 cm arc from B crosses the other ray twice, because the shortest distance from B to it is 5 cm)
None: C3 (3 + 6 = 9, less than 10), C5 (the angles add to 190°), C8 (95° + 90° = 185°)
Discussion Questions
C2 and C7 both give more than one triangle. How are the triangles in C2 different from each other, and how are the two triangles in C7 different?
C4 and C8 both give two angles and a side. Why does one give a triangle and the other none?
Which card was hardest to predict without drawing it?
Challenge Variation
Groups change one number on each "none" card so that it gives exactly one triangle, and trade cards with another group to check.
3
Geometry App Drag Test
15 minPairs
Pairs use a free geometry app (for example GeoGebra or Desmos Geometry) to test the case of two sides and an angle that is not between them. In an app, a slider is a bar you drag to change a number, and the picture updates at once.
Procedure
Draw a ray from point A. Draw a second ray from A so that the angle between them is 40°
Put point B on the second ray, 7 cm from A
Make a slider r from 1 to 10, and draw a circle with center B and radius r
Set r to 3, 4, 5, 6, 8 and 9. Each time, count the places where the circle crosses the first ray (not counting A itself), and write the number of triangles
Then build a four-sided shape with every side 5 cm and drag one corner. Write down whether the shape stays the same
Answer Key
The shortest distance from B to the first ray is about 4.5 cm. r = 3 and r = 4: no crossings, no triangle. r = 5 and r = 6: two crossings, two triangles. r = 8 and r = 9: one crossing, one triangle. The four-sided shape changes its angles as you drag, so four equal sides give more than one shape.
Discussion Questions
For which values of r did the circle cross the ray twice? Why did the crossings drop to one once r was longer than AB?
Why do two sides with the angle between them never give two triangles, even in the app?
Modification Without Devices
Pairs draw the 40° angle and the 7 cm side on paper and swing compass arcs of 3, 4, 5, 6, 8 and 9 cm from B.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Do the Arcs Meet? Three Sides
Left: the 9 cm side is drawn first, and arcs of 5 cm and 6 cm from its ends cross at the third vertex, so there is exactly one triangle. Right: arcs of 3 cm and 4 cm from the ends of an 8 cm side stop 1 cm apart, so there is no triangle. Drawn to scale, 30 pixels per centimeter.
Diagram 2: Two Sides and an Angle Not Between Them
A 30° angle at A and AB = 8 cm are fixed. A 5 cm arc from B crosses the other ray at C₁ and C₂, so triangles ABC₁ and ABC₂ both fit the measures. The dashed line is the shortest distance from B to the ray, 4 cm. Drawn to scale, 40 pixels per centimeter.
04
Homework Assignment
~30 min
7.G.A.2 Homework: Constructing Triangles from Three Measures
Directions: Use a ruler, protractor and compass, or a geometry app where a problem says so. Sketch each figure freehand and label the measures before you construct it. For every problem, write whether the measures give exactly one triangle, more than one triangle or no triangle, and explain why.
Part 1: Three Sides (Problems 1-2)
Decide whether each set of lengths makes a triangle. Show the test you used. (a) 6 cm, 8 cm, 13 cm (b) 5 cm, 9 cm, 14 cm (c) 2.5 cm, 4 cm, 6 cm (d) 7 cm, 7 cm, 15 cm
Construct a triangle with sides 6 cm, 7 cm and 8 cm using a ruler and compass. Measure its largest angle with a protractor, to the nearest degree. Explain why a classmate who follows the same steps gets a triangle that matches yours.
Part 2: Angles (Problems 3-4)
Construct a triangle with a 40° angle and a 75° angle, and a 9 cm side between them. Find the third angle without measuring, then measure it to check. How many different triangles fit these measures?
Sketch freehand a triangle for each set of angles, or explain why none exists. Then say whether each set gives exactly one, more than one or no triangle. (a) 30°, 60°, 90° (b) 45°, 55°, 90° (c) 20°, 20°, 140°
Part 3: Two Sides and an Angle (Problems 5-6)
Use a geometry app, or a compass on paper. Draw a 45° angle at A and mark B 10 cm from A on one ray. The shortest distance from B to the other ray is about 7.1 cm. Swing arcs from B with radius 6 cm, 8 cm and 12 cm. For each radius, how many triangles ABC have BC equal to that radius?
A triangle has sides of 9 cm and 4 cm. (a) List every whole-number length, in centimeters, that the third side could have. (b) If the triangle has two equal sides, how long is the third side? Explain why the other choice does not work.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sketch and Labels
Freehand sketch with every given measure in the right place
Sketch present, one measure missing or misplaced
No sketch
Construction
Sides within 1 mm and angles within 1°, arcs shown
Small errors, or arcs erased
Construction missing or does not match
How Many Triangles
Correct result (one, more than one, none) for every problem
One or two results wrong
Most results wrong
Reasoning
Each result explained with the side test, the angle sum or the arc crossings
Some reasons missing
No reasons given
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Keep a ruler, protractor and compass nearby for sketches. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
Which set of side lengths makes a triangle?
Answer: D
The two shorter sides must add to more than the longest: 5 + 7 = 12, which is more than 11. Choice A fails because 2 + 3 = 5 is less than 6; it comes from testing only 3 + 6 > 2, a sum that includes the longest side. Choice B fails because 4 + 4 = 8 is less than 9. Choice C gives 6 + 6 = 12, exactly the longest side, so the pieces lie flat.
Question 2 of 20 · Multiple Choice
How many triangles have angles of 40°, 60° and 80°?
Answer: B
40° + 60° + 80° = 180°, so the angles work, but no side length is given. A small triangle and a large triangle can both have these angles, so there are many. Choice A assumes that fixing the angles fixes the size. Choice C would be right only if the angles did not add to 180°. Choice D mixes this up with two sides and an angle not between them, the case that can give two.
Question 3 of 20 · Multiple Choice
How many triangles have angles of 35°, 65° and 90°?
Answer: C
35° + 65° + 90° = 190°, which is more than 180°, so no triangle has these angles. Choice B sees three angles and forgets to add them: three angles that do add to 180° give more than one triangle. Choice A treats the right angle as if it fixed the triangle.
Question 4 of 20 · Multiple Choice
A triangle has sides of 6 cm and 9 cm with a 50° angle between them. How many triangles fit these measures?
Answer: A
Draw the 9 cm side, draw a 50° angle at one end, and mark 6 cm on the new ray. There is only one way to join the ends, so there is exactly one triangle. Choice D confuses this with an angle that is not between the two sides. Choice B would need a measure to be missing, but two sides and the angle between them fix the shape. Choice C has no reason: 50° is less than 180°.
Question 5 of 20 · Multiple Choice
Two sides of a triangle are 4 cm and 10 cm. Which could be the length of the third side?
Answer: B
The third side must be longer than 10 - 4 = 6 cm and shorter than 10 + 4 = 14 cm, and 12 cm is between them: 4 + 10 = 14 is more than 12. Choice A treats "equal" as enough: 4 + 6 = 10, so the sides lie flat. Choice C does the same on the long end: 4 + 10 = 14 exactly. Choice D is longer than 4 + 10 = 14, so the two sides cannot reach.
Question 6 of 20 · Multiple Choice
A triangle has angles of 30° and 110° with a 6 cm side between them. What is the third angle, and how many triangles fit?
Answer: A
The third angle is 180° - 30° - 110° = 40°. The 6 cm side fixes the size, and the two angles at its ends fix the shape, so there is exactly one triangle. Choice B forgets that the 6 cm side fixes the size. Choice C writes the sum of the two given angles, 30° + 110° = 140°, as the third angle. Choice D subtracts from 360°, the angle sum of a four-sided shape: 360° - 140° = 220°.
Question 7 of 20 · Multiple Choice
A triangle is supposed to have angles of 100° and 85° with a 5 cm side between them. What can you conclude?
Answer: D
100° + 85° = 185°, more than 180° before the third angle is even added, so the rays from the ends of the 5 cm side spread apart and never meet. Choice A computes 180° - 185° = -5° and drops the negative sign. Choice B ignores the angle sum. Choice C uses the rule for two angles and the side between them without checking that the two angles add to less than 180°.
Question 8 of 20 · Multiple Choice
Angle A is 30° and side AB is 12 cm, so the shortest distance from B to the other ray of angle A is 6 cm. Side BC is across from angle A. Which length of BC gives two different triangles?
Answer: B
An arc of 10 cm from B is longer than the 6 cm shortest distance and shorter than AB = 12 cm, so it crosses the ray twice: two triangles. Choice A is shorter than 6 cm, so the arc never reaches the ray: no triangle. Choice C is exactly AB, so the second crossing is A itself and only one triangle is left. Choice D is longer than AB, so the arc crosses the ray only once: one triangle.
Question 9 of 20 · Multiple Choice
Which set of measures gives exactly one triangle?
Answer: A
Two sides and the angle between them always fix one triangle. Choice B has the angle across from the short side. Draw it: with a 30° angle at one end of the 9 cm side, the other ray passes 4.5 cm from the far end at its closest, so a 3 cm arc cannot reach it and there is no triangle. Choice C adds to 180° but fixes no size, so it gives many triangles. Choice D fails the side test: 3 + 5 = 8, less than 9.
Question 10 of 20 · Multiple Choice
Which of these triangles is impossible to draw?
Answer: D
Two right angles already add to 90° + 90° = 180°, which leaves 0° for the third angle, so the two sides never meet. Choice A is possible, with angles 90°, 45° and 45°. Choice B is possible, for example 120°, 30° and 30°. Choice C is possible, for example 50°, 60° and 70°. Picking A comes from thinking a right angle must be next to two sides of different lengths.
Question 11 of 20 · Multiple Choice
Sam wants to construct a triangle with angles of 35° and 75° and an 8 cm side between them. Which steps work?
Answer: B
The side between the two angles joins their vertices, so one angle goes at each end, and the rays are extended until they cross. Choice A puts both angles at the same vertex. Choices C and D build a triangle with two 8 cm sides and a 35° angle between them, whose other two angles are 72.5° each, not 75°.
Question 12 of 20 · Multiple Choice
In a geometry app, Priya keeps two sides at 5 cm and 6 cm and drags the angle between them from 30° to 120°. What does she see?
Answer: C
As the angle between the sides opens, their far ends move apart, so the third side grows (from about 3 cm at 30° to about 9.5 cm at 120°). For each angle there is only one triangle, because two sides and the angle between them fix it. Choice A would mean the angle does not matter. Choice B reverses the effect. Choice D confuses this with an angle that is not between the sides.
Question 13 of 20 · Multiple Choice
A triangle with two equal sides has one side of 5 cm and one side of 12 cm. How long is its third side?
Answer: C
The third side must equal one of the two given sides. If it were 5 cm, the sides would be 5, 5 and 12, and 5 + 5 = 10 is less than 12: no triangle. So it is 12 cm: 5 + 12 = 17 is more than 12. Choice A skips the side test. Choice B subtracts 12 - 5. Choice D adds 5 + 12.
Question 14 of 20 · Multiple Choice
Leo cuts a 24 cm straw into pieces of 5 cm, 7 cm and 12 cm. What happens when he tries to make a triangle with them?
Answer: D
5 + 7 = 12, exactly the length of the longest piece, so the two short pieces lie flat along it. The triangle inequality needs a sum greater than the longest side. Choice A treats "equal" as enough. Choice B confuses side lengths with angles, which do not fix the size. Choice C thinks the position matters, but turning the pieces does not change their lengths.
Question 15 of 20 · Short Answer
Do lengths of 7 cm, 9 cm and 15 cm make a triangle? If they do, how many different triangles have these sides? Explain.
Yes, exactly one triangle. The two shorter sides add to 7 + 9 = 16 cm, which is more than 15 cm. Three side lengths fix a triangle: draw the 15 cm side, swing arcs of 7 cm and 9 cm from its ends, and they cross in one place above the side (the matching point below gives the same triangle flipped over).
Question 16 of 20 · Short Answer
Two sides of a triangle are 5 cm and 11 cm. List every whole-number length the third side could have.
The third side must be longer than 11 - 5 = 6 cm and shorter than 11 + 5 = 16 cm. The whole numbers are 7, 8, 9, 10, 11, 12, 13, 14 and 15 cm, nine lengths. Check the ends: 5 + 6 = 11 lies flat, and 5 + 11 = 16 lies flat too.
Question 17 of 20 · Short Answer
Two angles of a triangle are 48° and 67°. Find the third angle. If no side lengths are given, how many triangles have these three angles?
The third angle is 180° - 48° - 67° = 65°. With no side length, there are more than one triangle: any size works, and each one is a scaled copy of the others.
Question 18 of 20 · Short Answer
Describe how to construct a triangle with sides of 6 cm and 4 cm and a 120° angle between them, using a ruler and protractor. How long is the third side, to the nearest millimeter?
Draw a 6 cm segment. At one end, use the protractor to draw a 120° angle, and mark a point 4 cm along the new ray. Join that point to the other end of the 6 cm segment. The third side measures about 8.7 cm. There is exactly one such triangle, because the angle is between the two given sides.
Question 19 of 20 · Short Answer
Mia draws a 60° angle at A and marks B 6 cm from A on one ray. She measures the shortest distance from B to the other ray: 5.2 cm. How many triangles ABC does she get if BC is 4 cm? 5.5 cm? 7 cm?
4 cm: no triangle, because the arc is shorter than 5.2 cm and cannot reach the ray. 5.5 cm: two triangles, because the arc is longer than 5.2 cm but shorter than AB = 6 cm, so it crosses the ray twice. 7 cm: one triangle, because the arc is longer than AB, so it crosses the ray only once; its other crossing lands behind A, off the ray.
Question 20 of 20 · Short Answer
Sketch freehand a triangle with two equal sides and a 100° angle between them. Find the other two angles. Could the 100° angle be one of the two equal angles instead? Explain.
The other two angles are equal and share 180° - 100° = 80°, so each is 40°. The 100° angle cannot be one of two equal angles: two 100° angles would already add to 200°, more than 180°.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.G.A.2 mean?
7.G.A.2 means students can draw shapes that meet given conditions and decide whether three measures give one triangle, more than one, or none. They draw freehand, with a ruler and protractor, and with a geometry app. For example, sides of 5 cm, 6 cm and 9 cm give exactly one triangle, while 3 cm, 4 cm and 8 cm give none.
What grade is 7.G.A.2, and what comes after it?
It is a grade 7 standard in the Geometry domain. In grade 8, students explain why the angles of a triangle add to 180° (8.G.A.5) and learn what congruent figures (same shape and same size) are (8.G.A.2). In high school geometry, students prove that three sides, two sides and the angle between them, or two angles and the side between them always fix one triangle (HSG.CO.B.8).
What is the triangle inequality?
The triangle inequality says the two shorter sides of a triangle must add up to more than the longest side. If they add up to less, the short sides cannot reach each other. If they add up to exactly the longest side, they lie flat on it. Students only need to test the two shorter sides against the longest one.
Why do three angles give more than one triangle?
Three angles fix the shape of a triangle but not its size. Two triangles with angles of 35°, 55° and 90° can have a shortest side of 2 cm or of 20 cm. They are scaled copies of each other, like a map and the land it shows. At least one side length is needed to fix the size.
When do two sides and an angle give two triangles?
Two triangles are possible only when the angle is not between the two sides. When the side across from the angle is longer than the shortest distance to the other ray, but shorter than the other given side, the compass arc crosses the ray twice. Students find this by drawing arcs, and the lesson does not ask them to memorize a rule.
Do students need a compass for 7.G.A.2?
The standard names freehand drawing, ruler and protractor, and technology. A compass is not required by the wording, but it is the easiest way to draw a triangle from three side lengths, because arcs show every point at a given distance. Straws, paper strips or a geometry app can replace it.
How is 7.G.A.2 tested?
Test items give three measures and ask whether they make one triangle, more than one or none, or ask which lengths could be the third side. Some ask students to construct a triangle and measure a side or angle. Class assessments should also include a real construction, since the standard is about drawing.
What mistakes should teachers watch for?
A common mistake is testing a short side plus the longest side, which always passes, instead of the two shorter sides. Others are putting a given angle at the wrong vertex, forgetting to add the angles before drawing, and assuming that two sides and any angle always give one triangle.
Why does the standard also mention other shapes?
The first sentence asks for geometric shapes in general, so students can also draw four-sided shapes (quadrilaterals) with given conditions, such as a rectangle 5 cm by 3 cm or a four-sided shape with all sides 4 cm. The second example shows that four equal sides do not fix one shape, which connects to the triangle work: a triangle cannot change shape once its three sides are fixed, but a four-sided shape can lean.
How can parents help with 7.G.A.2 at home?
Cut pieces of dry spaghetti or paper strips to different lengths and ask your child which sets of three make a triangle, before trying them. Then ask why a set does or does not work. A good answer compares the two shorter pieces with the longest one.
07
Related Standards
6 standards
These standards connect to 7.G.A.2: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
4.MD.C.6Prerequisite
Measure angles in whole-number degrees with a protractor and sketch given angles
Lesson coming soon
5.G.B.4Prerequisite
Classify two-dimensional figures in a hierarchy based on their properties
Lesson coming soon
Alongside
7.G.A.1Parallel
Solve problems with scale drawings, including actual lengths and areas
