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Learning objectives

What you should be able to do

Represent and solve construct triangles from given measurements problems using precise mathematical notation, models, and reasoning.
Explain why a method for construct triangles from given measurements works, interpret the result in context, and verify it independently.

Prerequisite check

Make sure the foundation is ready.

Can you explain and use prerequisite MAT-06-GEO-001 without relying on a keyword?

The new lesson depends on this prerequisite as active knowledge.

Learn

Build the idea from meaning, not memorization.

Mathematical meaning

Construct Triangles from Given Measurements connects measurement to structure. Formulas are justified by decomposition, rearrangement, similarity, coordinate reasoning, or invariance.

Represent the relationship

The Math Box uses geometry workspace to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.

Calculate, interpret, and verify

Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.

Math Box · Interactive lesson

Touch the math. Change it. See what stays true.

Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.

Angle & Rotation Lab

Change the turn and connect degree measure to the angle classification.

Angle in degrees60

60°

60° is acute.

Try these actions

Model one example of construct triangles from given measurements in the Math Box.
Change one input, predict the effect, and test the prediction.
State the relationship or invariant that explains what stayed mathematically consistent.

Worked examples

Twenty different ways to see the concept work.

Can side lengths 4, 5, and 5 form a triangle?

9 > 5, 9 > 5, 10 > 4; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 5, 6, and 6 form a triangle?

11 > 6, 11 > 6, 12 > 5; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 6, 7, and 7 form a triangle?

13 > 7, 13 > 7, 14 > 6; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 7, 8, and 8 form a triangle?

15 > 8, 15 > 8, 16 > 7; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 8, 9, and 9 form a triangle?

17 > 9, 17 > 9, 18 > 8; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 9, 10, and 10 form a triangle?

19 > 10, 19 > 10, 20 > 9; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 10, 11, and 11 form a triangle?

21 > 11, 21 > 11, 22 > 10; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 11, 5, and 12 form a triangle?

16 > 12, 23 > 5, 17 > 11; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 12, 6, and 13 form a triangle?

18 > 13, 25 > 6, 19 > 12; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 13, 7, and 14 form a triangle?

20 > 14, 27 > 7, 21 > 13; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 14, 8, and 15 form a triangle?

22 > 15, 29 > 8, 23 > 14; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 15, 9, and 16 form a triangle?

24 > 16, 31 > 9, 25 > 15; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 16, 10, and 17 form a triangle?

26 > 17, 33 > 10, 27 > 16; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 17, 11, and 18 form a triangle?

28 > 18, 35 > 11, 29 > 17; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 18, 5, and 19 form a triangle?

23 > 19, 37 > 5, 24 > 18; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 19, 6, and 20 form a triangle?

25 > 20, 39 > 6, 26 > 19; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 20, 7, and 21 form a triangle?

27 > 21, 41 > 7, 28 > 20; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 21, 8, and 22 form a triangle?

29 > 22, 43 > 8, 30 > 21; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 22, 9, and 23 form a triangle?

31 > 23, 45 > 9, 32 > 22; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 23, 10, and 24 form a triangle?

33 > 24, 47 > 10, 34 > 23; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Practice

Guided practice with hints

01Can side lengths 24, 11, and 25 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 35 > 25, 49 > 11, 36 > 24; yes

02Can side lengths 25, 5, and 26 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 30 > 26, 51 > 5, 31 > 25; yes

03Can side lengths 26, 6, and 27 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 32 > 27, 53 > 6, 33 > 26; yes

04Can side lengths 27, 7, and 28 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 34 > 28, 55 > 7, 35 > 27; yes

05Can side lengths 28, 8, and 29 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 36 > 29, 57 > 8, 37 > 28; yes

06Can side lengths 29, 9, and 30 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 38 > 30, 59 > 9, 39 > 29; yes

Practice

Independent practice

01Can side lengths 30, 10, and 31 form a triangle?

Answer: 40 > 31, 61 > 10, 41 > 30; yes

02Can side lengths 31, 11, and 32 form a triangle?

Answer: 42 > 32, 63 > 11, 43 > 31; yes

03Can side lengths 32, 5, and 33 form a triangle?

Answer: 37 > 33, 65 > 5, 38 > 32; yes

04Can side lengths 33, 6, and 34 form a triangle?

Answer: 39 > 34, 67 > 6, 40 > 33; yes

05Can side lengths 34, 7, and 35 form a triangle?

Answer: 41 > 35, 69 > 7, 42 > 34; yes

06Can side lengths 35, 8, and 36 form a triangle?

Answer: 43 > 36, 71 > 8, 44 > 35; yes

07Can side lengths 36, 9, and 37 form a triangle?

Answer: 45 > 37, 73 > 9, 46 > 36; yes

08Can side lengths 37, 10, and 38 form a triangle?

Answer: 47 > 38, 75 > 10, 48 > 37; yes

Common mistakes

Learn to catch the error, not just the answer.

Using a formula without identifying dimensions.

Label the figure and match each measure to the formula.

Using linear units for area or volume.

Use square units for area and cubic units for volume.

Applications & challenge

Use the idea beyond a single exercise.

Applications

Estimate material, floor space, packaging, distance, surface coverage, and capacity.
Use diagrams and coordinates to justify measurements rather than relying on appearance.

Challenge problems

Can side lengths 38, 11, and 39 form a triangle?

49 > 39, 77 > 11, 50 > 38; yes

Can side lengths 39, 5, and 40 form a triangle?

44 > 40, 79 > 5, 45 > 39; yes

Can side lengths 40, 6, and 41 form a triangle?

46 > 41, 81 > 6, 47 > 40; yes

Can side lengths 41, 7, and 42 form a triangle?

48 > 42, 83 > 7, 49 > 41; yes

Flashcards

Retrieve it from memory.

Flashcards

Card 1 of 4

Game mode

Grade 7 Reasoning Sprint

Solve construct triangles from given measurements problems accurately and justify each result with a valid check.

Round 1 · Score 0Can side lengths 38, 11, and 39 form a triangle?

1 point for an accurate result and 1 point for a mathematically valid explanation or check.

Practice

Mastery check

01Can side lengths 42, 8, and 43 form a triangle?

Answer: 50 > 43, 85 > 8, 51 > 42; yes

02Can side lengths 43, 9, and 44 form a triangle?

Answer: 52 > 44, 87 > 9, 53 > 43; yes

03Can side lengths 44, 10, and 45 form a triangle?

Answer: 54 > 45, 89 > 10, 55 > 44; yes

04Can side lengths 45, 11, and 46 form a triangle?

Answer: 56 > 46, 91 > 11, 57 > 45; yes

05Can side lengths 46, 5, and 47 form a triangle?

Answer: 51 > 47, 93 > 5, 52 > 46; yes

06Can side lengths 47, 6, and 48 form a triangle?

Answer: 53 > 48, 95 > 6, 54 > 47; yes

07Can side lengths 48, 7, and 49 form a triangle?

Answer: 55 > 49, 97 > 7, 56 > 48; yes

08Can side lengths 49, 8, and 50 form a triangle?

Answer: 57 > 50, 99 > 8, 58 > 49; yes

Terminology

Words to know

dimension: A measurable extent such as length, width, or height.
area: Two-dimensional measure in square units.
surface area: Total area of exterior faces or surfaces.
volume: Three-dimensional space measured in cubic units.

Continue learning

Connected concepts

Curriculum references

Standards and instructional references

California Department of Education

California Mathematics Framework (2023)

Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.

Common Core State Standards Initiative

Common Core State Standards for Mathematics — Grade 7

Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.

National Council of Teachers of Mathematics

Principles to Actions: Ensuring Mathematical Success for All

Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.

California Mathematics Framework ↗

Math curriculum Next lesson →

Learning objectives

What you should be able to do

Represent and solve construct triangles from given measurements problems using precise mathematical notation, models, and reasoning.
Explain why a method for construct triangles from given measurements works, interpret the result in context, and verify it independently.

Prerequisite check

Make sure the foundation is ready.

Can you explain and use prerequisite MAT-06-GEO-001 without relying on a keyword?

The new lesson depends on this prerequisite as active knowledge.

Learn

Build the idea from meaning, not memorization.

Mathematical meaning

Construct Triangles from Given Measurements connects measurement to structure. Formulas are justified by decomposition, rearrangement, similarity, coordinate reasoning, or invariance.

Represent the relationship

The Math Box uses geometry workspace to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.

Calculate, interpret, and verify

Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.

Math Box · Interactive lesson

Touch the math. Change it. See what stays true.

Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.

Angle & Rotation Lab

Change the turn and connect degree measure to the angle classification.

Angle in degrees60

60°

60° is acute.

Try these actions

Model one example of construct triangles from given measurements in the Math Box.
Change one input, predict the effect, and test the prediction.
State the relationship or invariant that explains what stayed mathematically consistent.

Worked examples

Twenty different ways to see the concept work.

Can side lengths 4, 5, and 5 form a triangle?

9 > 5, 9 > 5, 10 > 4; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 5, 6, and 6 form a triangle?

11 > 6, 11 > 6, 12 > 5; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 6, 7, and 7 form a triangle?

13 > 7, 13 > 7, 14 > 6; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 7, 8, and 8 form a triangle?

15 > 8, 15 > 8, 16 > 7; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 8, 9, and 9 form a triangle?

17 > 9, 17 > 9, 18 > 8; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 9, 10, and 10 form a triangle?

19 > 10, 19 > 10, 20 > 9; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 10, 11, and 11 form a triangle?

21 > 11, 21 > 11, 22 > 10; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 11, 5, and 12 form a triangle?

16 > 12, 23 > 5, 17 > 11; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 12, 6, and 13 form a triangle?

18 > 13, 25 > 6, 19 > 12; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 13, 7, and 14 form a triangle?

20 > 14, 27 > 7, 21 > 13; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 14, 8, and 15 form a triangle?

22 > 15, 29 > 8, 23 > 14; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 15, 9, and 16 form a triangle?

24 > 16, 31 > 9, 25 > 15; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 16, 10, and 17 form a triangle?

26 > 17, 33 > 10, 27 > 16; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 17, 11, and 18 form a triangle?

28 > 18, 35 > 11, 29 > 17; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 18, 5, and 19 form a triangle?

23 > 19, 37 > 5, 24 > 18; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 19, 6, and 20 form a triangle?

25 > 20, 39 > 6, 26 > 19; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 20, 7, and 21 form a triangle?

27 > 21, 41 > 7, 28 > 20; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 21, 8, and 22 form a triangle?

29 > 22, 43 > 8, 30 > 21; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 22, 9, and 23 form a triangle?

31 > 23, 45 > 9, 32 > 22; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Can side lengths 23, 10, and 24 form a triangle?

33 > 24, 47 > 10, 34 > 23; yes

All three triangle inequalities hold, so the lengths can form a triangle.

Practice

Guided practice with hints

01Can side lengths 24, 11, and 25 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 35 > 25, 49 > 11, 36 > 24; yes

02Can side lengths 25, 5, and 26 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 30 > 26, 51 > 5, 31 > 25; yes

03Can side lengths 26, 6, and 27 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 32 > 27, 53 > 6, 33 > 26; yes

04Can side lengths 27, 7, and 28 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 34 > 28, 55 > 7, 35 > 27; yes

05Can side lengths 28, 8, and 29 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 36 > 29, 57 > 8, 37 > 28; yes

06Can side lengths 29, 9, and 30 form a triangle?

Hint: Represent the relationship first, keep units visible, and identify a second verification method.

Answer: 38 > 30, 59 > 9, 39 > 29; yes

Practice

Independent practice

01Can side lengths 30, 10, and 31 form a triangle?

Answer: 40 > 31, 61 > 10, 41 > 30; yes

02Can side lengths 31, 11, and 32 form a triangle?

Answer: 42 > 32, 63 > 11, 43 > 31; yes

03Can side lengths 32, 5, and 33 form a triangle?

Answer: 37 > 33, 65 > 5, 38 > 32; yes

04Can side lengths 33, 6, and 34 form a triangle?

Answer: 39 > 34, 67 > 6, 40 > 33; yes

05Can side lengths 34, 7, and 35 form a triangle?

Answer: 41 > 35, 69 > 7, 42 > 34; yes

06Can side lengths 35, 8, and 36 form a triangle?

Answer: 43 > 36, 71 > 8, 44 > 35; yes

07Can side lengths 36, 9, and 37 form a triangle?

Answer: 45 > 37, 73 > 9, 46 > 36; yes

08Can side lengths 37, 10, and 38 form a triangle?

Answer: 47 > 38, 75 > 10, 48 > 37; yes

Common mistakes

Learn to catch the error, not just the answer.

Using a formula without identifying dimensions.

Label the figure and match each measure to the formula.

Using linear units for area or volume.

Use square units for area and cubic units for volume.

Applications & challenge

Use the idea beyond a single exercise.

Applications

Estimate material, floor space, packaging, distance, surface coverage, and capacity.
Use diagrams and coordinates to justify measurements rather than relying on appearance.

Challenge problems

Can side lengths 38, 11, and 39 form a triangle?

49 > 39, 77 > 11, 50 > 38; yes

Can side lengths 39, 5, and 40 form a triangle?

44 > 40, 79 > 5, 45 > 39; yes

Can side lengths 40, 6, and 41 form a triangle?

46 > 41, 81 > 6, 47 > 40; yes

Can side lengths 41, 7, and 42 form a triangle?

48 > 42, 83 > 7, 49 > 41; yes

Flashcards

Retrieve it from memory.

Flashcards

Card 1 of 4

Game mode

Grade 7 Reasoning Sprint

Solve construct triangles from given measurements problems accurately and justify each result with a valid check.

Round 1 · Score 0Can side lengths 38, 11, and 39 form a triangle?

1 point for an accurate result and 1 point for a mathematically valid explanation or check.

Practice

Mastery check

01Can side lengths 42, 8, and 43 form a triangle?

Answer: 50 > 43, 85 > 8, 51 > 42; yes

02Can side lengths 43, 9, and 44 form a triangle?

Answer: 52 > 44, 87 > 9, 53 > 43; yes

03Can side lengths 44, 10, and 45 form a triangle?

Answer: 54 > 45, 89 > 10, 55 > 44; yes

04Can side lengths 45, 11, and 46 form a triangle?

Answer: 56 > 46, 91 > 11, 57 > 45; yes

05Can side lengths 46, 5, and 47 form a triangle?

Answer: 51 > 47, 93 > 5, 52 > 46; yes

06Can side lengths 47, 6, and 48 form a triangle?

Answer: 53 > 48, 95 > 6, 54 > 47; yes

07Can side lengths 48, 7, and 49 form a triangle?

Answer: 55 > 49, 97 > 7, 56 > 48; yes

08Can side lengths 49, 8, and 50 form a triangle?

Answer: 57 > 50, 99 > 8, 58 > 49; yes

Terminology

Words to know

dimension: A measurable extent such as length, width, or height.
area: Two-dimensional measure in square units.
surface area: Total area of exterior faces or surfaces.
volume: Three-dimensional space measured in cubic units.

Continue learning

Connected concepts

Curriculum references

Standards and instructional references

California Department of Education

California Mathematics Framework (2023)

Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.

Common Core State Standards Initiative

Common Core State Standards for Mathematics — Grade 7

Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.

National Council of Teachers of Mathematics

Principles to Actions: Ensuring Mathematical Success for All

Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.

California Mathematics Framework ↗

Math curriculum Next lesson →